A derative-free algorithm for systems of nonlinear ... · Recently a new derivative-free algorithm...

A derative-free algorithm for systems of nonlinear inequalities

Giampalo LiuzziStefano Lucidi

Technical Report n. 42007

Dipartimento di Informatica e Sistemistica “Antonio Ruberti”Università degli Studi di Roma “La Sapienza”

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A derivative-free algorithm forsystems of nonlinear inequalities

G. Liuzzi∗, S. Lucidi∗

∗Università degli Studi di Roma “La Sapienza”Dipartimento di Informatica e Sistemistica “A. Ruberti”

Via Buonarroti 12 - 00185 Roma - Italy

e-mail (Liuzzi): [email protected] (Lucidi): [email protected]

Abstract

Recently a new derivative-free algorithm [6] has been pro-posed for the solution of linearly constrained finite minimaxproblems. Such an algorithm can also be used to solve sys-tems of nonlinear inequalities in the case when the deriva-tives are not available and provided that a suitable reformu-lation of the system into a minimax problem is carried out.In this paper we show an interesting property of the algo-rithm proposed in [6] when it is applied to the solution ofsystems of nonlinear inequalities. Namely, under some mildconditions regarding the regularity of the functions definingthe system, it is possible to prove that the algorithm locatea solution of the problem after a finite number of iterations.Furthermore, under a weaker regularity condition, it is pos-sible to show that an accumulation point of the sequencegenerated by the algorithm exists which is a solution of thesystem. The reported numerical results, though prelimi-nary, confirm the good properties of the method.

AMS subject classification. 65K05, 90C30, 90C56

Keywords: Derivative-free methods, nonlinear inequality sys-tems

2 G. Liuzzi, S. Lucidi

1 Introduction

We consider the following system of linear and nonlinear inequal-ities

{g(x) ≤ 0Ax ≤ b, (1)

where x ∈

Systems of nonlinear inequalities 3

in the paper. In Section 3 we recall the approach followed in [6]and prove some new results which are needed by the convergenceanalysis. Section 4 is devoted to the new convergence analysis ofthe method. In Section 5 we explain how to extend the proposedapproach to handle also linear and nonlinear equalities. Finally, inSection 6 some numerical results are reported which confirm thegood properties of the method.Before concluding this section, we introduce some notation andbackground material. We denote by D the polyhedral set definedby the linear inequality constraints, namely

D = {x ∈ d

},

where

B(x) ={

i ∈ {0, 1, . . . ,m} : gi(x) = maxj=0,1,...,m

{gj(x)}}

.

2 Definitions and assumptions

We begin this section by introducing the usual definition of sta-tionarity for Problem (2) which can be given as follows.


Definition 1 A point x̄ ∈ D is a stationary point of Problem (2)if

Df(x̄, d) ≥ 0, for all d ∈ T (x̄),where

T (x) = {d ∈ j d ≤ 0, for all j s.t. a>j x = bj} (5)

is the cone of feasible directions with respect to the linear inequalityconstraints.

By exploiting the particular structure of Problem (2), that is, theexpression of the max function f(x), it is possible to state a dif-ferent characterization of its stationary points (see [6]).

Proposition 1 A point x̄ ∈ D is a stationary point of Problem(2) if and only if λi, i ∈ B(x̄), exist such that

λi ≥ 0, i ∈ B(x̄),∑

i∈B(x̄)λi = 1, (6)

∑

i∈B(x̄)λi∇gi(x̄)

>

d ≥ 0, for all d ∈ T (x̄). (7)

Now we introduce some assumptions which are standard assump-tions in a constrained context.

Assumption 1 The functions gi :


Assumption 3 At any point x ∈ D, a vector z ∈ T (x) existssuch that

∇gi(x)>z < 0, for all i s.t. gi(x) ≥ 0.

We require Assumption 1 to hold true throughout the paper. As-sumptions 2 and 3 will be explicitly recalled when needed.Assumptions 2 and 3 both implies that the feasible region definedby system (1) is not empty, but the second one is stronger inthat it implies that F have a non-empty interior as showed by thefollowing proposition.

