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Integer and Combinatorial Optimization: Introduction John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA November 2018 Mitchell Introduction 1 / 18

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Integer and Combinatorial Optimization:Introduction

John E. Mitchell

Department of Mathematical SciencesRPI, Troy, NY 12180 USA

November 2018

Mitchell Introduction 1 / 18

Integer and Combinatorial Optimization

Outline

1 Integer and Combinatorial OptimizationComplexitySolution techniquesQuadratic constraints and objective functions

2 Cutting planes

3 Branch-and-bound: An example

Mitchell Introduction 2 / 18

Integer and Combinatorial Optimization

Combinatorial optimization

An optimization problem is a problem of the form

minx f (x)subject to x 2 S

where f (x) is the objective function and S is the feasible region.

A combinatorial optimization problem is one where there is only afinite number of points in S.

For example: colorings of the countries of a map, so that no

two adjacent coutries have the same color.

Mitchell Introduction 3 / 18

Integer and Combinatorial Optimization

Integer optimization

In an integer optimization problem, the variables are constrained totake integer values.

In a mixed integer optimization problem, some of the variables arerequired to take integer values, and the remainder can take fractionalvalues.

Mitchell Introduction 4 / 18

Integer and Combinatorial Optimization

Integer optimizationFor example, a mixed integer linear optimization problem has theform

minx2Rn,y2Rp cT x + dT y

subject to Ax + By g

x , y � 0, y integer

Here, A and B are given matrices, A 2 Rm⇥n, B 2 Rm⇥p.

The remaining parameters c, d , g are vectors, c 2 Rn, d 2 Rp, g 2 Rm.

All vectors are understood to be column vectors in this course.

Transpose is denoted by the superscript T .

The dot product between two vectors c and x can be denoted cT x .

In a mixed integer nonlinear optimization problem, the objectivefunction and/or the constraints may be nonlinear functions.

Mitchell Introduction 5 / 18

Integer and Combinatorial Optimization Complexity

Outline

1 Integer and Combinatorial OptimizationComplexitySolution techniquesQuadratic constraints and objective functions

2 Cutting planes

3 Branch-and-bound: An example

Mitchell Introduction 6 / 18

Integer and Combinatorial Optimization Complexity

Computational complexity

Linear optimization problems can be solved in polynomial time. Thisimplies the required runtime doesn’t grow too quickly as the problemsize grows.

By contrast, integer optimization problems are often NP-hard (to bedefined later), which means that the runtime can grow very quickly inthe problem size in the worst case.

So mixed integer linear optimization problems are a lot harder to solveto global optimality than linear optimization problems.

Mitchell Introduction 7 / 18

Integer and Combinatorial Optimization Solution techniques

Outline

1 Integer and Combinatorial OptimizationComplexitySolution techniquesQuadratic constraints and objective functions

2 Cutting planes

3 Branch-and-bound: An example

Mitchell Introduction 8 / 18

Integer and Combinatorial Optimization Solution techniques

Solution techniques

Our emphasis will be on methods for finding global optimal solutionsto integer optimization problems. If we have a global minimizer x⇤ thenthere is no other feasible point x 2 S with f (x) < f (x⇤).

In order to prove optimality, we will look at relaxations.

The simplest relaxation of a mixed integer linear optimization problemis to ignore integrality, giving a linear optimization problem. Theprincipal solution techniques we then examine are

cutting planes: try to improve the relaxationbranching: subdivide the feasible region.

Mitchell Introduction 9 / 18

Integer and Combinatorial Optimization Solution techniques

Solution techniques

Our emphasis will be on methods for finding global optimal solutionsto integer optimization problems. If we have a global minimizer x⇤ thenthere is no other feasible point x 2 S with f (x) < f (x⇤).

In order to prove optimality, we will look at relaxations.

The simplest relaxation of a mixed integer linear optimization problemis to ignore integrality, giving a linear optimization problem. Theprincipal solution techniques we then examine are

cutting planes: try to improve the relaxationbranching: subdivide the feasible region.

Mitchell Introduction 9 / 18

Integer and Combinatorial Optimization Quadratic constraints and objective functions

Outline

1 Integer and Combinatorial OptimizationComplexitySolution techniquesQuadratic constraints and objective functions

2 Cutting planes

3 Branch-and-bound: An example

Mitchell Introduction 10 / 18

Integer and Combinatorial Optimization Quadratic constraints and objective functions

Some combinatorial optimization problems can be formulated withquadratic constraints and/or a quadratic objective function. For otherproblems, it can be helpful to look at a quadratic relaxation.

These problems can sometimes be attacked as semidefiniteoptimization problems, which have a matrix of variables that must besymmetric and positive semidefinite.

A useful observation: we can replace the requirement that x is a binaryvariable by the requirement that x = x2.

