Transformation Te hniques inGlobal Optimization

Andreas Lundell, May 2007Master's Thesis, 12 reditsDepartment of Mathemati sÅbo Akademi UniversitySupervisor: Professor Tapio Westerlund

Contents1 Introdu tion 42 Basi de�nitions and theorems 62.1 Convex sets and fun tions . . . . . . . . . . . . . . . . . . . . 62.2 Quasi- and pseudo onvex fun tions . . . . . . . . . . . . . . . 102.3 Convexity of signomial fun tions . . . . . . . . . . . . . . . . . 112.4 The MINLP problem . . . . . . . . . . . . . . . . . . . . . . . 133 Transformations of signomial terms 153.1 Convexi� ation . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.1 Negative signomial terms . . . . . . . . . . . . . . . . . 153.1.2 Positive signomial terms . . . . . . . . . . . . . . . . . 163.2 Underestimation . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Negative signomial terms . . . . . . . . . . . . . . . . . 183.2.2 Positive signomial terms . . . . . . . . . . . . . . . . . 193.3 Pie ewise linear fun tions . . . . . . . . . . . . . . . . . . . . 203.3.1 Formulation using binary variables . . . . . . . . . . . 213.3.2 Formulation using spe ial ordered sets . . . . . . . . . 223.4 Examples of the transformation approa h . . . . . . . . . . . . 244 Optimization of the transformation approa h 284.1 Optimization of the power transformations . . . . . . . . . . . 284.1.1 Linear onditions for negative terms . . . . . . . . . . . 324.1.2 Linear onditions for positive terms . . . . . . . . . . . 334.2 Improving the numeri al stability of signomial terms . . . . . 351

4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3.1 Des ription of the simulations . . . . . . . . . . . . . . 364.3.2 Comments on the results . . . . . . . . . . . . . . . . . 365 The GGPECP algorithm 435.1 Des ription of the algorithm . . . . . . . . . . . . . . . . . . . 435.2 Some numeri al examples . . . . . . . . . . . . . . . . . . . . 445.2.1 A two dimensional MINLP problem . . . . . . . . . . . 445.2.2 A geometri programming example . . . . . . . . . . . 506 Con lusions 527 Swedish summary 537.1 Transformationstekniker inom global optimering . . . . . . . . 537.1.1 Transformering av signomialfunktioner . . . . . . . . . 537.1.2 Optimering av potenstransformationerna . . . . . . . . 557.1.3 Förbättring av den numeriska stabiliteten . . . . . . . 587.1.4 Slutsatser . . . . . . . . . . . . . . . . . . . . . . . . . 58Referen es 59Appendi es 61A The Compiled Optimization Method . . . . . . . . . . . . . . 61B The MILP formulation in Se tion 5.2.1 . . . . . . . . . . . . . 632

ForewordThe work for this thesis was arried out at the Pro ess Design Labora-tory at Åbo Akademi University under the supervision of Professor TapioWesterlund.

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1 Introdu tionThe pro edures available today for solving global optimization problems, i.e.,problems where the obje tive is to �nd the globally optimal solution, havegone through a gradual evolution over the years, and have, in the pro ess,be ome appli able to a larger s ope of problems. The on-going growth inavailable omputational power has lately in reased the interest in developingalgorithms for numeri ally solving su h problems, sin e it is often hard, ifnot impossible, to �nd an exa t solution.The mixed integer nonlinear programming (MINLP) problem formulationis a very general formulation of a nonlinear optimization problem. As thename states, a MINLP problem an ontain both integer and real variables,as well as, linear and nonlinear onstraints. In this thesis, the onstraintsare also allowed to ontain so- alled signomial fun tions, further generalizingthe formulation. Sin e the fun tions in the onstraints an be nonlinear, theproblem an be non onvex, making it more di� ult to solve.A method for solving MINLP problems ontaining signomial fun tions toglobal optimality is the generalized geometri programming extended uttingplane (GGPECP) algorithm. The GGPECP algorithm solves the MINLPproblems as a sequen e of onvexi�ed and overestimated subproblems. How-ever, sin e the algorithm requires that the signomial fun tions are onvex,transformation te hniques are needed to onvexify the fun tions. There-fore, power transformations applied to the individual variables are used totermwise transform the non onvex fun tions.The performan e of the MINLP solver is, to a large extent, dependent onthe transformations that are used, so it would be bene� ial to optimize thetransformations applied to the signomial terms. A new method for a om-plishing this is presented in this thesis. The method involves the formulationof a mixed integer linear problem (MILP), with the obje tive being to de-termine the optimal set of transformations required for transformation of allthe non onvex signomial terms in the MINLP problem.4

In Chapter 2, basi de�nitions and theorems from onvex analysis are pre-sented. Furthermore, signomial fun tions and the onditions for their on-vexity, as well as, a mathemati al formulation of the MINLP problem isdes ribed. Also, some examples, showing how it is possible to determinewhether a (signomial) fun tion is onvex or not, are given.The topi of Chapter 3 is the transformation, i.e., onvexi� ation and un-derestimation, of signomial terms. The underestimation step requires theuse of pie ewise linear fun tions, of whi h some alternative formulations arepresented. A few examples of the transformation approa h are also providedfor lari� ation.In Chapter 4, the method for optimizing the transformations is introdu ed.The linear onditions in the generated MILP problem are explained, andthe results from simulations of the method on groups of randomly gener-ated signomial fun tions are presented. Also, a te hnique for in reasing thenumeri al stability of the transformations is des ribed.The GGPECP algorithm is presented in Chapter 5 and it is shown, throughexamples, how the method for optimizing the transformations an be usedas an integrated part of the GGPECP algorithm.

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2 Basi de�nitions and theoremsConvexity is an integral part of global optimization, and in this hapter someof the basi de�nitions and theorems of onvexity are introdu ed. These anbe found in most standard literature in onvex analysis or global optimiza-tion, for example [BV04℄ and [Flo00℄. Furthermore, signomial fun tions,whi h will be the main topi of interest in the rest of the thesis, are de�nedand onditions regarding their onvexity are presented. Finally, a generalformulation of a mixed integer nonlinear programming (MINLP) problem ontaining signomial fun tions, is des ribed. To larify ertain aspe ts of thetheory presented in this hapter, examples and �gures are provided whenneeded.2.1 Convex sets and fun tionsOne of the assumptions in many of the de�nitions and theorems presentedin this hapter is that a set is onvex. A onvex set an be de�ned as:De�nition 2.1. A set C in Rn is onvex if it ontains all line segmentsjoining any two points in the set, i.e., if for all ve tors x and y in the set Cand any λ su h that 0 ≤ λ ≤ 1, the following riteria is satis�ed

λx + (1 − λ)y ∈ C.Examples of a onvex and a non onvex set in two dimensions are presentedin Figure 2.1. Observe that the multidimensional ase is analogous to thetwo-dimensional ase.Figure 2.1: Left. A non onvex set. Right. A onvex set.6

In most of the algorithms used in global optimization, a requirement for thefun tions is that they are onvex. A onvex fun tion an be de�ned as:De�nition 2.2. A fun tion f , de�ned on a onvex set C, is onvex for allve tors x and y in C and for all λ su h that 0 ≤ λ ≤ 1 iff(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y), (2.1)and stri tly onvex iff(λx + (1 − λ)y) < λf(x) + (1 − λ)f(y). (2.2)Conversely, if the inequalities in the expressions (2.1) and (2.2) are reversed,

f is on ave and stri tly on ave respe tively.A geometri interpretation of this is that for a onvex fun tion f , the linesegment between any two points on the graph of f lies above the graph. Fora on ave fun tion f , this line segment lies below the graph of f . Note that,for a linear fun tion there is always equality in expression (2.1), so all linearfun tions are both onvex and on ave. Furthermore, a onvex fun tion isalways ontinuous on the relative interior of its domain, see [BSM93℄.The following result follows naturally from De�nition 2.2.Lemma 2.3. If the fun tion f is onvex, then −f is on ave, and if f is on ave, then −f is onvex.It an be seen from De�nition 2.2 that if a fun tion f is onvex, then so isαf for any positive value of α, and if f1 and f2 are onvex fun tions, then sois their sum f1 + f2. This an be generalized into the following result:Lemma 2.4. If w1, w2, . . . , wn are nonnegative onstants, then the weightedsum of the onvex fun tions f1, f2, . . . , fn, i.e.,

f = w1f1 + w2f2 + . . . + wnfn,is a onvex fun tion. 7

Figure 2.2: Left. A onvex fun tion in two dimensions and one of its tangents.Right. A onvex fun tion in three dimensions and one of its tangent planes.Now an example of how the onvexity of a fun tion an be determined usingDe�nition 2.2.Example 2.5. All norms are onvex, sin e if the fun tion f : Rn 7→ R is anorm and 0 ≤ λ ≤ 1, then

f(λx + (1 − λ)y) ≤ f(λx) + f((1 − λ)y) = λf(x) + (1 − λ)f(y).The inequality follows from the triangle inequality and the equality from thehomogeneity of a norm.There are better ways to he k whether a di�erentiable fun tion is onvexor not, than using De�nition 2.2, for example, the �rst- or se ond-order onvexity requirement an be used.Theorem 2.6 (First-order onvexity requirement). Suppose the fun -tion f is di�erentiable on the onvex set C ∈ Rn. Then f is onvex if andonly if

f(y) ≥ f(x) + ∇f(x)T (y − x)holds for all ve tors x and y in the set C.Proof. See [BV04, p. 69℄.Sin e f(x) + ∇f(x)T (y − x) is the tangent hyperplane of f at x, Theorem2.6 states that the graph of a onvex fun tion always lies above its tangenthyperplanes. For the two- and three-dimensional ase, see Figure 2.2.8

Theorem 2.7 (Se ond-order onvexity requirement). Assume the fun -tion f is twi e di�erentiable on the open onvex set C ∈ Rn. Then f is onvexif and only if its Hessian matrix

H(f) =

∂2f∂x1

2

∂2f∂x1∂x2

. . . ∂2f∂x1∂xn

∂2f∂x2∂x1

∂2f∂x2

2 . . . ∂2f∂x2∂xn... ... . . . ...