Proposition 2 Assume that D 6= φ.(i) If Assumption 2 holds, then the feasible set F is not empty.(ii) If Assumption 3 holds, then a point x̃ ∈

d ≥ 0, for all d ∈ T (x̃). (8)

We show that x̃ is such that g(x̃) ≤ 0 and, by contradiction, let ussuppose that at least an index i exists such that gi(x̃) > 0. Then,every i ∈ B(x̃) is such that gi(x̃) > 0.By Assumption 2 we know that for every index i ∈ {i : gi(x̃) > 0},a vector z ∈ T (x̃) exists such that ∇gi(x̃)>z < 0 which constitutesa contradiction with relation (8) thus proving point (i).

Now we prove point (ii). Since Assumption 3 implies Assumption2, by point (i) we can conclude that the feasible set F is not empty.Let x̄ ∈ F , that is a point such that g(x̄) ≤ 0. By Assumption 3


we know that a vector z ∈ T (x̄) exists such that, for all i ∈ {i :gi(x) ≥ 0}, ∇gi(x̄)>z < 0. Let us consider the point x(²) = x̄+²z.Given the fact that z ∈ T (x̄), we get that an ²̄ > 0 exists suchthat, for all ² ∈ (0, ²̄],

a>j x(²) = a>j x̄ + ²a

>j z ≤ bj , j = 1, . . . , p.

By using a second order Taylor’s expansion we can write

gi(x(²)) = gi(x̄) + ²∇gi(x̄)>z + η(², z), i = 1, . . . , m (9)

where η(², z) is such that lim²↓0 η(², z)/² = 0. For every indexi ∈ {1, . . . , m} only two cases are possible: either gi(x̄) < 0 orgi(x̄) = 0. Assume first that index i is such that gi(x̄) < 0. In thiscase, by considering (9), an ²i > 0 exists such that gi(x(²i)) < 0.Let now i be such that gi(x̄) = 0. By assumption and by consid-ering again (9), we know that ²i > 0 exists such that gi(x(²i)) =²i

(∇gi(x̄)>z + η(²i, z)/²i

)< 0.

Hence, the point x̃ = x(²̃) where ²̃ = min{²̄, ²1, . . . , ²m} is suchthat g(x̃) < 0 and Ax̃ ≤ b. 2In the case of convex functions, the following proposition showsthat Assumptions 2 and 3 are minimal requirements for the feasi-bility of system (1).

Proposition 3 Assume that functions g1, . . . , gm are convex.

(i) If the feasible set F is not empty, then Assumption 2 holds.(ii) If a point x̃ ∈ j x = b. We can write

a>j (x̃− x) = a>j x̃− b ≤ 0,

which, by taking into account that index j was arbitrary, means(x̃− x) ∈ T (x).


Now we prove point (i). By the convexity assumption we havethat, for all i = 1, . . . ,m,

0 ≥ gi(x̃) ≥ gi(x) +∇gi(x)>(x̃− x).

This implies that, if gi(x) > 0, we must have ∇gi(x)>(x̃− x) < 0.Therefore, letting z = x̃− x, we obtain

∇gi(x)>z < 0, for all i s.t. gi(x) > 0.

Now we prove point (ii). To this aim, let x̃ ∈ F be such thatg(x̃) < 0. By the convexity assumption we have that, for alli = 1, . . . , m,

0 > gi(x̃) ≥ gi(x) +∇gi(x)>(x̃− x).

This implies that, if gi(x) ≥ 0, we must have ∇gi(x)>(x̃− x) < 0.Therefore, letting z = x̃− x, we obtain

∇gi(x)>z < 0, for all i s.t. gi(x) ≥ 0

thus completing the proof. 2

We end this section by recalling the following lemma which is aslight modification of Theorem 2.2 of reference [4] and which willbe used in subsequent sections.

Lemma 1 Let x̂ be any given point in D. If Assumption 3 holds,then an open neighborhood B(x̂; ρ) of x̂ and a direction d ∈ T (x̂)exist such that, for all i ∈ Iπ(x̂), we have

∇gi(x)>d ≤ −1, ∀ x ∈ B(x̂; ρ). (10)

3 Smooth approximation and preliminaryresults

To the aim of tackling with the non differentiability of functionf(x), in [6] it is used the following twice continuously differentiablesmoothing function (described in [1, 9])


f(x; µ) = µ ln

(1 +

m∑

i=1

exp(

gi(x)µ

))= µ ln

m∑

i=0

exp(

gi(x)µ

)

whose gradient is given by

∇f(x;µ) =m∑

i=0

λi(x; µ)∇gi(x), (11)

where

λi(x; µ) =exp

(gi(x)

µ

)

1 +m∑

j=1

exp(

gj(x)µ

) ∈ (0, 1), i = 0, 1, . . . , m, (12)

andm∑

i=0

λi(x; µ) = 1.