Mitchell Introduction 11 / 18

f (x) = I x t I x t Q x

: : : : : : : : : : - 1¥

Cutting planes

Outline

1 Integer and Combinatorial OptimizationComplexitySolution techniquesQuadratic constraints and objective functions

2 Cutting planes

3 Branch-and-bound: An example

Mitchell Introduction 12 / 18

Cutting planes

A cutting plane example

Initial relaxation:

Solve linear optimization relaxation

min{cT x : Ax b, x � 0}

Solution to linear optimization relaxation

Mitchell Introduction 13 / 18

¥- c

Cutting planes

A cutting plane example

Initial relaxation:

Solve linear optimization relaxation

min{cT x : Ax b, x � 0}

Solution to linear optimization relaxation

Cutting plane aT

1 x = b1

Mitchell Introduction 13 / 18

Cutting planes

Add a cutting plane and reoptimize

Solve linear optimization relaxation

min{cT x : Ax b, x � 0, aT

1 x b1}

Solution to linear optimization relaxation

Cutting plane aT

1 x = b1

Mitchell Introduction 14 / 18

Cutting planes

Add a cutting plane and reoptimize

Solve linear optimization relaxation

min{cT x : Ax b, x � 0, aT

1 x b1}

Solution to linear optimization relaxation

Cutting plane aT

1 x = b1

Cutting plane aT

2 x = b2

Mitchell Introduction 14 / 18

Cutting planes

Add a cutting plane and reoptimize

Solve linear optimization relaxation

min{cT x : Ax b, x � 0, aT

1 x b1, aT

2 x b2}

Solution to linear optimization relaxation

Cutting plane aT

1 x = b1

Cutting plane aT

2 x = b2

Solution to relaxation is integral, so it solves the integer optimizationproblem.

Mitchell Introduction 15 / 18

Branch-and-bound: An example

Outline

1 Integer and Combinatorial OptimizationComplexitySolution techniquesQuadratic constraints and objective functions

2 Cutting planes

3 Branch-and-bound: An example

Mitchell Introduction 16 / 18

Branch-and-bound: An example

An exampleWe look at the integer optimization problem

minx2R3 5x1 + 5x2 � 13x3subject to x1 + x2 + x3 6

10x1 � 8x2 15 (IOP)�6x1 � x2 + 9x3 9

xi � 0, xi integer, i = 1, 2, 3

It’s easy to find a feasible solution for this problem: x = (0, 0, 0).

Hence we can initialize with zu = 0.

We let

F = {x 2 R3 : x1 + x2 + x3 6, 10x1 � 8x2 15,�6x1 � x2 + 9x3 9,

x1, x2, x3 � 0}.

Mitchell Introduction 17 / 18

Branch-and-bound: An example

The tree for the example

Root node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

Mitchell Introduction 18 / 18

Branch-and-bound: An example

The tree for the example

Root node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

x1 1x1 � 2

Mitchell Introduction 18 / 18

Branch-and-bound: An example

The tree for the example

Root node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x1 = (1, 0, 1 2

3 ),z

1 = �16 23

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

x1 1x1 � 2

Mitchell Introduction 18 / 18

Branch-and-bound: An example

The tree for the exampleRoot node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x1 = (1, 0, 1 2

3 ),z

1 = �16 23

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

infeasible,z

6 = +1

fathom byinfeasibility

x1 1x1 � 2

x3 � 3

Mitchell Introduction 18 / 18

Branch-and-bound: An example

The tree for the exampleRoot node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x1 = (1, 0, 1 2

3 ),z

1 = �16 23

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

x5 = (2, 5

8 , 2),z

5 = �12 78

infeasible,z

6 = +1

fathom byinfeasibility

x1 1x1 � 2

x3 2x3 � 3

Mitchell Introduction 18 / 18

Branch-and-bound: An example

The tree for the exampleRoot node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x1 = (1, 0, 1 2

3 ),z

1 = �16 23

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

x3 = (0, 0, 1),z

3 = �13

fathom byintegrality

x5 = (2, 5

8 , 2),z

5 = �12 78

fathom bybounds

infeasible,z

6 = +1

fathom byinfeasibility

x1 1x1 � 2

x3 1 x3 2x3 � 3

Mitchell Introduction 18 / 18

¥*.

Branch-and-bound: An example

The tree for the exampleRoot node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x1 = (1, 0, 1 2

3 ),z

1 = �16 23

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

x3 = (0, 0, 1),z

3 = �13

fathom byintegrality

x4 = (1, 3, 2),z

4 = �6

fathom bybounds

x5 = (2, 5

8 , 2),z

5 = �12 78

fathom bybounds

infeasible,z

6 = +1

fathom byinfeasibility

x1 1x1 � 2

x3 1x3 � 2

x3 2x3 � 3

Mitchell Introduction 18 / 18

Branch-and-bound: An example

The tree for the exampleRoot node:min{c

Tx :

x 2 F}

x0 = (1.5, 0, 2),z

0 = �18.5

x1 = (1, 0, 1 2

3 ),z

1 = �16 23

x2 = (2, 5

8 , 22972 ),

z2 = �18 1

9

x3 = (0, 0, 1),z

3 = �13

fathom byintegrality

x4 = (1, 3, 2),z

4 = �6

fathom bybounds

x5 = (2, 5

8 , 2),z

5 = �12 78

fathom bybounds

infeasible,z

6 = +1

fathom byinfeasibility

x1 1x1 � 2

x3 1x3 � 2

x3 2x3 � 3

The optimal solution to (IOP) is x⇤ = (0, 0, 1), with value z⇤ = �13.Mitchell Introduction 18 / 18