∂2f∂xn∂x1

∂2f∂xn∂x2

. . . ∂2f∂xn

2

is positive semide�nite, i.e., whenever the matrix H(f) only has nonnega-tive eigenvalues. Conversely, the fun tion f is on ave if H(f) is negativesemide�nite, i.e., whenever H(f) only has nonpositive eigenvalues.Proof. See [Eme04, p. 15℄.Note that for a fun tion f : R 7→ R, Theorem 2.7 redu es to the onditionsf ′′(x) ≥ 0 for onvexity and f ′′(x) ≤ 0 for on avity.Here a few examples are given of how the se ond-order onvexity require-ments an be used to determine whether a fun tion is onvex or not.Example 2.8. Some elementary onvex and non onvex fun tions.(a) The exponential fun tion f(x) = eax is onvex on R for a ∈ R, sin e

f ′′(x) = a2eax is always nonnegative.(b) The negative logarithm fun tion f(x) = − log x is onvex on R+, sin ef ′′(x) = 1

x2 ≥ 0 for all x ∈ R+.( ) The power fun tion f(x) = xa is onvex on R+ for a ≤ 0 or a ≥ 1 and on ave on R+ for 0 ≤ a ≤ 1, sin e f ′′(x) = a(a−1)xa−2 is nonnegativefor a ≤ 0 or a ≥ 1 and nonpositive for 0 ≤ a ≤ 1.(d) The power fun tion −xa is onvex on R+ for 0 ≤ a ≤ 1, sin e thefun tion −f is onvex if f is on ave.A ording to the se ond-order onvexity requirement, onvexity for a fun tionof more than two variables an be determined using the Hessian matrix.9

Example 2.9. The quadrati -over-linear fun tion f(x, y) = x2/y, y > 0 is onvex sin e its Hessian matrixH(f) =

2

y3

[

y2 −xy

−xy x2

]

has the nonnegative eigenvalues λ0 = 0 and λ1 = x2 + y2.2.2 Quasi- and pseudo onvex fun tionsQuasi- and pseudo onvexity an be regarded as extensions to onvexity, andmany of the properties of onvex fun tions have ounterparts for quasi- orpseudo onvex fun tions as well.De�nition 2.10. A di�erentiable fun tion f : C 7→ R, where C ∈ Rn is anonempty onvex set, is quasi onvex if and only if, for all ve tors x and y in

C and 0 ≤ λ ≤ 1, the following inequality holdsf(λx + (1 − λ)) ≤ max{f(x), f(y)}.This states, that the value of the fun tion at a segment does not ex eed themaximum of the values at the endpoints. Note that all onvex fun tionsare quasi onvex, but the onverse is not true. It an also be shown that afun tion is quasi onvex if at most one of the eigenvalues of the Hessian ispositive, see [BV04, p. 101℄.De�nition 2.11. A di�erentiable fun tion f : C 7→ R, where C ∈ R

n is anonempty onvex set, is pseudo onvex if the statementf(y) < f(x) ⇒ ∇f(x)T (y − x) < 0is true for all ve tors x and y in C.A quasi onvex fun tion an have lo al minimums that are not global ones,whereas all lo al minimums of a pseudo onvex fun tion are global. This isillustrated in Figure 2.3. 10

Figure 2.3: Left. A quasi onvex fun tion. Right. A pseudo onvex fun tion.2.3 Convexity of signomial fun tionsSin e power, bilinear and trilinear fun tions all an be regarded as signomialfun tions, signomials are quite ommon in mathemati al models. Hen e, itis essential in global optimization problems, to be able to determine whethera signomial fun tion is onvex or not.De�nition 2.12. A signomial fun tion is de�ned as the sum of signomialterms, where ea h term onsists of produ ts of power fun tions, i.e.,σ(z) =

J∑

j=1

cj

I∏

i=1

zpji

i , (2.3)where cj, pji ∈ R for all indi es i and j. In this thesis, it is also assumed thatthe variables z are positive.For J equal to one, the fun tion σ(z) simpli�es to a power fun tion of onevariable if I is equal to one, and to a bilinear or trilinear fun tion if I is equalto two or three respe tively. Note that this is dependent on the values of thepowers p; if exa tly one of the powers is equal to one and the rest zero, thenthe signomial fun tion simpli�es to a linear fun tion.When all onstants c are positive, the fun tion (2.3) is alled a posynomial,and hen e, it is always possible to write a signomial fun tion as the di�eren ebetween two posynomials by grouping the terms in groups where either cj > 0or cj < 0.The following theorem, from [MF95℄, gives onditions for when a signomialterm is onvex or on ave. 11

Theorem 2.13. A signomial term s(z) = c∏I

i=1 zpi

i , where c > 0, is(a) onvex in RI if one of the following statements is true(i) pi ≤ 0, ∀i = 1, . . . , I,(ii) ∃k : pk +

∑

i6=k pi ≥ 1, where pk > 0 and pi ≤ 0, ∀i 6= k,i = 1, . . . , I,(b) on ave in R

I if the following statements are truepi ≥ 0, ∀i = 1, . . . , I and ∑I

i=1 pi ≤ 1.In the proof of this theorem, the Hessian matrix of the fun tion s is used todetermine under what onditions the fun tion is onvex. However, the proofis quite long, and is therefore not presented here.The following result for negative signomial terms follows naturally from The-orem 2.13 (b).Lemma 2.14. A signomial term s(z) = c∏I

i=1 zpi

i , where c < 0, is onvex ifthe following onditions are truepi ≥ 0, ∀i = 1, . . . , I and

∑Ii=1 pi ≤ 1.Proof. By Lemma 2.3, a fun tion −f is onvex if f is on ave.The transformations, used to onvexify non onvex signomial terms, presentedin Chapter 3, are based on the results from Theorem 2.13 and Lemma 2.14.However, �rst a few examples on how to use these results to determine if asignomial fun tion is onvex.Example 2.15. The signomial term s(z1, z2) = czp1

1 zp2

2 is onvex if(a) c > 0, p1 = −0.5 and p2 = −2, sin e p1, p2 ≤ 0,(b) c > 0, p1 = 2 and p2 = −1, sin e p1 > 0, p2 < 0 and p1 + p2 ≥ 1,( ) c < 0, p1 = 0.5 and p2 = 0.5, sin e p1, p2 > 0 and p1 + p2 ≤ 1.12

(a) c > 0 (b) c < 0Figure 2.4: Classi� ation of the onvexity of a signomial term czp1

1 zp2

2 .Note that the fun tion s in Example 2.15 (b) is the same fun tion as inExample 2.9. This shows how useful Theorem 2.13 and Lemma 2.14 anbe, sin e it is not ne essary to al ulate the Hessian and its eigenvalues todetermine whether the fun tion is onvex or not.By using the de�nitions of onvexity, quasi onvexity and pseudo onvexityit is possible to onstru t Figure 2.4. These �gures omply fully with theresults from Theorem 2.13 and Lemma 2.14 when the limit I is equal to two.2.4 The MINLP problemA general formulation of a global optimization problem is the mixed integernonlinear programming (MINLP) problem formulation. A MINLP problem an be expressed as:min f(z), z = (z1, z2, . . . , zI),s.t. Az = a, Bz ≤ b,

gn(z) ≤ 0, n = 1, 2, . . . , Jn,

qm(z) + σm(z) ≤ 0, m = 1, 2, . . . , Jm.

(2.4)13

Figure 2.5: The MINLP problem spa eGlobal optimization problems an be ategorized into sub lasses in regardto the problem spa e, i.e., the fun tion and variable spa e, ontaining theproblem. A somewhat simpli�ed overview of the MINLP problem spa e isprovided in Figure 2.5.Depending on the method used to solve the MINLP problem, the di�eren-tiable fun tion f to be minimized, i.e., the obje tive fun tion, an be either onvex or pseudo onvex. In the GGPECP algorithm presented in Chapter5, pseudo onvex obje tive fun tions are allowed. Furthermore, the ve tor zmay onsist of both ontinuous variables in the ompa t subset X of a �nitedimensional Eu lidian spa e and integer variables in a �nite dimensional in-teger set Y . The fun tions g, q and σ are di�erentiable real pseudo onvex, onvex and signomial fun tions respe tively. Finally, the matri es A andB, as well as, the ve tors a and b should onsist of onstants and be ofappropriate dimensions.However, to be able to solve the MINLP problem (2.4) using most methodsavailable today, the signomial fun tions σ must be onvex. A method fortransforming non onvex signomial fun tions to onvex form is presented inthe following two hapters.

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3 Transformations of signomial terms3.1 Convexi� ationThere are many di�erent transformation approa hes to onvexifying non on-vex signomial terms in a nonlinear optimization problem. For example, theexponential, inverse and power transformations have previously been studiedin [Bjö02℄ and [Pör00℄. In this thesis, power transformations from [Wes05℄are used in the onvexi� ation pro ess.However, using power transformations to onvexify the signomial terms onlymoves the non onvexities from the individual terms to the onstraints intro-du ed by the inverse transformations. Even so, by approximating the inversepower transformations with pie ewise linear fun tions, the whole problem an be onvexi�ed.Sin e the onvexity requirements for positive signomial terms in Theorem2.13 are di�erent from the requirements for negative signomial terms inLemma 2.14, di�erent onditions for the power transformations applied tothe individual variables are needed depending on the sign of the term.3.1.1 Negative signomial termsA ording to Lemma 2.14, a negative signomial term is onvex if all powersare positive and the sum of the powers is less or equal to one. Therefore,it is always possible to onvexify a nonnegative signomial term by applyingpower transformations of the formzi = ZQi

i ⇒ Zi = z1/Qi

ion ea h of the original variables zi, where the powers Q ful�ll the following onditions

Qi > 0, if pi > 0,Qi < 0, if pi < 0. (3.1)15

After the transformations, the signomial term will look likec

I∏

i=1

ZpiQi

i ,and this transformed term will be onvex under the ondition that the fol-lowing inequality is ful�lledI

∑

i=1

piQi ≤ 1. (3.2)Note that ea h term piQi in expression (3.2) is nonnegative, so the sum of theterms is always positive. Therefore, depending on the values of the powersQ, ondition (3.2) an always be satis�ed, as long as, the powers are hosenarbitrary lose to zero from either the positive or the negative side.3.1.2 Positive signomial termsTheorem 2.13 states that a positive signomial term is onvex if either allvariables z have negative powers, or one power is positive while the rest arenegative and the sum of the powers is greater than or equal to one. Therefore,the same power transformation

zi = ZQi

i ⇒ Zi = z1/Qi

i ,as in the ase with a negative signomial term, an be used to onvexify theterm. However, the onditions for the powers Q are di�erent:

Qi > 0, if pi > 0 ∧ i = k,Qi < 0, if pi > 0 ∧ i 6= k,Qi = 1, if pi < 0. (3.3)For the variables originally having a negative power, no transformation isneeded, so the power Q are de�ned to be equal to one be ause z = Z.16

The index k, where 1 ≤ k ≤ I, whi h may or may not exist, orrespondsto the power remaining positive after the transformation, i.e., the produ tpkQk is positive, while the produ ts piQi are negative for all other indi esi 6= k. Furthermore, if the index k exists, the sum of the powers must alsobe greater than or equal to one, i.e.,

I∑

i=1

piQi ≥ 1. (3.4)Using the power transformations mentioned above, it is always possible to onvexify a positive signomial term, and after the transformation, the trans-formed signomial term will be of the formc