By using Lemma 1, it is possible to show that the smoothing func-tion f(x;µ) has a strong connection with the system of inequalities(1). In particular, the following proposition shows a property ofthe directional derivative of the smoothing function onto infeasiblepoints.

Proposition 4 Let x̂ be any given point in D. If Assumption 3holds, then µ(x̂) > 0 and σ(x̂) > 0 exist such that a directiond ∈ T (x̂) exists for which, for all x ∈ B(x̂;σ(x̂))∩D and g(x) 6≤ 0,µ ∈ (0, µ(x̂)], it results

∇f(x;µ)>d ≤ − 12(m + 1)

.

Proof. By Assumption 3, we have that the hypotheses of Lemma1 are satisfied at x̂ for I = Iπ(x̂). Let B(x̂; ρ) and d be the neigh-borhood and the direction considered in Lemma 1. By continuity,we can find a neighborhood B(x̂; σ(x̂)) ⊆ B(x̂; ρ) such that fori 6∈ Iπ(x̂) and x ∈ B(x̂;σ(x̂)), we have gi(x) < 0; it follows thatIπ(x) ⊆ Iπ(x̂) and Iν(x̂) ⊆ Iν(x), for x ∈ B(x̂; σ(x̂)).


Let now x ∈ B(x̂; σ(x̂)) ∩ D be an infeasible point with respectto the nonlinear inequality constraints; then, there must existsat least an index i ∈ Iπ(x) such that gi(x) > 0 which impliesIπ(x) 6= φ. By recalling expression (11), we can write∇f(x; µ)>d =

∑

i∈Iν(x̂)λi(x; µ)∇gi(x)>d +

∑

i∈Iπ(x̂)λi(x;µ)∇gi(x)>d.

By Lemma 1 we have that ∇gi(x)>d ≤ −1, i ∈ Iπ(x̂), so that wecan write

∇f(x; µ)>d ≤∑

i∈Iν(x̂)λi(x; µ)∇gi(x)>d−

∑

i∈Iπ(x̂)λi(x; µ). (13)

Let ı̄ ∈ Iπ(x̂) be an index such that gı̄(x) = maxi∈Iπ(x̂){gi(x)}.Since

exp(gi(x)/µ)exp(gı̄(x)/µ)

≤ 1, for all i ∈ {1, . . . , m},then

∑

i∈Iπ(x̂)λi(x;µ) ≥ λı̄(x; µ) ≥ (14)

≥ 11 + 1 +

m∑

i=1,i6=ı̄

exp(gi(x)/µ)exp(gı̄(x)/µ)

≥ 11 + m

.

By considering (13) and (14), we get

∇f(x; µ)>d ≤∑

i∈Iν(x̂)λi(x; µ)∇gi(x)>d− 11 + m. (15)

Now, since Iν(x̂) ⊆ Iν(x), for x ∈ B(x̂; σ(x̂)), by expression (12),it follows that, for any given x ∈ B(x̂; σ(x̂)) ∩ D not feasible,

limµ→0+

λi(x;µ) = 0, i ∈ Iν(x̂).

Hence, by the boundedness of ∇gi(x)>d over B(x̂; σ(x̂)), a µ(x̂) >0 exists such that for all µ ∈ (0, µ(x̂)] we have

∑

i∈Iν(x̂)λi(x; µ)∇gi(x)>d < 12(m + 1) . (16)

The result follows from (16) and (15). 2


4 DF Algorithm and convergence proper-ties

In this section we briefly recall the method proposed in [6] andprove its convergence properties in relation to the system of non-linear inequalities (1). Following [6], we employ the smooth ap-proximating problem

minx∈D

f(x; µ), (17)

which, as µ → 0, approximate the non-smooth Problem (2). Theapproximating parameter µ will be adaptively reduced by the al-gorithm during the optimization process.At this point, we briefly recall the main idea which is behind thederivative-free method. Specifically, the lack of first-order infor-mation can be overcome by sampling the smoothing function alonga suitable set of search directions. By this we mean that, at eachiteration, the set of search directions must satisfy the followingassumption.

