4.3 brief discussion on ilp problems 4.3.1 lp problems optical...

Optical Networks

Poompat Saengudomlert

Session 9

Background of ILP

P. Saengudomlert (2017) Optical Networks Session 9 1 / 15

4.3 Brief Discussion on ILP Problems4.3.1 LP Problems

General form of a linear programming (LP) problem

minimize c1x1 + . . .+ cNxN

subject to a11x1 + . . .+ a1NxN ≥ b1...

aM1x1 + . . .+ aMNxN ≥ bM

x1, . . . , xN are decision variables.

c1, . . . , cN are objective/cost coefficients.

b1, . . . , bM and a11, . . . , a1N , a21, . . . , aMN are constraint coefficients.


More compact representation

minimize cTx

subject to Ax ≥ b

where

x =

x1...xN

, c =

c1...cN

,b =

b1...

bM

,A =

a11 · · · a1N...

. . ....

aM1 · · · aMN

NOTE: Justification of “general” form.

x1 + x2 ≤ 1 ⇒ − x1 − x2 ≥ −1x1 + x2 = 1 ⇒ x1 + x2 ≥ 1 and − x1 − x2 ≥ −1


Diet Problem

Minimize the diet cost subject to nutritional constraints

apple juice orange juice min intake

vitamin A (unit/glass) 1 2 2

vitamin B (unit/glass) 2 1 2

cost (unit/glass) 3 1

Let x1 and x2 be amounts (in glass) of apple juice and orange juice.

minimize 3x1 + x2

subject to x1 + 2x2 ≥ 22x1 + x2 ≥ 2x1, x2 ≥ 0


From feasible set and contour lines, corner point (0,2) is optimal.

feasible setoptimalsolution

direction ofcost increase

Cost gradient is perpendicular to contour line.

An optimal solution (if exists) is at some corner point(if exists) of feasible set.


Existence of Optimal Solution for Linear Optimization

Three possibilities for existence of optimal solution1 Unique optimal solution (c)2 No optimal solution with optimal cost equal to −∞ (c′)3 Infinitely many optimal solutions (c′′)

feasible set

Simplex algorithm: Start with some corner point and iteratively visitsanother corner point with a better cost until no further improvement.


4.3.2 ILP Problems

The general form of an integer linear programming (ILP) problem is

minimize cTx

subject to Ax ≥ bx ∈ ZN

Compared to LP, the additional constraint x ∈ ZN makes the problemmuch more difficult.

Exploiting LP is not straightforward, as will be seen next.

Solving the problem by ignoring x ∈ ZN is called relaxation. Thecorresponding linear optimization problem is called the relaxedproblem.


Relaxation

Let f ∗ and frelax be optimal costs for the original problem and for therelaxed problem.

1 2 3 4

1

2

1 2 3 4

1

2

If an optimal solution from relaxation is an integer point, it is alsooptimal for the original problem (case (a)).

Otherwise (case (b)), frelax is a lower bound on f∗.


Integer Rounding

Consider relaxation followed by integer rounding, i.e. using thenearest integer point. Let frounding be the optimal cost obtained.

1 2 3 4

1

2

Rounding to nearest integer point may be infeasible.

Rounding to nearest feasible integer point may not be optimal, butcan yield an upper bound.


Heuristics

Consider relaxation followed by integer rounding and by local search,i.e. checking neighbors of the current solution.

For example, neighbors of (x1, x2) can be taken as (x1 ± 1, x2) and(x1, x2 ± 1) that are feasible.While local search yields a local optimal solution, it does not guaranteea global optimal solution.The previous figure provides an example of a local minimum at(x1, x2) = (3, 0); the global minimum is at (1, 2).

Several ILP problems are NP-complete and have no exact solutions sofar. For practical engineers, approximated solutions are usuallyobtained for NP-complete problems using heuristics.

⇒ Local search is one example.


Branch-and-Bound

Branch-and-bound is commonly used to obtain exact optimalsolutions of ILP problems.

Let F be the feasible set. If the problem with F is too difficult,break F into subsets Fi ’s to form (hopefully easier) subproblems.An unsolved subproblem is called active.

Notations

A: set of active subproblemsFi : feasible set of subproblem ixirelax: optimal solution obtained from relaxation with Fif irelax: optimal cost obtained from relaxation with Fixbest: best integer solution found so far

fbest: best cost found so far from an integer solution


Iteration of Branch-and-Bound

Initialize A = {F} and fbest = ∞.1 Remove one active subproblem, say subproblem i , from A.2 If subproblem i is infeasible, i.e. Fi = ∅, discard the subproblem and

go to step 5. Otherwise, compute xirelax and firelax through relaxation

and go to step 3.

3 If f irelax ≥ fbest, discard the subproblem and go to step 5. Otherwise,go to step 4.

4 If xirelax is an integer point, set xbest = xirelax and go to step 5.

Otherwise, partition Fi into smaller subsets, add these subsets to A,and go to step 5.

5 If A ̸= ∅, go to step 1. Otherwise, terminate by returning xbest andfbest as an optimal solution and the optimal cost respectively.


One simple partitioning of Fi is to use a noninteger component ofxirelax. Given x

irelax,j is not an integer, two subproblems are created

with additional constraints

xj ≤ ⌊x irelax,j⌋ or xj ≥ ⌈x irelax,j⌉.

Example 4.1:

Consider

minimize x1 − 2x2subject to − 4x1 + 6x2 ≤ 9

x1 + x2 ≤ 4x1, x2 ≥ 0x1, x2 ∈ Z


Example 4.1 (continued):

1 2 3 4

1

2

3

Fi xirelax f irelax next stepF (1.5,2.5) −3.5 create subproblemsF with x2 ≥ 3 infeasible infeasible discardF with x2 ≤ 2 (0.75,2) −3.25 create subproblemsF with x2 ≤ 2, x1 ≥ 1 (1,2) −3 update UF with x2 ≤ 2, x1 ≤ 0 (0,1.5) −3 discard

In conclusion, f ∗ = −3 and (x∗1 , x∗2 ) = (1, 2).P. Saengudomlert (2017) Optical Networks Session 9 14 / 15

Additional Comments on Branch-and-Bound

The order of selecting Fi from A is unspecified. Possibilities arebreadth-first search or depth-first search with respect to a treecontaining these subproblems.

Subproblems of a problem become children in this tree.

The running time depends on whether we can find a good integersolution early. If we can find a small fbest early, lots of subproblemscan be ignored.

For a large problem, the computer may run out of memory to storethe set A of active subproblems, leading to no exact solution. In suchcases, xbest may be used as an approximated solution.


4.3 brief discussion on ilp problems 4.3.1 lp problems optical...

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