I∏

i=1

ZpiQi

i ,where the powers Q ful�ll the onditions (3.12).3.2 UnderestimationApplying the transformations mentioned above onvexi�es the signomialfun tion, but only by moving the non onvexities from the signomial terms tothe onstraints introdu ed by the inverse power transformations. However,by approximating the nonlinear transformations with pie ewise linear fun -tions, whi h are pie ewise onvex, the whole problem will be onvexi�ed onthe ondition that the approximation of ea h of the transformed signomialterms underestimates the original term. An illustration of pie ewise linearapproximations of a onvex and a on ave fun tion are shown in Figure 3.1.Sin e the onditions for the powers Q used in the power transformationsare di�erent for positive and negative signomial terms, the underestimation onditions will also di�er in these two ases.17

Figure 3.1: Left. A pie ewise linear approximation of a onvex fun tion. Right.A pie ewise linear approximation of a on ave fun tion.3.2.1 Negative signomial termsThe inverse transformations, whi h will be approximated by pie ewise linearfun tions, are of the formZi = z

1/Qi

i . (3.5)When repla ing the inverse transformation Z with its pie ewise linear ap-proximation Z, the following inequality should hold if the onvexi�ed termis to underestimate the original one:c

I∏

i=1

ZpiQi

i ≤ c

I∏

i=1

ZpiQi

i . (3.6)For a negative signomial term (c < 0) all the powers are positive, so expres-sion (3.6) gives that ea h approximated variable Z should be greater or equalto the original variable Z, i.e.,Zi ≥ Zi (3.7)should hold for all indi es i. Be ause Z is the pie ewise linear approximationof the fun tion Z = z1/Q, the inequality (3.7) is true if the fun tions Zi are onvex. This an be seen from Figure 3.1.The inverse transformation (3.5) is onvex if the powers Q are greater thanzero and less than or equal to one, or if the powers Q are negative. The om-bination of these requirements with the onvexi� ation requirements (3.1),gives that a negative signomial term is onvexi�ed and underestimated if the18

powers Q, used in the power transformations, ful�ll the following onditions:

0 < Qi ≤ 1, if pi > 0,Qi < 0, if pi < 0, (3.8)under the extra ondition that the inverse transformations (3.5) are approx-imated with pie ewise linear fun tions.3.2.2 Positive signomial termsA positive signomial term, with at least one positive power, is underestimatedin a similar way as a negative signomial term. The inverse transformationsof the power transformations used to onvexify the positive terms, are

Zi = z1/Qi

i ,where the powers Q ful�ll the following onditions

Qi > 0, if pi > 0 ∧ i = k,Qi < 0, if pi > 0 ∧ i 6= k,Qi = 1, if pi < 0.The index k orresponds to the power remaining positive after the transfor-mations, if su h a power exists. Hen e, the following inequality should holdfor the transformed signomial termc

I∏

i=1

ZpiQi

i ≤ cI

∏

i=1

ZpiQi

i , (3.9)where the fun tions Z are the pie ewise linear approximations of the inversetransformations Z.Sin e the variables having negative powers do not need to be transformed,there is no need to approximate them with pie ewise linear fun tions. There-19

fore, they an be removed from both sides of the inequality (3.9) withouta�e ting it, sin e they are positive (assuming z > 0).The ondition for the power Qk, used on the variable remaining with a pos-itive power after the transformation, was that Qk is positive, so Zk = z1/Qk

kis an in reasing fun tion and the following ondition should be ful�lledZpkQk

k ≤ ZpkQk

k ⇔ Zk ≤ Zk. (3.10)For the rest of the transformed variables, the ondition was Qi < 0, whi himplies that the fun tions Zi (i 6= k) are de reasing fun tions. So for allindi es i 6= k the following should hold:1

Zpi|Qi|i

≤1

Zpi|Qi|i

⇔ Zi ≥ Zi. (3.11)Thus, sin e the onstant c is positive, the inequality (3.9) is true, i.e., thesignomial term is underestimated, as long as, the onditions (3.10) and (3.11)are ful�lled.Equation (3.10) is generally true if Zk is a on ave fun tion, and equation(3.11) if Zi is a onvex fun tion. Combining the fa t that the fun tionZ = z1/Q is onvex if Q ≤ 1 and on ave if Q ≥ 1 with the onvexity re-quirements (3.3), gives the following onditions for the transformed signomialterm to be onvex, as well as, underestimated:

Qi ≥ 1, if pi > 0 ∧ i = k,Qi < 0, if pi < 0 ∧ i 6= k,Qi = 1, if pi < 0.

(3.12)3.3 Pie ewise linear fun tionsAs mentioned previously, the transformed signomial terms are underesti-mated on the ondition that the nonlinear inverse transformations, used to onvexify the problem, are approximated with pie ewise linear fun tions.20

This is always possible, with the quality of the approximations determinedby the number of breakpoints in luded in the linearization. There are manydi�erent te hniques for approximating a fun tion with a pie ewise linear fun -tion; here two methods are presented, one using binary variables and theother using so- alled spe ial ordered sets. These methods, in ombinationwith the transformation pro edure presented earlier, an be used to onvex-ify and overestimate the MINLP problem (2.4).3.3.1 Formulation using binary variablesA variable z, whi h assumes the values z1, . . . , zK at K onse utive break-points (z1 < z2 < . . . < zK), an be expressed asz =

K−1∑

k=1

(zkbk + (zk+1 − zk)sk), (3.13)where bk is a binary variable, i.e., bk ∈ {0, 1}, and sk is a real variable su hthat 0 ≤ sk ≤ bk. Furthermore, the following ondition must be validK−1∑

k=1

bk = 1,that is, exa tly one binary variable at a time is allowed to be nonzero. Thiswill divide the original domain of z into K − 1 onvex subdomains. Byusing the linearity of the sum (3.13), the fun tion Z = f(z) an then beapproximated by a pie ewise linear fun tion Z, expressed asZ =

K−1∑

k=1

(Zkbk + (Zk+1 − Zk)sk),where Zk is the value of the fun tion f at the breakpoints zk, i.e., Zk = f(zk).A pie ewise linear approximation with �ve equidistant breakpoints is illus-trated in Figure 3.2. Other ways of de�ning pie ewise linear fun tions usingbinary variables an, for example, be found in [FP01℄.21

Figure 3.2: A pie ewise linear approximation Z of the fun tion f(z) in four stepsusing binary variables.3.3.2 Formulation using spe ial ordered setsThe other method for expressing pie ewise linear fun tions presented here,uses so- alled spe ial ordered sets (SOS). This method is often omputation-ally more e� ient than the one using binary variables. There are varioustypes of spe ial ordered sets, e.g., SOS Type 1 and SOS Type 2.De�nition 3.1. A spe ial ordered set is a set of variables (integers, ontin-uous or mixed integer and ontinuous), whi h ful�ll the following riteria:• for SOS type 1, at most one variable in the set may be nonzero,• for SOS type 2, at most two variables may be nonzero, and if twovariables are nonzero, they must be adja ent in the set.The following de�nition for pie ewise linear fun tions using spe ial orderedsets is from [BF76℄. All variables z in an interval [zk, zk+1] an be written as

z = zkwk + zk+1wk+1, (3.14)where the sum of the positive variables wk and wk+1 is one, i.e., wk+wk+1 = 1.Sin e expression (3.14) is a linear equation, a pie ewise linear approximation22

Figure 3.3: A pie ewise linear approximation Z of the fun tion f(z) in four stepsusing SOS type 2.Z of the fun tion Z = f(z) in the interval [zk, zk+1], an be expressed as

Z = Zkwk + Zk+1wk+1,where Zi is the value of the fun tion Z at the breakpoints zi, i.e., Zi = f(zi).In the general ase, a pie ewise linear approximation of the fun tion f(z)with the values Zi at the breakpoints zi, for i = 1, . . . , K, an be written asZ =

K∑

k=1

Ziwi, wi ≥ 0,wherez =

K∑

k=1

zkwi,

K∑

k=1

wi = 1,and {wk}Kk=1 is a spe ial ordered set of type 2 with the weights {zk}

Kk=1. Theweights are used by the MILP algorithm to order the variables, and therefore,all weights must have di�erent values. Here, this requirement means that

zi 6= zj for all i 6= j. A pie ewise linear fun tion with �ve breakpoints(K = 5) is illustrated in Figure 3.3.23

3.4 Examples of the transformation approa hIn this se tion, some simple examples are presented to illustrate how thetransformations are applied to signomial terms and how the inverse transfor-mations are approximated and repla ed with pie ewise linear fun tions.Example 3.2. Transformation of the negative signomial term −x2y−0.5.Using the power transformations x = XQ1 and y = Y Q2 results in−x2y−0.5 ⇒ −X2Q1Y −0.5Q2.The onditions Q1 > 0, Q2 < 0 and 2Q1 −0.5Q2 ≤ 1 must be ful�lled for theterm to be onvexi�ed and underestimated. Thus, for example by hoosing

Q1 = 0.5 and Q2 = −1, the term is onvexi�ed, as long as, the inversetransformations X(x) = x2 and Y (y) = y−1 are repla ed by the pie ewiselinear approximations X and Y .The method using binary variables from Se tion 3.3.1 gives the followingapproximation in one step from x = 1 to x = 4 and y = 2 to y = 5

X = X(1)bx + (X(4) − X(1))sx = bx + 15sx,

Y = Y (2)by + (Y (5) − Y (2))sy = 0.5by − 0.3sy,

x = bx + 3sx,

y = 2by + 3sy.However, sin e these are pie ewise approximations in one step only, the bi-naries bx and by are always one and the previous expressions simpli�es toX = 1 + 15sx,

Y = 0.5 − 0.3sy,

x = 1 + 3sx,

y = 2 + 3sy,with 0 ≤ sx ≤ 1 and 0 ≤ sy ≤ 1. 24

Example 3.3. Transformation of the positive signomial term x2y0.5z−1.Applying the transformations x = XQ1, y = Y Q2 and z = ZQ3 to the signo-mial term givesx2y0.5z−1 ⇒ X2Q1Y 0.5Q2Z−Q3.Here, either Q1 or Q2 an be hosen to be positive (and larger than or equalto one) and the other negative, or both an be hosen to be negative. In thisexample Q1 and Q2 are hosen so that Q1 ≥ 1 and Q2 < 0. Sin e z has anegative power, it does not need to be transformed and Q3 = 1. Hen e, the onvexity requirement be omes

2Q1 + 0.5Q2 − Q3 = 2Q1 + 0.5Q2 − 1 ≥ 1.By hoosing Q1 = 2 and Q2 = −1, the positive signomial term aboveis onvexi�ed. It is also underestimated when the inverse transformationsX(x) = x0.5 and Y (y) = y−1 are approximated by the pie ewise linear fun -tions X and Y .Using the approa h with spe ial ordered sets of type 2, explained in Se tion3.3.2, and assuming that 1 ≤ x ≤ 4 and 2 ≤ y ≤ 5, the pie ewise linearapproximations an be written as