Assumption 4 Let {xk} be a sequence of points belonging to Dand {Dk} be a sequence of sets of search directions. Then, for allk, Dk = {dik : ‖dik‖ = 1, i = 1, . . . , rk}, and, for some constantν̄ > 0,

cone{Dk ∩ T (xk; ν)} = T (xk; ν) ∀ν ∈ [0, ν̄].

where T (x; ν) = {d ∈ j d ≤ 0, ∀j s.t. a>j x ≥ bj − ν}.Moreover,

⋃∞k=0 Dk is a finite set and rk is bounded.

An example on how to compute sets of directions satisfying theabove assumption can be found in [8].Here, for the sake of clarity, we report the algorithm proposed in[6] for the solution of linearly constrained finite minimax problems.For a detailed description and analysis of the method we refer theinterested reader to [6].


DF Algorithm

Data. x0 ∈ D, µ0 > 0, init step0 > 0, γ > 0, θ ∈ (0, 1),ν > 0, k := 0.

While ( g(xk) 6≤ 0 ) thenChoose a set of directions Dk = {d1k, . . . , drkk }satisfying Assumption 4;

i := 1; yik := xk; α̃ik := init stepk;

RepeatCompute the maximum steplength ᾱiksuch that A(yik + ᾱ

ikd

ik) ≤ b;

α̂ik := min{ᾱik, α̃ik};If (α̂ik > 0 and f(yik + α̂ikdik; µk) ≤ f(yik; µk)−

γ(α̂ik)2) then

compute αik by theExpansion Step(ᾱik, α̂

ik, y

ik, d

ik;α

ik);

α̃i+1k := αik;

elseαik := 0; α̃

i+1k := θα̃

ik;

End If

yi+1k := yik + α

ikd

ik; i := i + 1;

Until ( i > rk )

Choose xk+1 ∈ D such that f(xk+1; µk) ≤ f(yik; µk);init stepk+1 := α̃ik;

µk+1 := min{µk, max

i=1,...,rk{(α̃ik)1/2, (αik)1/2}

};

k := k + 1;

End While


Expansion Step (ᾱik, α̂ik, y

ik, d

ik; α

ik).

Data. γ > 0, δ ∈ (0, 1).α := α̂ik;

Repeat

αik := α; α := min{ᾱik, (α/δ)};Until (αik < ᾱ

ik and f(y

ik + αd

ik; µk) ≤ f(yik; µk)− γα2)

As showed in reference [6], a significant role in carrying out theconvergence analysis of Algorithm DF is played by the index setK defined as follows

K = {k : µk+1 < µk}. (18)

Proposition 5 Let {xk} be the sequence produced by AlgorithmDF, and let {xk}K be the subsequence corresponding to the subsetof indices K. Let x̄ be any accumulation point of {xk}K , that is,an index set K̄ ⊆ K exists such that limk→∞,k∈K̄ xk = x̄. Then,

(i) x̄ is a stationary point of Problem (2);

(ii) for every d ∈ T (x̄) limk→∞,k∈K̄

∇f(xk; µk)>d ≥ 0.

Proof. Point (i) directly follows from Corollary 1 of reference [6].As regards point (ii), its proof follows with minor modificationsfrom the proofs of Theorem 1 and Proposition 4 of reference [6].In fact, in Theorem 1 it is shown that the sequences produced bythe algorithm satisfies the hypothesis of Proposition 4. Then, theproof of Proposition 4 is done by showing that

limk→∞,k∈K̄

∇f(xk; µk)>d ≥ 0

holds for every d ∈ T (x̄). See, for instance, relations (21) and (29)of reference [6]. 2


Proposition 6 Under Assumption 2, Algorithm DF either

(i) stops after finitely many iterations on a feasible point x̄ or

(ii) it produces an infinite sequence of points {xk} such that everylimit point x̄ of the subsequence {xk}K , where K is definedby (18), is feasible for system (1).

Proof. If an index k̄ exists such that g(xk̄) ≤ 0 then AlgorithmDF returns the feasible point x̄ = xk̄. If this is not the case, thenan infinite sequence {xk} is produced and we can apply point (i) ofProposition 5 to conclude that every limit point of the subsequence{xk}K is stationary for Problem (2). Therefore, from Proposition1 we know that λi, i ∈ B(x̄), exist such that

λi ≥ 0, i ∈ B(x̄),∑

i∈B(x̄)λi = 1, (19)

∑


>

d ≥ 0, for all d ∈ T (x̄). (20)

Let us suppose by contradiction, that x̄ is not feasible, that is,at least an index i ∈ {1, . . . ,m} exists such that gi(x̄) > 0. ByAssumption 2, we know that z ∈ T (x̄) exists such that

∇gi(x̄)>z < 0, for all i s.t. gi(x̄) > 0.

