Optimization for Communications and Networks

Poompat Saengudomlert

Session 1

Introduction & Outline

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Course Information

Instructor: Poompat Saengudomlert (poompat.s@bu.ac.th)

Class website: http://bucroccs.bu.ac.th/courses/

Grading policy:

Assignments & in-class quizzes & projects 30%Mid-semester exam 30%Final exam 40%

Exam policy: Closed-book exams with sheets of notes allowed

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Textbook

P. Saengudomlert,Optimization for Communicationsand Networks. Science Publishers,Enfield, NH, USA, 2011

A list of corrections will be keptand made available.

More powerful and easy-to-useoptimization softwares will bediscussed in class.

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1 Introduction

1.1 Components of Optimization Problems

Objective function: real function f

Variables: N real unknowns x = (x1, . . . , xN)1

Unconstrained optimization problem:

minimize f (x)

subject to x ∈ RN

Solution: value of x

Optimal solution: a solution x∗ that minimizes f

Optimal cost: the minimum value f ∗ of objective function f

1Each complex unknown z can be thought of as zR + izI, wherezR and zI are real.

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Example Unconstrained Optimization Problem

minimize f (x) = e−|x | sin(x)

subject to x ∈ R

df (x∗)

x= 0

⇒ x∗ = ±π

4,±5π

4. -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-8 -6 -4 -2 0 2 4 6 8

f(x)

x

By inspection, x∗ = −π

4is the optimal solution.2

The associated optimal cost is f ∗ = −e−π/4

√2

.

2−π/4 is the global minimum while 5π/4 is a local minimum.P. Saengudomlert (2015) Optimization Session 1 5 / 15

Constrained Optimization

Constrained optimization problem:

minimize f (x)

subject to x ∈ F

Feasible set F (continuous or discrete)

Constraint: a condition expressed in terms of x, e.g., x ≥ 0.

Assume that F can be expressed using a set (or a subset) ofconstraints of the following form.

∀l ∈ {1, . . . , L}, gl(x) ≤ 0

∀m ∈ {1, . . . ,M}, hm(x) = 0

x ∈ ZN

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Example Constrained Optimization Problem

minimize f (x) = (x + 1)2

subject to x ≥ 0

unconstrained

minimum 0

1

feasible set

constrained

minimum

By inspection, x∗ = 0 . The optimal cost is f ∗ = 1 .

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Example Constrained Optimization Problem

minimize f (x) = (x1 + 1)2 + (x2 + 1)2

subject to x ≥ 0

unconstrained

minimum

1

1

feasible

set

constrained

minimum

contour lines

By inspection of contour lines, x∗ = 0 and f ∗ = 2 .

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Example Constrained Optimization Problem

Minimize the cost of a diet subject to nutritional requirements

apple juice orange juice minimum (unit)

vit. A (unit/glass) 1 2 2

vit. B (unit/glass) 2 1 2

cost (unit/glass) 3 1

Let x1 and x2 be the variables denoting the amounts (in glass) of appleand orange juice. The optimization problem is

minimize 3x1 + x2

subject to x1 + 2x2 ≥ 2

2x1 + x2 ≥ 2

x1, x2 ≥ 0

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The feasible set is illustrated below.

feasible set

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feasible setoptimalsolution

direction ofcost increase

By inspection of contour lines, x∗ = (0, 2) and f ∗ = 2 .

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1.2 Classes of Optimization Problems

Consider 3 different classes in this course.3

Convex optimization: convex feasible set F and convex objectivefunction f

Linear optimization: feasible set F defined by linear constraints andlinear objective function f

Integer linear optimization: similar to linear optimization but withinteger variables

NOTE:

Linear optimization is a special case of convex optimization.

The term “programming” is often used instead of “optimization”.

3Not considered is the most general class of nonlinearoptimization problems. However, algorithms for nonlinear problemsare often constructed from those for convex and linear problems.

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Course Outline

Convex optimization (50%)

Linear optimization (25%)

Integer linear optimization (25%)

Applications

Transmit power allocation in multi-carrier systemsRouting in wireline circuit/packet-switched networksRouting in wireless sensor networksRouting and wavelength assignment in optical networksNetwork topology design

Optimization softwares: Octave (close to MATLAB), Python/PuLP

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Example Optimization Project

WDMPlanner v1.0 for routing optimization in WDM networks, created byAIT students

Demo of program in class

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Another Example Optimization Project

Sudoku solver to be developed in class (if time permits)

Demo of program in class

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