X = X(1)wx,1 + X(4)wx,2 = wx,1 + 2wx,2,

Y = Y (2)wy,1 + Y (5)wy,2 = 0.5wy,1 + 0.2wy,2,

x = wx,1 + 4wx,2,

y = 2wy,1 + 5wy,2,where wx,1 + wx,2 = 1 and wy,1 + wy,2 = 1.Example 3.4. The non onvex fun tionf(x) = (x4 + 79.5x2 − 170x + 120) − 15x3, 1 ≤ x ≤ 6,is made up of a onvex part, as well as, a non onvex part onsisting of the lastterm −15x3. The fun tion is onvexi�ed and underestimated by applying the25

power transformation x = XQ with Q = 1/3, and approximating the inversepower transformation X(x) = x3 with the pie ewise linear fun tion X takingvalues Xk at the breakpoints xk.The transformed fun tion f(x, X) and the expressions for the pie ewise linearapproximations an then be written as:f(x, X) = (x4 + 79.5x2 − 170x + 120) − 15X,

X =K

∑

k=1

Xkwk,

x =

K∑

k=1

xkwk,

K∑

k=1

wk = 1,where wk ≥ 0 belong to a spe ial ordered set of type 2.The onvex underestimations f(x, X) of the fun tion f(x) are illustrated inFigure 3.4, with linear approximations of X using two, three, �ve and ninebreakpoints. From the �gure it an be seen that the approximation improveswhen the number of breakpoints is in reased.However, the transformed MINLP problem grows in omplexity for ea hadded breakpoint, so there should be a tradeo� between the quality of theapproximation, and the size of the problem. Another approa h to redu ingthe omplexity of the onvexi�ed and overestimated optimization problem,would be to optimize the power transformations used. This will be the topi of the next hapter.

26

1 1.5 2 2.5 3 3.5 4 4.5 5

−500

−400

−300

−200

−100

0

(a) K = 2

1 1.5 2 2.5 3 3.5 4 4.5 5−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

20

(b) K = 3

1 1.5 2 2.5 3 3.5 4 4.5 5−50

−40

−30

−20

−10

0

10

20

( ) K = 5

1 1.5 2 2.5 3 3.5 4 4.5 5−50

−40

−30

−20

−10

0

10

20

(d) K = 9Figure 3.4: The fun tion f(x) = (x4 +79.5x2−170x+120)−15x3 and the onvexunderestimations f(x, X) with K = 2, 3, 5, 9.27

4 Optimization of the transformation approa h4.1 Optimization of the power transformationsThe power transformations used in the onvexi� ation and underestimationpro ess of signomial terms des ribed in the previous hapter an be appliedtermwise to signomial fun tions of the typeσm(z) =

J∑

j=1

cj

I∏

i=1

zpji

i .The transformations allows for some degrees of freedom regarding how thepowers Q in the transformations z = ZQ are hosen. The fa t that no trans-formation o urs if the power Q is equal to one, an be used to devise amethod optimizing the transformations needed to onvexify and underesti-mate signomial fun tions, and in the pro ess, determining whi h variablesneed to be transformed and what transformations an be used.In the rest of this thesis, it is assumed that the index j orresponds to thej-th non onvex signomial term in the whole MINLP problem having a totalof JT non onvex signomial terms. The onvex signomial terms are in ludedin the fun tions q and the non onvex signomial terms in the fun tions σ inthe generalized signomial onstraints in problem (2.4). The onvexity of thesignomial terms an be he ked by using Theorem 2.13 and Lemma 2.14.A binary variable bji, whi h is equal to one if the variable zi in the j-th termis transformed by a power transformation and equal to zero otherwise, isintrodu ed. The transformed signomial fun tion an then be written as

σm(z) =

JT∑

j=1

cj

I∏

i=1

z(1−bji)pji

i · ZjibjipjiQji,be ause the expression z(1−b)p · ZbpQ simpli�es to zp if the binary b is equalto zero and to ZpQ if b is equal to one.28

The next step is to formalize a mixed integer linear programming (MILP)problem with the obje tive being to optimize the set of transformations re-quired to onvexify the signomial terms.The total number of transformations when onvexifying and underestimatingthe JT non onvex signomial terms is the sumJT∑

j=1

I∑

i=1

bji, (4.1)sin e ea h binary b is equal to one if and only if the orresponding variableis transformed.However, when approximating the onvexi�ed signomial fun tion with apie ewise linear fun tion, some of the expressions and variables used in theapproximations an be reused in the approximation of the variable z in allthe terms where z is found, even if the power transformations used are notthe same. For example, if the linear approximations are expressed with themethod from Se tion 3.3.1, the same binary variables b and real variables s an be used in all the approximations involving the same variable, as long as,the breakpoints are the same. Therefore, in some ases, it ould be more ben-e� ial to minimize the total number of original variables transformed, ratherthan the total number of transformations, and in the pro ess, simplifying thetransformed MINLP problem.To a omplish this, a new binary variable B is introdu ed for ea h of theoriginal variables found in the non onvex signomial terms. The binary Bi isequal to one if the i-th variable is transformed by a power transformationin any of the terms where it is found, and equal to zero otherwise. This ondition an be expressed asJT∑

j=1

bji ≤ JT Bi. (4.2)29

The total number of transformed original variables an then be expressed asI

∑

i=1

Bi. (4.3)As mentioned previously, the onvexi� ation pro ess of the signomial termsalso underestimates the onvexi�ed terms when the inverse power transfor-mations are approximated with pie ewise linear fun tions. Sin e the powersused in the transformations an be both small or large, numeri al di� ulties an arise when al ulating the pie ewise linear approximations.For example, if the power Q = 0.1 is used in a transformation, the inversepower transformation be omes Z(z) = z1/Q = z10, whi h an ause problemswhen al ulating the values of the inverse transformation at the breakpoints.Therefore, sin e values of Q loser to +1 or −1 are numeri ally more stablethan values lose to zero or large negative values, an additional penalty fa toris in luded in the obje tive fun tion. This penalty fun tion, onsisting of thesum of the absolute deviations ∆ from +1 if Q is positive and −1 if Q isnegative, isJT∑

j=1

I∑

i=1

∆ji. (4.4)To be able to express the absolute deviations from either +1 or −1, a binaryvariable β, being equal to one whenever the power Q is positive and equal tozero whenever Q is negative, is introdu ed. These onditions an be writtenas the following linear onditions, using so- alled big-M formulation

Qji ≥ M(βji − 1),

Qji ≤ Mβji,⇒

βji = 0 : −M ≤ Qji ≤ 0,

βji = 1 : 0 ≤ Qji ≤ M,(4.5)where M is a large positive number. Furthermore, the onditions for the

30

absolute deviations from +1 or -1 ∆ an be expressed as:

Qji − ∆ji + Mβji ≤ M + 1,

Qji − ∆ji − Mβji ≤ −1,

−Qji − ∆ji + Mβji ≤ M − 1,

−Qji − ∆ji − Mβji ≤ 1.

(4.6)The obje tive fun tion of the MILP problem used to optimize the transfor-mations will now onsist of a sum of three parts; �rst the total number oftransformations (4.1), then the total number of transformed variables (4.3),and �nally the penalty fa tor (4.4).To allow for di�erent strategies for optimizing the transformations, ea h ofthe three parts onstituting the obje tive fun tion is multiplied with theweights δ1, δ2 and δ3. Thus, the obje tive fun tion used in the methodbe omes

δ1

I∑

i=1

Bi + δ2

JT∑

j=1

I∑

i=1

bji + δ3

JT∑

j=1

I∑

i=1

∆ji. (4.7)Furthermore, the onditions (4.2), (4.5) and (4.6) must be in luded in theMILP problem, along with requirements guaranteeing orre t power trans-formations. The transformation requirements are di�erent for negative andpositive signomial terms and will be presented in Se tions 4.1.1 and 4.1.2.In on lusion, a few remarks on erning how the weights in the obje tivefun tion an be hosen. For example, if the number of transformations isof most on ern, then δ1 should be larger than the other weights, and if itis important to minimize the total number of transformed variables, then δ2should be larger than δ1 and δ3. Finally, if numeri ally good transformationsis the most signi� ant property, then δ3 should be the largest weight.31

4.1.1 Linear onditions for negative termsAs previously on luded, the powers used in the transformation of variablesin negative signomial terms must ful�ll the requirements

0 < Qji ≤ 1, if pji > 0,

Qji < 0, if pji < 0,(4.8)if the term is to be onvexi�ed and underestimated.For a positive power p in a negative signomial term, the linear onditions in- luded in the MILP problem must guarantee that if a transformation o urs(b = 1), then the power Q must be between zero and one, and if no transfor-mation o urs (b = 0), then Q must be equal to one. These onditions anbe formulated as:

Qji ≥ 1 − bji,

Qji ≤ 1 − ǫ bji,

Qji ≥ ǫ,

⇒

bji = 0 : Qji = 1,

bji = 1 : ǫ ≤ Qji < 1,(4.9)where M is a large positive number and ε = 1/M .For a negative power p in a negative signomial term, a transformation isalways ne essary and the power Q used in the transformation is always neg-ative, so the linear onditions to be in luded in the MILP problem are

bji = 1,

−M ≤ Qji ≤ −ǫ.(4.10)Finally, the sum of the powers in the transformed signomial term shouldbe less than or equal to one a ording to equation (3.2), so the following ondition

I∑

i=1

pjiQji ≤ 1, (4.11)must also be added to the generated MILP problem.32

4.1.2 Linear onditions for positive termsA ording to Theorem 2.13, there are two ways to onvexify a non onvexpositive signomial term: Either all powers should be negative after the trans-formations, or one power should be positive and the rest negative while thesum of the powers is greater than or equal to one.A binary variable αji orresponding to whether the i-th variable in the j-th signomial term is left positive (α = 1) or negative (α = 0) after thetransformation is introdu ed. Sin e at most one variable per term is allowedto have a positive power after the transformation, the following onditionmust also be in luded:I

∑

i=1

αji ≤ 1. (4.12)A ording to the onditions (3.12), the power Q should be greater or equalto one when the binary α is equal to one, and less than zero when α is equalto zero. These statements an be expressed using the following inequalities:

Qji ≤ αji M − ε(1 − αji),

Qji ≥ −M + αji(M + 1),⇒

αji = 0 : −M ≤ Qji ≤ −ǫ,

αji = 1 : 1 ≤ Qji ≤ M.(4.13)Furthermore, the binary b should indi ate whether or not a transformationo urs. The only ase when a variable in a positive signomial term is nottransformed, and b is equal to zero, is when both α and Q are equal to one,otherwise b should be equal to one. This an be written as the followinginequalities:

bji ≥ 1 − αji,

bji ≥ ε(Qji − 1),

bji ≤ (1 − ε)Qji + M(1 − αji),

⇒

αji = 0 ∧ Qji < 0 : bji = 1,

αji = 1 ∧ Qji = 1 : bji = 0,

αji = 1 ∧ Qji ≥1

1−ǫ: bji = 1,(4.14)where the same value on the large positive number M an be used as in the ase with the negative signomial terms and ε = 1/M .33