Now, by considering that B(x̄) ⊆ {i : gi(x̄) > 0}, the above rela-tion along with the non-negativity of the multipliers λi yields

∑


>

z < 0, (21)

which contradicts (20). 2

Proposition 7 Under Assumption 3, Algorithm DF stops afterfinitely many iterations on a point x̄ which solves system (1).


Proof. Reasoning by contradiction, let us suppose that an infinitesequence {xk} is produced such that g(xk) 6≤ 0, for all integers k.Now let x̄ be an accumulation point of the subsequence {xk}K ,that is, an index set K̄ ⊆ K exists such that limk→∞,k∈K̄ xk = x̄.By virtue of Proposition 4 we have that a k̂ ∈ K̄ and a directiond̂ ∈ T (x̄) exist such that

∇f(xk;µk)>d̂ ≤ − 12(m + 1) , (22)

for all k ≥ k̂, k ∈ K̄. Relation (22) contradicts point (ii) ofProposition 5 thus completing the proof. 2

By using Proposition 3 and Propositions 6 and 7, it is possible tostate the following corollaries.

Corollary 1 Assume that functions g1, . . . , gm are convex andthat the feasible set F is not empty. Then, Algorithm DF ei-ther stops after finitely many iterations on a feasible point x̄ or itproduces an infinite sequence of points {xk} such that every limitpoint x̄ of the subsequence {xk}K , where K is defined by (18), isfeasible for system (1).

Corollary 2 Assume that functions g1, . . . , gm are convex andthat a point x̃ ∈


We redefine D to denote the feasible set w.r.t. linear equality andinequality constraints, that is,

D = {x ∈ z = 0, for all j = 1, . . . , m̃.

6 Numerical Experience

In this section we report some preliminary numerical results ob-tained by Algorithm DF. Far from being an extensive testing, thecomputational experience is principally intended to show that, be-sides the interesting theoretical properties, Algorithm DF is reli-able. Algorithm DF is implemented in double precision Fortran 90


and run on a computer with an Intel 3.2GHz Pentium IV processorwith 1GB RAM.As concerns the test problems they were all selected from the well-known CUTEr [3] (Constrained and unconstrained testing envi-ronment, revisited) test set. Among all the CUTEr test problemswe selected 83 problems according to the following criteria.

- The provided initial point is (obviously) infeasible but itmust satisfy the linear constraints (if any). This becauseAlgorithm DF handles directly the linear constraints by ex-plicitly forcing feasibility of all iterates with respect to them.To this aim, the initial point x0 is required to belong to theset D.

- The problem should have at least one nonlinear inequalityconstraint. Indeed, many problems of the CUTEr collec-tion have only nonlinear equality constraints so that, find-ing a feasible point amounts to solving a system on nonlinearequalities. This latter problems have long been studied inthe literature and some very efficient and reliable methodsexist for their solution.

- The number of variables should not exceed 200 for fixed di-mension problems and, more or less, 50 for variably dimen-sioned problems. The rationale for this particular choice hasbeen that of selecting a consistent number of problems but,at the same time, of guaranteeing that many of them havea dimension suitable for a derivative-free code.

As regards the actual implementation of the code, we slightly mod-ified the version used in the paper [6] by introducing the stoppingcondition on the feasibility of the current iterate g(xk) ≤ 0 andtightening the tolerance on the step length. In particular, we useas an alternate stopping condition init stepk ≤ 10−5 in the linearlyconstrained case, and maxi=1,...,n α̃ik ≤ 10−5 in the unconstrainedcase. We also impose a limit on the required number of functionevaluations which must not exceed 2 · 105.Let P be the set of selected test problems. Following the approachproposed in [2], let tp, p ∈ P , denote a performance measure of


Algorithm DF on the set P of problems. We say that Dt(τ) is aperformance profile with respect to the performance measure tp if,for every τ ≥ 0, Dt(τ) = |{p ∈ P : tp ≤ τ}|. For each problemp ∈ P , we define

np = number of variables,mp = number of nonlinear constraints (if nonlinear

equalities are present, then mp = m̂ + m̃),²p = feasibility violation on the starting point x0,²∗p = feasibility violation on the solution point x

∗,φ∗p = number of evaluations of f(x;µ) required to get

convergence.

