Optimization Background for Network Design

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<b>Optimization Background for </b>

<b>Optimization Background for </b>

<b>Network Design</b>

<b>Network Design</b>

<b>David Tipper</b>

<b>Associate Professor</b>

<b>Associate Professor</b>

Department of Information Science

and

Telecommunications
University of Pittsburgh

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<b></b>

<b>Slides 5</b>

<b>Slides 5</b>

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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
( , , )<i>x y z</i> <i>x</i> <i>y</i> <i>z</i>

<i>x</i> <i>y</i> <i>z</i>

φ φ φ

φ ∂ ∂ ∂

∇ = + +

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Constrained Optimization

Maximize (or minimize):
Subject to:

Constraints
Objective

• General Symbolic Model

…

(

<i>x</i> <i>x</i> <i>xn</i>

)

<i>f</i> 1, 2K

( ) { }

( ) { }

1 1 2 1

2 1 2 2

, , ,
, , ,

<i>n</i>
<i>n</i>

<i>g</i> <i>x x</i> <i>x</i> <i>b</i>

<i>g</i> <i>x x</i> <i>x</i> <i>b</i>

≤ ≥ =
≤ ≥ =
K

K

( 1, 2 ) { , , }

<i>m</i> <i>n</i> <i>m</i>

<i>g</i> <i>x x</i>_K<i>x</i> ≤ ≥ = <i>b</i>

… where <i>x</i>1,<i>x</i>2K<i>xn</i> are the <b>decision variables</b>

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Mathematical Programming

• Types of Mathematical Programs:

<b>– Linear Programs (LP):</b>the objective and constraint functions
are linear and the decision variables are continuous.

<b>– Integer (Linear) Programs (IP):</b>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.

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Linear Programming

Maximize:
Subject to:
Constraints
Objective
Bounds

… where <i>a_ij</i>,<i>b_j</i>,<i>c_j</i> are the model parameters.

• General Symbolic Form

…

{ }

{ }

11 1 12 2 1 1
21 1 22 2 2 2

, ,
, ,

<i>n</i> <i>n</i>

<i>n</i> <i>n</i>

<i>a x</i> <i>a x</i> <i>a x</i> <i>b</i>

<i>a x</i> <i>a x</i> <i>a x</i> <i>b</i>

+ + + ≤ ≥ =

+ + + ≤ ≥ =

K
K

{ }

1 1 2 2 , ,

0 , 1, ,

<i>m</i> <i>m</i> <i>mn</i> <i>n</i> <i>m</i>

<i>j</i>

<i>a x</i> <i>a</i> <i>x</i> <i>a</i> <i>x</i> <i>b</i>

<i>x</i> <i>j</i> <i>n</i>

+ + + ≤ ≥ =
≤ =
K
K
<i>n</i>
<i>nx</i>
<i>c</i>
<i>x</i>
<i>c</i>
<i>x</i>

<i>c</i>1 1+ 2 2+_K

<i>x</i>
<i>c</i>

<i>Maximize</i> _: <i>T</i>

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Linear Programming

Maximize:

Subject to: Constraints
Objective

Bounds

• Can be written in matrix formulation

<i>x</i>

<i>c</i>

<i>T</i>

<i>b</i>

<i>Ax</i>

=

<i>j</i>
<i>x_j</i> ∀

≤
0

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

- for link 4-5 ... ?

1
5
2
4
3
1
5
2 ₄

3
f2
f3
f4
10
20
20
5
8
20 10
20
5
8
2 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 do 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:
<i>k</i>
<i>i</i>
δ
12

1 (flow1 crosses span 12)
1

δ =

35 _{1 (flow3 crosses span 35)}
3

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)

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