No transformation is needed for variables with negative powers (p < 0) inpositive signomial terms. Hen e, the power Q is set equal to one, and thebinaries α and b are both equal to zero, i.e., the following onditions

Qji = 1,

bji = 0,

αji = 0,

(4.15)must be added to the MILP problem.The onvexity ondition for the j-th non onvex signomial term is a ordingto (3.4)I

∑

i=1

pjiQji ≥ 1,if one of the powers in the term is positive after transformation, i.e., whenever∑I

i=1 αji = 1. If all powers are negative, no additional onvexity onditionshould be imposed on the term. Therefore, by in luding the onditionI

∑

i=1

pjiQji − MI

∑

i=1

αji ≥ 1 − M, (4.16)in the MILP problem for all positive non onvex signomial terms, only powertransformations guaranteeing onvexity is allowed.The solution of the MILP problem des ribed in Se tion 4.1, with the obje tivefun tion (4.7) and the linear onditions (4.2), (4.5), (4.6), (4.8) - (4.11) and(4.12) - (4.16), will indi ate the number of transformations, as well as, whattransformations an be used to onvexify and underestimate the non onvexsignomial terms.Sin e the re eived set of power transformations an be optimized in di�erentways depending on the values of the weights δ1, δ2 and δ3 in the obje tivefun tion, the onvexi�ed and underestimated MINLP problem an be solvedmore e� iently, ompared to just using the transformation methods men-tioned in Chapter 3. 34

To make it easier for the reader to obtain an overview of the method, thewhole MILP problem formulation is ompiled in Appendix A.4.2 Improving the numeri al stability of signomial termsBoth the onvexi�ed and the originally onvex signomial terms an in ludevariables with both small and large powers, whi h an in numeri al di� ultieswhen the MINLP problem is solved. However, by rewriting the signomialterms in a numeri ally more stable form, it is possible to over ome thisproblem.To simplify the following expressions, the indi es i orresponding to variableszi involved in power transformations are in luded in the index set Ij. Thej-th onvexi�ed and underestimated signomial term an then be written as

sj(z) = cj

∏

i∈Ij

ZjipjiQji ·

∏

i/∈Ij

zpji

i , (4.17)where Z orresponds to the pie ewise linear approximations of the inversetransformations Z.Sin e the variables z are positive, taking the logarithm and then the exponen-tial of the signomial term (4.17), will yield the following equivalent expressionfor the onvexi�ed and underestimated termsj(z) = exp

log(cj) +∑

i∈Ij

(pjiQji) · log(Zji) +∑

i/∈Ij

pji · log(zi)

, (4.18)where the pie ewise linear approximations Z are de�ned, for example, a - ording to one of the methods presented in Se tion 3.3.35

4.3 Simulations4.3.1 Des ription of the simulationsTo study how well the method works, simulations on randomly generatedsignomial fun tions were performed. A program generating the linear prob-lem de�ned in Appendix A was designed, and the MILP problems were thensolved with the solver ILOG CPLEX 10.0.Simulations on two di�erent problems sizes were performed, the smaller one onsisting of groups of 25 signomial terms and a total of 50 variables, and thelarger one of groups of 50 signomial terms and 100 variables. Also, sin e thetransformation pro ess is di�erent for positive and negative signomial terms,simulations on only negative, only positive and mixed negative and positivesignomial terms were performed.The signomial terms was generated so that most of them were produ tsof between one and three variables, and no one was a produ t of more thanseven variables. The distributions of the number of variables in the signomialterms are illustrated in the �gures orresponding to the simulations, and theyare approximately the same in all simulations. Furthermore, to demonstratethe impa t of the size of the powers in the signomial terms, the powerswere normally distributed with varian e four in some simulations and one inothers. The mean of the powers was zero.The strategy used in the simulations was to minimize the total number oftransformed original variables, so the values on the weights δ in the obje tivefun tion (4.7) were hosen to be δ1 = 0.01, δ2 = 1 and δ3 = 0.0001. Theparameters of the simulations performed are listed in Table 4.1 and the resultsare presented in Table 4.2 and Figures 4.1 - 4.4.4.3.2 Comments on the resultsThe results from the �rst and se ond simulations, with 50,000 signomialgroups onsisting of 25 positive and 25 negative terms ea h, and a total36

Simulation # Variables # Terms Type Varian e Figure1 50 25 positive 4 4.12 50 25 negative 43 100 50 positive 4 4.24 100 50 negative 45 50 25 mixed 1 4.36 100 50 mixed 17 50 25 mixed 4 4.48 100 50 mixed 4Table 4.1: The parameters of the simulations.of 50 variables, are presented in Figure 4.1. These results indi ate thatnegative signomial terms appear to require more transformed variables andmore power transformations than positive terms. The median number oftransformed variables and the total number of transformations are 24 and 28in the positive ase, and 42 and 52 in the negative ase.The reason why negative signomial terms require more transformations, andhen e more transformed variables, are that the sum of the powers for anegative signomial term always should be greater than zero and smaller orequal to one. Therefore, variables with negative powers, as well as, largepositive powers (larger than one) always have to be transformed in negativeterms. For positive signomial terms, either all powers should be negative,or at most one should be positive and the sum of the powers greater thanor equal to one. This gives more freedom regarding how to hoose whi hvariables should be transformed and whi h transformations ould be usedand, in the pro ess, allows for less transformed variables and transformations.The fa t that positive signomial terms, in general, require less transforma-tions than negative signomial terms, an be seen from the sub�gures repre-senting the distribution of the number of transformations required per termin Figure 4.1. For example, over 75 per ent of positive signomial terms ofone variable require no transformation, while the orresponding number fornegative signomial terms is only about 25 per ent.37

Simulation Transformed variables Transformations1 24 282 42 523 49 574 84 1065 29 336 58 687 34 408 67 81Table 4.2: The median number of transformed variables and transformations.The results from Simulations 3 and 4 are presented in Figure 4.2. The num-bers of terms and variables were doubled from 25 terms and 50 variables to 50terms 100 variables. The results are, however, quite similar, sin e the mediannumber of original variables transformed was 49 in the positive ase, and 84in the negative ase, approximately double that of the smaller problems. Thenumber of power transformations required was 57 and 106 in Simulations 3and 4, ompared to 28 and 52 from Simulations 1 and 2.The results for groups of mixed positive and negative signomial terms of thesame two problem sizes as earlier, i.e., 25 terms and 50 variables, as wellas, 50 terms and 100 variables, are presented in Figures 4.3 and 4.4. InSimulations 5 and 6, the powers have the varian e one and in Simulations7 and 8 varian e four. When omparing the orresponding problem sizeswith ea h other, it an be seen that a larger varian e indi ates that moretransformations, and hen e more transformed variables, are needed. One ofthe reasons for this is that for a negative signomial term, the sum of thepowers must be between zero and one, so a power larger than one always orresponds to a transformation, while a power between zero and one maynot need any transformation at all. Furthermore, the distributions of thenumber of power transformations required per term are also almost identi alin the two simulations.38

Figure 4.1: The results from 50,000 simulations of the method on groups of 25 positive (left) and 25 negative (right)signomial terms with a total of 50 variables. The powers were generated to have varian e four and mean zero.

39

Figure 4.2: The results from 50,000 simulations of the method on groups of 50 positive (left) and 50 negative (right)signomial terms with a total of 100 variables. The powers were generated to have varian e four and mean zero.

40

Figure 4.3: The results from 50,000 simulations of the method on groups of 25 signomial terms and 50 variables (left) andgroups of 50 signomial terms and 100 variables (right). The terms are mixed positive and negative and the powers weregenerated to have varian e one and mean zero.

41

Figure 4.4: The results from 50,000 simulations of the method on groups of 25 signomial terms and 50 variables (left) andgroups of 50 signomial terms and 100 variables (right). The terms are mixed positive and negative and the powers weregenerated to have varian e four and mean zero.

42

5 The GGPECP algorithm5.1 Des ription of the algorithmOne of the methods available today for solving non onvex MINLP problemsof the type (2.4) to global optimality is the GGPECP algorithm presented in[WW03℄. The method in orporates the transformation te hniques for signo-mial fun tions des ribed in Chapter 3 with the generalized αECP (extended utting plane) algorithm from [WP02℄. By integrating the transformationstep into the MINLP solver, results from previous iterations an be utilized,making the solution pro ess more e� ient.The GGPECP algorithm onsists of an initial onvexi� ation step, where thenon onvex signomial terms ontained in the generalized signomial onstraintsare onvexi�ed and underestimated using pie ewise linear fun tions, as de-s ribed in Chapter 3. Be ause the signomial onstraints are underestimated,the feasible region of the original problem is overestimated.This leads to the fa t that, if a solution satisfying the original onstraints ofthe overestimated MINLP problem is found, it is also the globally optimalsolution. Otherwise, in the next iterations, the pie ewise linear approxima-tions are improved by adding additional breakpoints and solving the updatedMINLP problem. The pro ess is terminated when all the signomial on-straints are ful�lled to an arbitrary a ura y ε, i.e., whenever the following riterion is satis�ed:max(qm(z) + σm(z)) ≤ ε.Di�erent strategies for sele ting the new grid points at ea h GGPECP iter-ation is dis ussed in [Wes05℄. For example, the previous solution values orthe midpoint of the interval between the two breakpoints where the solutionpoint is found an be used. Furthermore, it is not ne essary to add newbreakpoints to the approximations of all the original variables transformed,but instead breakpoints for only the variables involved in the generalizedsignomial onstraints that are violated the most an be added.43

Figure 5.1: A simpli�ed �ow hart of the GGPECP algorithm.By in orporating the method for optimizing the power transformations fromChapter 4 into the initial transformation step of the GGPECP algorithm, theperforman e when solving the overestimated subproblems an be in reased.A simpli�ed �ow hart of the GGPECP algorithm is shown in Figure 5.1.5.2 Some numeri al examples5.2.1 A two dimensional MINLP problemTo illustrate how the GGPECP algorithm works, it is applied to the followingtwo-dimensional MINLP problem:min y − 3x,s.t. y + 5x ≤ 36, −y + 0.25x ≤ −1,

(2y2 − 2y0.5 + 11y + 8x − 39) − 2x0.5y2 + 0.1x1.5y1.5 ≤ 0,

1 ≤ x ≤ 7, 1 ≤ y ≤ 7, x ∈ R+, y ∈ Z

+.