Note that, even though np,mp and ²p do not properly denote aperformance measure just because they represent a property of theproblem itself, it makes sense to consider profiles Dn(τ), Dm(τ)and D²(τ). In particular, functions Dn(τ) and Dm(τ), plotted inFigure 1, tells us, for every choice of τ > 0, how many problemshave np ≤ τ variables and mp ≤ τ nonlinear constraints, respec-tively, thus giving us information about the composition of thetest set.

0 20 40 60 80 100 120 14010

20

30

40

50

60

70

80

90

τ

n

100

101

102

103

0

10

20

30

40

50

60

70

80

90

τ

Dn(τ) Dm(τ)

(a) (b)Figure 1: Test set composition

Similarly, for every value of τ > 0, D²(τ) denotes the number ofproblems that have an initial feasibility violation at most equalto τ . Figure 2 reports the shapes of profiles D²(τ), D²∗(τ) and


10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

0

10

20

30

40

50

60

70

80

90

τ

n

100

101

102

103

104

105

0

10

20

30

40

50

60

70

80

90

τ

Dǫ(τ)

Dǫ∗(τ) Dφ∗(τ)

(a) (b)

Figure 2: Performances of Algorithm DF

Dφ∗(τ), respectively. Again, for any given value of τ > 0, D²(τ)represents the number of problems Algorithm DF is able to solvewith a reported feasibility violation no greater than τ . Whereas,Dφ(τ) represents the number of problems that get solved in nomore than τ function evaluations.Figure 2 and in particular profile D²∗(τ) confirms the theoreticalproperty of Algorithm DF that locates a strictly feasible point formore that 70 problems. Moreover, almost all the 83 problems aresolved with a feasibility violation no greater that 10−2. As con-cerns profile Dφ∗(τ), it suggests us some interesting conclusions aswell. First of all, Algorithm DF is able to solve more than halfthe problems by using 100 function evaluations. Furthermore, theinefficiencies inherent to the derivative-free approach are also ap-parent from Figure 2. Indeed, on approximately 15% of the prob-lems Algorithm DF required more than 1000 function evaluationsto converge.

7 Conclusions

We have proved that the algorithm proposed in [6] for finite mini-max problems has some additional properties when applied to thesolution of systems of nonlinear inequalities when derivatives arenot available. Under suitable assumptions we have shown that Al-


gorithm DF is able to locate a solution of system (1) after a finitenumber of iterations. The reported numerical results show thatAlgorithm DF can be a viable tool for the solution of systems ofnonlinear inequalities.

References

[1] D. P. Bertsekas, Constrained Optimization and LagrangeMultipliers Methods, Academic Press, New York, 1982.

[2] E. D. Dolan and J. J. Moré, Benchmarking optimizationsoftware with performance profiles, Math. Programming, 91(2002), pp. 201–213.

[3] N. I. M. Gould, D. Orban and Ph. L. Toint, CUTEr andSifDec: a constrained and unconstrained testing environment,revisited, ACM T. Math. Software, 29 (2003), pp. 373-394.

[4] S.P. Han and O.L. Mangasarian, Exact penalty functionsin nonlinear programming, Math. Programming, 17 (1979), pp.251–269.

[5] T.G. Kolda, R.M. Lewis and V. Torczon, Optimizationby Direct Search: New Perspectives on Some Classical andModern Methods, SIAM Review, 45 (2003), pp. 385–482.

[6] G. Liuzzi, S. Lucidi and M. Sciandrone, A derivative-free algorithm for linearly constrained finite minimax problems,SIAM J. Optimization, 16 (2006), pp. 1054–1075.

[7] S. Lucidi, M. Sciandrone and P. Tseng, Objective-derivative-free methods for constrained optimization, Math.Programming Ser. A, 92 (2002), pp. 37–59.

[8] R.M. Lewis and V. Torczon, Pattern search methods forlinearly constrained minimization, SIAM J. Optimization, 10(2000), pp. 917–941.

[9] S. Xu, Smoothing method for minimax problems, Computa-tional Optimization and Appl., 20 (2001), pp. 267–279.

A derative-free algorithm for systems of nonlinear ... · Recently a new derivative-free algorithm...

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