Optimization Background for Network Design

– “A Mathematical Programming Model is a mathematical decision model for planning (programming) decisions that optimize an objective function and satisfy limitations imposed by mathem[r]

Optimization Background for Optimization Background for

Network Design Network Design

David Tipper

Associate Professor

Associate Professor Department of Information Science

and

Telecommunications University of Pittsburgh

tipper@tele.pitt.edu

tipper@tele.pitt.edu

Slides 5

Slides 5

http://www.sis.pitt.edu/~dtipper/2110.html

http://www.sis.pitt.edu/~dtipper/2110.html

Network Design Tools

Network Design Tools

• Optimization formulation to try and minimize cost

– Metro and WANS Designed using computer aid tools

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• Variety of tools available

– WANDL, VPISystems, OPNET, RSOFT, etc – trend is to develop tools for internal use only make money on consulting

Network Design Tools

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Optimization Review

• Optimization Techniques

– Seek to find maximum or minimum of a objective function

– Set of unknown decision variables

– Constraintslimit the possible values for the variables

• Definition

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Types of Optimization Problems

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Simple Continuous Optimization

• If Unconstrained

– objective function F(X)is continuous function of xand x is continuous

– Find MAX/MIN of F(x)by differentiation (set derivative = 0) – Determine if MAX or MIN by second derivative

– If multi-dimensional – calculate gradient – use numerical gradient search methods

– Newton's method gives rise to a wide and important class of algorithms that require computation of the gradient vector

ˆ ˆ ˆ ( , , )x y z x y z

x y z

φ φ φ

φ ∂ ∂ ∂

∇ = + +

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Constrained Optimization

Maximize (or minimize): Subject to:

Constraints Objective

• General Symbolic Model

…

(x x xn)

f 1, 2K

( ) { }

( ) { }

1

2 2

, , , , , ,

n n

g x x x b

g x x x b

≤ ≥ = ≤ ≥ = K

K

( 1, ) { , , }

m n m

g x xKx ≤ ≥ = b

… where x1,x2Kxn are the decision variables

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Mathematical Programming

• Types of Mathematical Programs:

– Linear Programs (LP):the objective and constraint functions are linear and the decision variables are continuous

– Integer (Linear) Programs (IP):one or more of the decision variables are restricted to integer values only and the functions are linear

• Pure IP: all decision variables are integer

• Mixed IP (MIP): some decision variables are integer, others are continuous

Telcom 2120 Linear Programming Maximize: Subject to: Constraints Objective Bounds

… where aij,bj,cj are the model parameters

• General Symbolic Form

…

{ }

{ }

11 12 1 21 22 2

, , , ,

n n

n n

a x a x a x b

a x a x a x b

+ + + ≤ ≥ =

+ + + ≤ ≥ =

K K

{ }

1 2 , ,

0 , 1, ,

m m mn n m

j

a x a x a x b

x j n

+ + + ≤ ≥ = ≤ = K K n nx c x c x

c1 1+ 2+K

x c

Maximize : T

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Linear Programming

Maximize:

Subject to: Constraints Objective

Bounds

• Can be written in matrix formulation

x cT

b Ax=

j xj ∀

≤

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What are the remaining constraints ? : - for link 3-5 ?:

- for link 4-5 ?

1 5 4 f2 f3 f4 10 20 20 20 10 20 2

c5: f3 + f2 <= 20; /*link 35 capacity */ c6: f4 + f1 <= 5; /*link 45 capacity */

f1

(note this makes prior constraint f4 + f1<=10 redundant )

Network Flow LP Formulations (8)

“flow assignment to routes” or “arc-path” approach - example (2)

Source: W D Grover, ECE 681, UofA, Fall 2004

• Note that the “indicator” parameters not appear explicitly in the executable model

• Really they just represent our knowledge of the topology and the routes being considered

• Implicitly above, we only wrote the flow variables that had non-zero coefficients Examples: k i δ 12

1 (flow1 crosses span 12)

δ =

35 1 (flow3 crosses span 35)

δ =

Hence f1 is in the first constraint Hence f3 is in the fifth constraint, etc

Network Flow LP Formulations (9)

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Sonet/STM Design Problem

Complexity - Solving Design Problems

• Real world Network Design problems are quite large (have many variables and constraints)

– Graph Theory and Optimization Based algorithms for network design are complex – when can one use a technique?

• Complexity of an algorithm usually denotes O(.) which denotes the order of time growth in the algorithm as a function of problem variables

– Dijkstra’s Algorithm for SPT O(N2) where N is number of nodes in graph

– Prim’s Algorithm for MST O(E log(N)) where N is # nodes, E # edges

• Problems that can be solved by a deterministic algorithm in a polynomial time complexity denoted P that is O(Nk)

• Problems that can not be solved with P complexity denoted NP and don’t scale well

– Linear Programming Problems have P complexity – Integer Programming Problems have NP complexity

• Still Branch and Bound can be used for small problems !

• In general for NP problems use Sub-optimal algorithms (meta-heuristics)