(5.1)44

Figure 5.2: The integer-relaxed feasible region of the MINLP problem (5.1).The problem onsists of a linear obje tive fun tion, two linear onstraintsand a generalized signomial onstraint. The generalized signomial onstraint,with a onvex and a non onvex part, is a fun tion of two variables x and y,where x is a real variable and y is a integer-valued variable. The last twosignomial terms make up the non onvex part and the rest of the terms the onvex part.When the variable y is relaxed to a real-valued variable, the feasible region isthe shaded area in Figure 5.2, onsisting of two disjoint regions. This ausesproblems for most MINLP solvers.Applying the transformation method from Chapter 4 to the two non onvexsignomial terms in the onstraints, gives that the transformationsy = Y 0.25

1 ⇒ Y1 = y4,

y = Y−1/32 ⇒ Y2 = y−3.will onvexify and underestimate the terms. Thus, the variable y is trans-formed by power transformations in both terms, while the variable x doesnot require any transformations at all. The MILP problem generated by themethod, and its solution, an be found in Appendix B.The variable y in the two non onvex terms in the generalized signomial on-straint in (5.1) are repla ed with the transformed variables Y1 and Y2, yielding45

Figure 5.3: The overestimated integer-relaxed feasible region of the MINLP prob-lem (5.1) before the se ond GGPECP iteration.the new expression for the onstraint(2y2 − 2y0.5 + 11y + 8x − 39) − 2x0.5Y 0.5

1 + 0.1x1.5Y −0.52 ≤ 0.The inverse transformations of the transformed variables, i.e., Y1 = y4 and

Y2 = y−3, are now repla ed with their pie ewise linear approximations in the onstraints (using the initial breakpoints y = 1 and y = 7). The expressionsneeded for the pie ewise linear fun tions using the SOS approa h from Se tion3.3 are:Y1 = 1w1 + 2401w2,

Y2 = 1w1 + 0.0029w2,

y = 1w1 + 7w2,

w1 + w2 = 1,where the positive real-valued variables w1 and w2 belong to a spe ial orderedset of type 2. The new onvexi�ed onstraint be omes(2y2 − 2y0.5 + 11y + 8x − 39) − 2x0.5Y 0.5

1 + 0.1x1.5Y −0.52 ≤ 0,and the new overestimated feasible region is illustrated in Figure 5.3.46

Figure 5.4: The overestimated integer-relaxed feasible region of the MINLP prob-lem (5.1) before the third GGPECP iteration.The onvexi�ed MINLP problem is now solved using the αECP method,giving the value of the obje tive fun tion as −16.8 for the values x = 6.6and y = 3. Be ause the original generalized signomial onstraint has thevalue 23.9035, this solution is not yet globally optimal, so further GGPECPiterations are needed. The pie ewise linear approximations of Y1 and Y2 areenhan ed by adding the urrent solution point y = 3 as a breakpoint. Theexpressions for the new pie ewise linear fun tions be ome:Y1 = 1w1 + 81w2 + 2401w3,

Y2 = 1w1 + 0.0370w2 + 0.0029w3,

y = 1w1 + 3w2 + 7w3,

w1 + w2 + w3 = 1.Note that for ea h breakpoint added to the linear approximations, only onevariable is added to the spe ial ordered set, and that the same set an beused in the expressions for both Y1 and Y2. However, if di�erent breakpointsare added to the linear approximations of Y1 and Y2, this is not the ase.When repla ing the previous approximation with this new one, the feasibleregion, as shown in Figure 5.4, be omes smaller, and will better approximatethat of the original problem. Also note that the previous solution point isnot in luded in the new feasible region.47

Figure 5.5: The overestimated integer-relaxed feasible region of the MINLP prob-lem (5.1) before the fourth GGPECP iteration.Solving the MINLP subproblem gives the obje tive value −15.2 at the pointx = 6.4 and y = 4. The original generalized signomial onstraint has thevalue 16.1984, so it is not yet satis�ed and more iterations are thereforerequired.As in the previous iteration, the solution value y = 4 is in luded in the setof breakpoints for the linear approximation of the inverse transformations Y1and Y2, giving the following equations for the pie ewise linear fun tions:

Y1 = 1w1 + 81w2 + 256w3 + 2401w4,

Y2 = 1w1 + 0.0370w2 + 0.0156w3 + 0.0029w4,

y = 1w1 + 3w2 + 4w3 + 7w4,

w1 + w2 + w3 + w4 = 1.Using these approximations for the inverse power transformations gives thefeasible region shown in Figure 5.5. Again, the previous solution point hasbeen ex luded, as the approximations are improved.The solution to this subproblem gives the value of the obje tive fun tionas -13.6, for the values x = 6.2 and y = 5 of the variables. The originalsignomial onstraint has the value 3.8889, so the optimal solution is not yetfound. 48

Figure 5.6: The overestimated integer-relaxed feasible region of the MINLP prob-lem (5.1) before the fourth and �nal GGPECP iteration.By ontinuing to use the same method, the previous solution y = 5 is added asa breakpoint to the pie ewise linear approximations, yielding the expressionsY1 = 1w1 + 81w2 + 256w3 + 625w4 + 2401w5,

Y2 = 1w1 + 0.0370w2 + 0.0156w3 + 0.0080w4 + 0.0029w5,

y = 1w1 + 3w2 + 4w3 + 5w4 + 7w5,

w1 + w2 + w3 + w4 + w5 = 1.Solving this MINLP subproblem, gives that for the values x = 6 and y = 6,the obje tive fun tion has the globally optimal value −12, sin e the value ofthe original generalized signomial onstraint is −12.6622, i.e., less than zero,whi h was the requirement. The overestimated feasible region in this lastiteration is shown in Figure 5.6.

49

5.2.2 A geometri programming exampleThe following example is a geometri programming problem from [RM78℄,and in ludes a signomial obje tive fun tion, as well as, seven signomial in-equality onstraints. In total eight positive real variables are used.min 2.0 z0.91 z−1.5

2 z−33 + 5.0 z−0.3

4 z2.65 + 4.7 z−1.8

6 z−0.57 z8,s.t. 7.2 z−3.8

1 z2.22 z4.3

3 + 0.5 z−0.74 z−1.6

5 + 0.2 z4.36 z−1.9

7 z8.58 ≤ 1,

10.0 z2.31 z1.7

2 z4.53 ≤ 1, 0.6 z−2.1

4 z0.45 ≤ 1,

6.2 z4.56 z−2.7

7 z−0.68 ≤ 1, 3.1 z1.6

1 z0.42 z−3.8

3 ≤ 1,

3.7 z5.44 z1.3

5 ≤ 1, 0.3 z−1.16 z7.3

7 z−5.68 ≤ 1.Sin e the obje tive fun tion ontains signomial terms, the previous problemhas to be reformulated to be of the same form as problem (2.4). This anbe done by repla ing the obje tive fun tion with an additional variable uand then in luding the di�eren e of the previous obje tive fun tion and thevariable u as an additional signomial onstraint, i.e.,min u,s.t. 2.0 z0.9

1 z−1.52 z−3

3 + 5.0 z−0.34 z2.6

5 + 4.7 z−1.86 z−0.5

7 z8 − u ≤ 0,

7.2 z−3.81 z2.2

2 z4.33 + 0.5 z−0.7

4 z−1.65 + 0.2 z4.3

6 z−1.97 z8.5

8 ≤ 1,

10.0 z2.31 z1.7

2 z4.53 ≤ 1, 0.6 z−2.1

4 z0.45 ≤ 1,

6.2 z4.56 z−2.7

7 z−0.68 ≤ 1, 3.1 z1.6

1 z0.42 z−3.8

3 ≤ 1,

3.7 z5.44 z1.3

5 ≤ 1, 0.3 z−1.16 z7.3

7 z−5.68 ≤ 1.The underlined terms are onvex, and an therefore be in luded in the onvexfun tions q in (2.4), while the rest of the terms are in luded in the non onvexsignomial fun tions σ. Using the transformation method from Chapter 4 onthe fun tions σ gives that 12 power transformations are required to onvexifyand underestimate the signomial terms, and that the variables z4 and z6 donot require any transformation at all. The power transformations given bythe method, as well as, the onvexi�ed terms are listed in Table 5.1.50

Original term Transformed term Transformations2.0 z0.9

1 z−1.52 z−3

3 2.0 Z−0.91,1 z−1.5

2 z−33 z1 = Z−1

1,1

4.7 z−1.86 z−0.5

7 z8 4.7 z−1.86 z−0.5

7 Z−12,8 z8 = Z−1

2,8

7.2 z−3.81 z2.2

2 z4.33 7.2 z−3.8

1 Z−2.23,2 Z7

3,3 z2 = Z−13,2 , z3 = Z1.6279

3,3

0.2 z4.36 z−1.9

7 z8.58 0.2 z4.3

6 z−1.97 Z−1.4

4,8 z8 = Z−0.164714,8

10.0 z2.31 z1.7

2 z4.53 10.0 Z−1.8

5,1 Z−1.75,2 z4.5

3 z1 = Z−0.782615,1 , z2 = Z−1

5,2

0.6 z−2.14 z0.4

5 0.6 z−2.14 Z−0.4

6,5 z5 = Z−16,5

3.1 z1.61 z0.4

2 z−3.83 3.1 Z5.2

7,1Z−0.47,2 z−3.8

3 z1 = Z3.257,1 , z2 = Z−1

7,2

3.7 z5.44 z1.3

5 3.7 z5.44 Z−1.3

8,5 z5 = Z−18,5

0.3 z−1.16 z7.3

7 z−5.68 0.3 z−1.1

6 Z−7.39,7 z−5.6

8 z7 = Z−19,7Table 5.1: The original and transformed terms, as well as, the obtained powertransformations to transform problem (5.1).Solving the onvexi�ed and underestimated problem with the GGPECP al-gorithm gives the optimal obje tive value 29.2291 for the following values onthe variables z:

z1 = 0.9688, z2 = 0.1990, z3 = 1.1213, z4 = 0.7844,

z5 = 1.0022, z6 = 0.7007, z7 = 1.0934, z8 = 0.9717.This solution is slightly better than the solution found in [RM78℄, where theobje tive value was 29.5985.

