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Chapter 1. Introduction Ex : Diet Problem

Daily requirements : energy(2000kcal), protein(55g), calcium(800mg)

Food

Servingsize

Energy(kcal)

Protein(g)

Calcium(mg)

Price per serving (cents)

Max servingallowed

Oatmeal 28g 110 4 2 3 4

Chicken 100g 205 32 12 24 3

Eggs 2 large 160 13 54 13 2

Wholemilk 237cc 160 8 285 9 8

Cherry pie 170g 420 4 22 20 2

Pork with beans

260g 260 14 80 19 2

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Formulation:

654321 1920913243Minimize xxxxxx

Subject to

20

20

80

20

30

40

6

5

4

3

2

1

x

x

x

x

x

x

800802228554122

55144813324

2000260420160160205110

654321

654321

654321

xxxxxx

xxxxxx

xxxxxx

)4,0( 11 xx

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Linear Programming Problem ( 선형계획법 문제 )

min

max

n

jjj xc

1

n

jijij mibxa

1,...,1,

jx j 0

Subject to

objective function ( 목적함수 )

Constraints ( 제약식 )

nonnegativity constraints ( 비음제약식 ) (may not exist for some variables, then they are called unrestricted or free variables)

right hand side ( 우변상수 )

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Unusual formulations

Cutting stock problem : (Refer Chapter 13)

Rolls of papers (called raws) with width to be cut into small pieces(finals).

pieces of width , need to be produced.

How to cut the rolls to meet demands while minimizing wastes?

minimize

s. t. ,

and integer for all

Consider possible cutting patterns for a raw.

: number of rolls to be cut using -th cutting pattern.

denotes the total number of possible patterns which can be a very large number.

if the number of finals produced in the j-th pattern is .

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ex) raws W=100 in.,

need 97 finals of width 45 in. 610 finals of width 36 in.

395 finals of width 31 in. 211 finals of width 14 in.

211

395

610

97

1

0

0

0

2

0

0

0

1

1

0

1

0

0

1

1

1

0

1

1

0

0

0

2

37364321 xxxxxx

Min x1 + x2 + x3 + … + x37

jx j integer and0

Note:• number of patterns grows fast as problem becomes large (We don’t solve the

problem with all columns in the model. We start with a few columns and solve the LP, and then identify and add a new column to the model and solve the LP again, … (column generation method).

• round down fractional optimal solution to LP to obtain integer solution, then use a few more raws to meet demands.

• extensions to 2-dimensional cutting stock (nesting problem), 3-D packing

Piecewise linear convex function:

Consider ,

(maximum of affine functions, called a piecewise linear convex function.)

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𝑓 (𝑥)

𝑐1 ’𝑥+𝑑1

𝑐 2’𝑥+𝑑2𝑐 3’ 𝑥+𝑑3

𝑥0

Linear Programming 2012

7

Thm: Let be convex functions. Then

is also convex.

pf)

=

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Minimization of piecewise linear convex function

𝑓 (𝑥)

𝑥   (𝑅𝑛)

𝑐1 ’𝑥+𝑑1

𝑐 2’𝑥+𝑑2𝑐 3’ 𝑥+𝑑3

, .

min

s. t. , s.t. ,

,

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ex) parallel processor scheduling problem

There are processors and jobs to be processed on any one of the processors.

: processing time of job on processor .

Assign jobs to processors so that overall finish time (makespan) is minimized. Formulation as an integer pro-gramming problem.

Let = 1 if job is assigned to processor , 0 otherwise.

minimize {}

s. t. ,

and integer for all

minimize

s. t. ,

,

and integer for all

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Linear programming relaxation of an integer programming problem is obtained by dropping the integrality requirement on the variables and considering only the linear constraints.

The optimal value of the linear programming relaxation provides a lower bound (for the minimization problem) on the optimal value of the integer programming problem. Hence it can be used impor-tantly in the algorithm for integer programming problem.

( Integer Program: min , , and integer.