51

6 Con lusionsSome transformation te hniques in global optimization for transforming non- onvex MINLP problems ontaining signomial terms to a onvex form over-estimating the original problem have been presented in this thesis.Chapter 2 ontained some basi de�nitions and theorems regarding onvexsets and fun tions. Furthermore, the signomial fun tion, as well as, theformulation of an MINLP problem were des ribed. In Chapter 3, a on-vexi� ation and underestimation pro edure for signomial terms based onpower transformations was presented. This pro edure was further devel-oped in Chapter 4, where a method for optimizing the transformations wasintrodu ed. Some simulations of the method were also performed to deter-mine how well it performed. Finally, in Chapter 5 the GGPECP algorithmwas brie�y dis ussed and two examples were also provided to show how theoptimization te hniques for the transformations an be integrated into thealgorithm.The method for optimizing the transformations, introdu ed in Chapter 4, an be used to �nd optimal sets of transformations for onvexifying and un-derestimating MINLP problems ontaining signomial terms. However, someaspe ts of the method were not fully studied. For example, the impa t ofdi�erent hoi es of the parameters δ1, δ2 and δ3, determining the optimiza-tion strategy, did not �t within the s ope of this thesis. When hoosing thevalues of these parameters, the strategy for adding the breakpoints in theGGPECP algorithm should also be taken into a ount; this ould perhapsbe a topi for further study. Also, the penalty term in the obje tive fun -tion, providing numeri ally more stable transformations, an be probably beimproved further by making it nonlinear, resulting in better transformations.Furthermore, a study of the omputational e� ien y of the di�erent typesof pie ewise linear fun tions should also be performed. The method usingspe ial ordered sets is probably more e� ient, depending on the algorithmsused by the MINLP solver, but sin e there are other ways to de�ne pie ewiselinear fun tions, a more systemati omparison should be performed.52

7 Swedish summary7.1 Transformationstekniker inom global optimeringSignomialfunktioner är vanliga i i ke-linjära globala optimeringsproblem ef-tersom exempelvis alla bilinjära o h trilinjära funktioner samt alla potens-funktioner kan ses som spe ialfall av signomialfunktioner. Ett i ke-linjärtoptimeringsproblem där variablerna tillåts vara både heltal o h reella tal kanformuleras som:min f(z), z = (z1, z2, . . . , zI),s.t. Az = a, Bz ≤ b,

gn(z) ≤ 0, n = 1, 2, . . . , Jn,

qm(z) + σm(z) ≤ 0, m = 1, 2, . . . , Jm.

(7.1)Objektfunktionen f tillåts vara av olika typer beroende på vilken metod somanvänds för att lösa problemet. Om GGPECP-algoritmen från [WW03℄ an-vänds, får objektfunktionen vara pseudokonvex. Vektorn z kan bestå av bådeheltalsvariabler o h reella variabler. Funktionerna g, q o h σ är di�erentier-bara pseudokonvexa o h konvexa funktioner respektive signomialfunktioner.Signomialfunktionerna de�nieras som summan av termer som består av pro-dukter av potensfunktioner, alltså

σ(z) =J

∑

j=1

cj

I∏

i=1

zpji

i .Här är cj o h pji reella tal för alla värden på indexen i o h j. Dessutom antasdet i fortsättningen även att variablerna z är positiva.7.1.1 Transformering av signomialfunktionerEn i ke-konvex signomialterm kan konvexi�eras genom att använda olikatyper av transformationer på de enskilda variablerna i termen. Här användspotenstransformationer av formen z = ZQ, vilka beskrivs närmare i [Wes05℄.53

Följande resultat angående signomialtermers konvexitet är från [MF95℄:(a) En positiv signomialterm är konvex om antingen alla potenser p är ne-gativa, eller en potens är positiv medan resten är negativa o h summanav potenserna större än eller lika med ett.(b) En negativ signomialterm är konvex om alla potenser är positiva o hsumman av potenserna är mindre än eller lika med ett.Genom att använda dessa resultat inser man att en godty klig signomialtermi ett optimeringsproblem av typen (7.1) alltid kan transformeras med hjälpav potenstransformationer av formen z = ZQ, där potenserna Q uppfyllervissa kriterium. Då man använder dessa transformationer, kommer den i ke-konvexa delen av funktionen däremot bara att �yttas från själva termentill de införda begränsningarna som motsvarar den inversa transformationenZ = z1/Q; om man däremot approximerar den inversa transformationen medsty kevis linjära funktioner, kommer det ursprungliga problemet att indelas ikonvexa delproblem. Eftersom det är viktigt att inga möjliga lösningspunkterförsvinner då approximationerna görs, är det nödvändigt att se till att detransformerade signomialtermerna alltid underskattar de ursprungliga. På såsätt ser man till att lösningsområdet för det approximativa problemet alltidär större än det ursprungliga. Detta åstadkoms genom att ytterligare villkorinförs för potenserna Q.I [Wes05℄ härleddes kraven på potenserna Q för korrekta transformationersom både konvexi�erar o h underestimerar signomialtermer. Eftersom kon-vexi�eringskraven är olika för positiva o h negativa signomialtermer, blir ävenvillkoren för värdena på potenserna Q annorlunda i dessa båda fall.För en negativ signomialterm gäller kraven

0 < Qi ≤ 1, om pi > 0,Qi < 0, om pi < 0, o h I

∑

i=1

piQi ≤ 1, (7.2)för potenstransformationer som konvexi�erar o h underskattar termen. Mot-54

svarande krav för en positiv signomialterm är

Qi ≥ 1, om pi > 0 ∧ i = k,Qi < 0, om pi < 0 ∧ i 6= k,Qi = 1, om pi < 0.

o h I∑

i=1

piQi ≥ 1. (7.3)Här motsvarar indexet k den variabel som har positiv potens efter transfor-mationen, om en sådan existerar, alltså pkQk > 0 medan piQi < 0 för varjeindex i 6= k. Olikheten, som anger att summan av de nya potenserna pQ skavara större än eller lika med ett, måste bara uppfyllas om indexet k existerar.7.1.2 Optimering av potenstransformationernaHär presenteras en metod för att optimera potenstransformationerna somkrävs för att konvexi�era o h underestimera alla signomialtermer i ett opti-meringsproblem av typen (7.1). Denna metod baseras på att följande linjäraheltalsproblem uppställs (där indexet j motsvarar de JT i ke-konvexa signo-mialtermerna o h indexet i variablerna zi i hela det ursprungliga problemet):min δ1

I∑

i=1

Bi + δ2

JT∑

j=1

I∑

i=1

bji + δ3

JT∑

j=1

I∑

i=1

∆ji, (7.4)s.t. ∀i :

JT∑

j=1

bji ≤ JT Bi, (7.5)∀i, j : pji 6= 0 :

Qji ≥ M(βji − 1),

Qji ≤ Mβji,

Qji − ∆ji + Mβji ≤ M + 1,

Qji − ∆ji − Mβji ≤ −1,

−Qji − ∆ji + Mβji ≤ M − 1,

−Qji − ∆ji − Mβji ≤ 1,

(7.6)55

∀j : cj < 0 :I

∑

i=1

pjiQji ≤ 1, (7.7)∀i : pji > 0 :

Qji ≥ 1 − bji,

Qji ≤ 1 − ǫbji,

Qji ≥ ǫ,

(7.8)∀i : pji = 0 :

{

Qji = 0,

bji = 0,(7.9)

∀i : pji < 0 :

{

−M ≤ Qji ≤ −ε,

bji = 1,(7.10)

∀j : cj > 0 :I

∑

i=1

pjiQji − MI

∑

i=1

αji ≥ 1 − M,I

∑

i=1

αji ≤ 1, (7.11)∀i : pji > 0 :

Qji ≤ αji M − ε(1 − αji),

Qji ≥ −M + αji(M + 1),

bji ≥ 1 − αji,

bji ≥ ε(Qji − 1),

bji ≤ (1 − ε)Qji + M(1 − αji), (7.12)∀i : pji = 0 :

Qji = 0,

bji = 0,

αji = 0,

(7.13)∀i : pji < 0 :

Qji = 1,

bji = 0,

αji = 0,

(7.14)∀i, j : Qji ∈ R, bji, αji ∈ {0, 1}; M, δ1, δ2, δ3 > 0; ε = 1/M.Objektfunktionen (7.4) består av tre delar. Den första delen motsvarar totalaantalet transformationer o h är summan av de binära variablerna bji. Dessavariabler är lika med ett om den i-te variabeln i den j-te i ke-konvexa sig-nomialtermen transformeras av en potenstransformation o h är annars likamed noll. 56

Den andra delen av objektfunktionen motsvarar antalet variabler som överhu-vudtaget transformeras. Då man approximerar den inverse transformationenZji = z

1/Qji

i med sty kevis linjära funktioner kan vissa variabler återanvändasi alla approximationer av samma variabel zi i olika termer, även om trans-formationerna (alltså värdet på Qji) kan vara olika. Därför är det möjligt attgenom att minimera antalet variabler som överhuvudtaget transformeras inågon term förenkla det transformerade optimeringsproblemet. Den binäravariabeln Bi som är lika med ett om variabeln zi transformeras i någon termo h lika med noll annars införs. Villkoret på denna variabel garanteras avdet linjära bivillkoret (7.5), som tvingar variabeln att anta värdet ett om denurspungliga variabeln zi transformeras i någon term.Den tredje delen av objektfunktionen inkluderas för att favorisera numerisktstabilare transformationer. Eftersom det är de inversa transformationerna,alltså Z = z1/Q, som ska approximeras med sty kevis linjära funktioner, kannumeriska svårigheter uppstå om värdet på Q är positivt o h nära noll, ellerom Q är ett stort negativt tal. Värden på potensen Q nära +1 eller −1är alltså att föredra, o h tredje delen av objektfunktionen består därför avsumman av de absoluta avvikelserna ∆ från +1 om Q är positiv o h −1 omQ är negativ. Avvikelserna uttry ks med hjälp av bivillkoren (7.6).De tre delarna i objektfunktionen multipli eras med vikterna δ1, δ2 o h δ3för att möjliggöra olika optimeringsstrategier. Genom att ge vikterna olikavärden kan man favorisera någon eller några av de tre egenskaperna: så fåtransformationer som möjligt, så få transformerade variabler som möjligteller numeriskt stabilare transformationer.För att uppfylla de krav på korrekta transformationer som �nns i uttry ken(7.2) o h (7.3) samt ge korrekta värden på binärerna b o h potenserna Q,inkluderades även bivillkoren (7.7) - (7.10) för de negativa signomialtermerna,o h (7.11) - (7.14) för de positiva signomialtermerna.

57

7.1.3 Förbättring av den numeriska stabilitetenEftersom signomialtermer, både transformerade o h ursprungligt konvexa,kan innehålla variabler med både stora o h små potenser, kan termerna varanumeriskt ostabila vid lösning av det transformerade problemet. Genom enenkel omskrivning, är det do k möjligt att drastiskt förbättra den numeriskastabiliteten. Den j-te transformerade signomialtermen, där Z motsvarar densty kevis linjära approximationen av den inversa transformationen Z, kanskrivas i formensj(z) = cj

∏

i∈Ij

ZjipjiQji ·

∏

i/∈Ij

zpji

i ,där indexen i för vilka variabeln zi transformerats inkluderas i indexmängdenIj . Eftersom variablerna z är positiva kan termen omskrivas i den numerisktstabilare formen

sj(z) = exp

log(cj) +∑

i∈Ij

(pjiQji) · log(Zji) +∑

i/∈Ij

pji · log(zi)

.7.1.4 SlutsatserGenom att lösa det linjära heltalsproblemet för optimering av potenstrans-formationerna som beskrevs ovan, fås optimala mängder av transformationerberoende på värdena av parametrarna δ1, δ2 o h δ3. Simuleringar som ut-fördes på slumvis genererade grupper av signomialtermer, med strategin attantalet transformerade ursprungliga variabler skulle minimeras, resulterade idata som visade att det med hjälp av metoden är möjligt att drastiskt minskapå antalet variabler som måste transformeras. Ta k vare färre transformatio-ner blir komplexitetsnivån för det transformerade problemet lägre o h dettakan lösas snabbare.Den främsta vinsten är do k den att metoden, förutom att indikera vilkavariabler som bör transformeras för att signomialtermerna ska konvexi�eras,även ger vilka transformationer som kan användas. Således kan den användassom en integrerad del i en lösare för optimeringsproblem av typen (7.1).58

Referen es[BF76℄ E.M.L. Beale and J.J.H. Forrest, Global optimization using spe ialordered sets, Mathemati al Programming 10 (1976), 52�69.[Bjö02℄ K.-M. Björk, A global optimization method with some heat ex- hanger network appli ations, Ph.D. thesis, Åbo Akademi Univer-sity, 2002.[BSM93℄ M. S. Bazaraa, H. D. Sherali, and Shetty C. M., Nonlinear pro-gramming: Theory and algorithms, Wiley-Inters ien e, 1993.[BV04℄ S. Boyd and L. Vandenberghe, Convex optimization, CambridgeUniversity Press, 2004.[Eme04℄ S. Emet, A omparative study of solving some non onvex MINLPproblems, Ph.D. thesis, Åbo Akademi University, 2004.[Flo00℄ C. A. Floudas, Deterministi global optimization. Theory, methodsand appli ations, Kluwer A ademi Publishers, 2000.[FP01℄ C. A. Floudas and P. M. Pardalos, En y lopedia of optimization,Kluwer A ademi Publishers, 2001.[MF95℄ C.D. Maranas and C.A. Floudas, Finding all solutions of nonlin-early onstrained systems of equations, Journal of Global Optimiza-tion 7 (1995), 143�182.[Pör00℄ R. Pörn, Mixed integer non-linear programming: Convexi� ationte hniques and algorithm development, Ph.D. thesis, Åbo AkademiUniversity, 2000.[RM78℄ M.J. Rij kaert and X.M. Martens, Comparison of generalized ge-ometri programming algorithms, Journal of Optimization Theoryand Appli ations 26 (1978), no. 2, 205�242.59

[Wes05℄ T. Westerlund, Some transformation te hniques in global opti-mization, Global Optimization: From Theory to Implementation(L. Liberti and N. Ma ulan, eds.), Springer, 2005, pp. 47�74.[WP02℄ T. Westerlund and R. Pörn, Solving pseudo- onvex mixed-integerproblems by utting plane te hniques, Optimization and Engineer-ing 3 (2002), 253�280.[WW03℄ T. Westerlund and J. Westerlund, GGPECP - An algorithm forsolving non- onvex MINLP problems by utting plane and trans-formation te hniques, Chemi al Engineering Transa tions 3 (2003),1045�1050.

60

Appendi esA The Compiled Optimization MethodBelow is the general MILP problem formulation for optimizing the powertransformations approa h from Chapter 4 ompiled.min δ1

I∑

i=1

Bi + δ2

JT∑

j=1

I∑

i=1

bji + δ3

JT∑

j=1

I∑

i=1

∆ji,s.t. ∀i :

JT∑

j=1

bji ≤ JT Bi,

∀i, j : pji 6= 0 :

Qji ≥ M(βji − 1),

Qji ≤ Mβji,

Qji − ∆ji + Mβji ≤ M + 1,

Qji − ∆ji − Mβji ≤ −1,

−Qji − ∆ji + Mβji ≤ M − 1,

−Qji − ∆ji − Mβji ≤ 1,

∀j : cj < 0 :

I∑

i=1

pjiQji ≤ 1,

∀i : pji > 0 :

Qji ≥ 1 − bji,

Qji ≤ 1 − ǫbji,

Qji ≥ ǫ,

∀i : pji = 0 :

{

Qji = 0,

bji = 0,

∀i : pji < 0 :

{

−M ≤ Qji ≤ −ε,

bji = 1,

61

∀j : cj > 0 :I

∑

i=1

pjiQji − MI

∑

i=1

αji ≥ 1 − M,

I∑

i=1

αji ≤ 1,

∀i : pji > 0 :

Qji ≤ αji M − ε(1 − αji),

Qji ≥ −M + αji(M + 1),

bji ≥ 1 − αji,

bji ≥ ε(Qji − 1),

bji ≤ (1 − ε)Qji + M(1 − αji),

∀i : pji = 0 :

Qji = 0,

bji = 0,

αji = 0,

∀i : pji < 0 :

Qji = 1,

bji = 0,

αji = 0,

∀i, j : Qji ∈ R, bji, αji ∈ {0, 1}; M, δ1, δ2, δ3 > 0; ε = 1/M.

62

B The MILP formulation in Se tion 5.2.1Below is the MILP problem to obtain the power transformations in the ex-ample in Se tion 5.2.1. The parameter values M = 10, δ1 = 0.01, δ2 = 1 andδ3 = 0.001 have been used.MinimizeBB(0,0) + BB(1,1)+ 0.01 b(0,0) + 0.01 b(0,1) + 0.01 b(1,0) + 0.01 b(1,1)+ 0.001 delta(0,0) + 0.001 delta(0,1) + 0.001 delta(1,0) + 0.001 delta(1,1)Subje t toEq_4.2(0): b(0,0) + b(1,0) - 2 BB(0,0) <= 0Eq_4.11(0): 0.5 Q(0,0) + 2 Q(0,1) <= 1Eq_4.5.1(0,0): -Q(0,0) + 10 beta(0,0) <= 10Eq_4.5.2(0,0): Q(0,0) - 10 beta(0,0) <= 0Eq_4.6.1(0,0): Q(0,0) - delta(0,0) + 10 beta(0,0) <= 11Eq_4.6.2(0,0): Q(0,0) - delta(0,0) - 10 beta(0,0) <= -1Eq_4.6.3(0,0): -Q(0,0) - delta(0,0) + 10 beta(0,0) <= 9Eq_4.6.4(0,0): -Q(0,0) - delta(0,0) - 10 beta(0,0) <= 1Eq_4.9.1(0,0): Q(0,0) + b(0,0) >= 1Eq_4.9.2(0,0): Q(0,0) + 0.1 b(0,0) <= 1Eq_4.5.1(0,1): -Q(0,1) + 10 beta(0,1) <= 10Eq_4.5.2(0,1): Q(0,1) - 10 beta(0,1) <= 0Eq_4.6.1(0,1): Q(0,1) - delta(0,1) + 10 beta(0,1) <= 11Eq_4.6.2(0,1): Q(0,1) - delta(0,1) - 10 beta(0,1) <= -1Eq_4.6.3(0,1): -Q(0,1) - delta(0,1) + 10 beta(0,1) <= 9Eq_4.6.4(0,1): -Q(0,1) - delta(0,1) - 10 beta(0,1) <= 1Eq_4.9.1(0,1): Q(0,1) + b(0,1) >= 1Eq_4.9.2(0,1): Q(0,1) + 0.1 b(0,1) <= 1Eq_4.2(1): b(0,1) + b(1,1) - 2 BB(1,1) <= 0Eq_4.12(1): alpha(1,0) + alpha(1,1) <= 1Eq_4.5.1(1,0): -Q(1,0) + 10 beta(1,0) <= 10Eq_4.5.2(1,0): Q(1,0) - 10 beta(1,0) <= 0Eq_4.6.1(1,0): Q(1,0) - delta(1,0) + 10 beta(1,0) <= 11Eq_4.6.2(1,0): Q(1,0) - delta(1,0) - 10 beta(1,0) <= -163

Eq_4.6.3(1,0): -Q(1,0) - delta(1,0) + 10 beta(1,0) <= 9Eq_4.6.4(1,0): -Q(1,0) - delta(1,0) - 10 beta(1,0) <= 1Eq_4.13.1(1,0): Q(1,0) - 10.1 alpha(1,0) <= - 0.1Eq_4.13.2(1,0): Q(1,0) - 11 alpha(1,0) >= -10Eq_4.14.1(1,0): b(1,0) + alpha(1,0) >= 1Eq_4.14.2(1,0): b(1,0) - 0.1 Q(1,0) >= -0.1Eq_4.14.3(1,0): b(1,0) + 10 alpha(1,0) - 0.9 Q(1,0) <= 10Eq_4.5.1(1,1): -Q(1,1) + 10 beta(1,1) <= 10Eq_4.5.2(1,1): Q(1,1) - 10 beta(1,1) <= 0Eq_4.6.1(1,1): Q(1,1) - delta(1,1) + 10 beta(1,1) <= 11Eq_4.6.2(1,1): Q(1,1) - delta(1,1) - 10 beta(1,1) <= -1Eq_4.6.3(1,1): -Q(1,1) - delta(1,1) + 10 beta(1,1) <= 9Eq_4.6.4(1,1): -Q(1,1) - delta(1,1) - 10 beta(1,1) <= 1Eq_4.13.1(1,1): Q(1,1) - 10.1 alpha(1,1) <= - 0.1Eq_4.13.2(1,1): Q(1,1) - 11 alpha(1,1) >= -10Eq_4.14.1(1,1): b(1,1) + alpha(1,1) >= 1Eq_4.14.2(1,1): b(1,1) - 0.1 Q(1,1) >= -0.1Eq_4.14.3(1,1): b(1,1) + 10 alpha(1,1) - 0.9 Q(1,1) <= 10Eq_4.16(1): 1.5 Q(1,0) + 1.5 Q(1,1)- 10 alpha(1,0) - 10 alpha(1,1) >= -9Bounds0.1 <= Q(0,0) <= 100.1 <= Q(0,1) <= 10-10 <= Q(1,0) <= 10-10 <= Q(1,1) <= 10BinariesBB(0,0) BB(1,1) b(0,0) b(0,1) b(1,0) b(1,1)beta(1,0) beta(1,1) beta(0,1) beta(0,0) alpha(1,0) alpha(1,1)EndSolving the MILP problem gives the minimum at 1.0214 with the followingnonzero values of the variables:BB(1,1)=1; b(0,1)=1; b(1,1)=1; beta(0,0)=1; beta(0,1)=1; beta(1,0)=1;alpha(1,0)=1; delta(0,1)=0.75; delta(1,1)=0.66667;Q(0,0)=1; Q(0,1)=0.25; Q(1,0)=1; Q(1,1)=-0.33333.64