Let , be the optimal value of the integer program (IP) and its LP re-laxation, respectively. Let be the optimal solution of IP, then is a feasible solution to LP relaxation. Hence .)

For the maximization problem, LP relaxation gives an upper bound on the optimal value of IP.

Note that maximize cannot be formulated as a linear programming problem. (maximizing a convex function)

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Special case of piecewise linear objective function : separable piecewise linear objective function.

function is called separable if .

If objective function is nonlinear, but separable we may approx-imate it by piecewise linear function. (need some caution)

The cost structure may be progressive, e.g.: electricity cost.

𝑥𝑖

𝑓 𝑖(𝑥𝑖)

𝑎 3𝑎 2𝑎10

𝑐 3

𝑐 2𝑐1

𝑐 4

slope:

𝑥1 𝑖 𝑥2 𝑖 𝑥3 𝑖 𝑥 4 𝑖

𝑐1<𝑐2<𝑐 3<𝑐 4

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Ex: Suppose = 1, = 2, = 3, = 4

= 5, = 10, = 15

If = 7, we want to have = + + + = 5 + 2 + 0 + 0

and objective function value is = 15 + 22 + 30 + 4 0 = 9

Replace in the constraints with , where

0 , 0 , 0 , 0

In the objective function, use : min

Since we solve minimization problem, it is guaranteed that if we get

in an optimal solution , have values at their upper bounds. Ex(continued): = 7 may be expressed 5 + 0 + 2 + 0 with = 5, = 0, =

2, = 0.

But then, objective function value is = 15 + 20 + 32 + 4 0 = 11, which is worse than the previous solution. (We have a minimization problem.)

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Terminology

Feasible solution ( 가능해 ) : vector that satisfies all constraints Optimal solution ( 최적해 ) : feasible solution that gives the largest

objective function value. Optimal value ( 최적값 ) : objective function value of an optimal so-

lution. Infeasible LP : LP that does not have a feasible solution Unbounded : LP is called unbounded if there exists a feasible solu-

tion that gives optimal value > for any finite > 0.

ex: {max , s.t. }

Every LP has one of the following three statuses: There exists an optimal solution with finite optimal value (There

may exist multiple optimal solutions) Infeasible Unbounded

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Brief History of LP

Solving systems of linear inequalities : Fourier, 1826, not efficient (Chapter 16)

Simplex method : G. B. Dantzig, 1947 Ellipsoid method : L. G. Khachian, 1979

First polynomial time algorithm (theoretically efficient algorithm) for LP, practically not good.

Interior point method : L. Karmarkar, 1984

polynomial time algorithm, many variations, practically good per-formance. Recently, used for some nonlinear programming prob-lems successfully (convex optimization)

Theory of LP provides important foundation for many other disci-plines like integer programming, networks and graphs, nonlinear programming, etc .

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Standard form

(In vector notation)maximize maximize s. t. , s.t.

,

Any LP problem can be expressed as the standard formMinimization problem : solve max , then take the negative of

the optimal objective value. Optimal solution is the sameEquality constraint : , Unrestricted variable substitute by ( in the objective func-

tion and in the constraints where .

(Simplex method finds an optimal solution with at most one of is positive)

Some people use min(max) as standard form.

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ex) max max

s. t. s. t.

Formulation when absolute values appear in the objective function

(Assume cj 0 for variables involving absolute values)

min min

s. t. , s.t. ,

unrestricted for all for all

Express free variable as () in the constraints ()Also in the objective function as () ().We want = , if

= , if We need in an optimal solution, i.e. at most one of is positive.If for all , this is guaranteed to hold at an optimal solution.

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Ex: Suppose = 3, = 5 = 0 – 5 in an optimal solution, hence objec-tive value is 3 (0 + 5) = 15. If we have = 2, = 7, then = 2 7 = – 5. But objective value is 3 (2 + 7) = 27, which is inferior compared to the previous solution. Hence at most one of is positive at an opti-mal solution.

Alternative formulation:

minimize s. t. ,

, ,