MULTI OBJECTIVE OPTIMIZATIONANDTRADE

OFFS USING PARETO OPTIMALITY

Amogh Mundhekar

Nikhil Aphale

[University at Buffalo, SUNY]

Multi-Objective Optimization

Multi-Objective Optimization

•Involves the simultaneous optimization of several

incommensurable and often competing objectives.

•These optimal solutions are termed as Pareto optimal

solutions.

•Pareto optimal sets are the solutions that cannot be

improved in one objective function without

deteriorating their performance in at least one of the

rest.

•Problems usually Conflicting in nature (Ex: Minimize

cost, Maximize Productivity)

•Designers are required to resolve Trade-offs.

Multi-Objective Optimization

Pareto Optimal means:

“Take from Peter to pay Paul”

Minimize {f1(x),…….,fn(x)}T

where fi(x) = ith objective function to be minimized,

n = number of objectives

Subject to:

g(x) ≤ 0;

h(x) = 0;

x min ≤ (x) ≤ (x max)

Typical Multi-Objective

Optimization Formulation

Basic Terminology

Search space or design space is the set of all possible

combinations of the design variables.

Pareto Optimal Solution achieves a trade off. They are

solutions for which any improvement in one objective results in

worsening of atleast one other objective.

Pareto Optimal Set: Pareto Optimal Solution is not unique,

there exists a set of solutions known as the Pareto Optimal Set.

It represents a complete set of solutions for a Multi-Objective

Optimization (MOO).

Pareto Frontier: A plot of entire Pareto set in the Design

Objective Space (with design objectives plotted along each

axis) gives a Pareto Frontier.

Dominated & Non- Dominated points

A Dominated design point, is the one for which there

exists at least one feasible design point that is better

than it in all design objectives.

Non Dominated point is the one, where there does not

exist any feasible design point better than it. Pareto

Optimal points are non-dominated and hence are also

known as Non-dominated points.

Challenges in the Multi Objective

Optimization problem.

Challenge 1: Populate the Pareto Set

Challenge 2: Select the Best Solution

Challenge 3: Find the corresponding Design Variables

Solution methods for Challenge 1

Methods discusses in earlier lectures:

Random Sampling

Weighting Method

Distance Method

Constrained Trade-off method

Solution methods for Challenge 1

Methods discusses to be discussed today:

Random Sampling

Weighting Method

Distance Method

Constrained Trade-off method

Normal Boundary Intersection method

Goal Programming

Pareto Genetic Algorithm

Weighted Sum Approach

Weighted Sum Approach

Uses weight functions to reflect the importance of each

objective.

Involves relative preferences.

Inter-criteria preference- Preference among several

objectives. (e.g. cost > aesthetique)

Intra-criterion preference- Preference within an objective.

(e.g. 100< mass <200)

Drawbacks of Weighted sum method

Finding points on the Pareto front by varying the weighting

coefficients yields incorrect outputs.

Small changes in ‘w’ may cause dramatic changes in the

objective vectors. Whereas large changes in ‘w’ may result in

almost unnoticeable changes in the objective vectors. This

makes the relation between weights and performance very

complicated and non-intuitive.

Uneven sampling of the Pareto front.

Requires Scaling.

Drawbacks of Weighted sum method..

For an even spread of the weights, the optimal solutions in the

criterion space are usually not evenly distributed

Weighted sum method is essentially subjective, in that a

Decision Maker needs to provide the weights.

This approach cannot identify all non-dominated solutions.

Only solutions located on the convex part of the Pareto front

can be found. If the Pareto set is not convex, the Pareto points

on the concave parts of the trade-off surface will be missed.

Does not provide the means to effectively specify intra-

criterion preferences.

Normal Boundary

Intersections (NBI)

Normal Boundary Intersections (NBI)

NBI is a solution methodology developed by Das and Dennis

(1998) for generating Pareto surface in non-linear

multiobjective optimization problems.

This method is independent of the relative scales of the

objective functions and is successful in producing an evenly

distributed set of points in the Pareto surface given an evenly

distributed set of parameters, which is an advantage compared

to the most common multiobjective approaches—weighting

method and the ε-constraint method.

A method for finding several Pareto optimal points for a

general nonlinear multi criteria optimization problem, aimed at

capturing the tradeoff among the various conflicting objectives.

Normal Boundary Intersections (NBI)

NBI is a solution methodology developed by Das and Dennis

(1998) for generating Pareto surface in non-linear

multiobjective optimization problems.

This method is independent of the relative scales of the

objective functions and is successful in producing an evenly

distributed set of points in the Pareto surface given an evenly

distributed set of parameters, which is an advantage compared

to the most common multiobjective approaches—weighting

method and the ε-constraint method.

A method for finding several Pareto optimal points for a

general nonlinear multi criteria optimization problem, aimed at

capturing the tradeoff among the various conflicting objectives.

Payoff Matrix (n x n)

Pareto Frontier using NBI

Where,

fN – Nadir point

fU – Utopia point

Convex Hull of Individual Minima (CHIM)

The set of points in objective space that are convex

combinations of each row of payoff table, is referred to as the

Convex Hull of Individual Minima (CHIM).

Formulation of NBI Sub Problem

Where,

n : Normal Vector from CHIM towards the origin

D : Represents the set of points on the normal.

Beta : Weight

The vector constraint F(x) ensures that the point x is actually

mapped by F to a point on the normal, while the remaining

constraints ensure feasibility of x with respect to the original

problem (MOP).

NBI algorithm

Advantages of NBI

Finds a uniform spread of Pareto points.

NBI improves on other traditional methods like goal

programming in the sense that it never requires any prior

knowledge of 'feasible goals'.

It improves on multilevel optimization techniques from

the tradeoff standpoint, since multilevel techniques

usually can only improve only a few of the 'most

important' objectives, leaving no compromise for the rest.

Goal Programmingzi

Mi

dGP dGP

Target value

+

-WGP

WGP

- +

HistoryGoal programming was first used by Charnes

and Cooper in 1955.

The first engineering application of goal

programming, was by Ignizio in 1962:

Designing and placing of the antennas on the

second stage of the Saturn V.

Requirements:

1) Choosing either Max or Min the objective

2) Setting a target or a goal value for each objective

3) Designer specifies &

Therefore, indicate penalties for deviating from either sides

Basic principle:

Minimize the deviation of each

design objective from its target value

Deviational variables &

-WGP +

WGP

How this method works?

dGP dGP- +

zi

Mi

dGP dGP

Target value

+

-WGP

WGP

- +

The Game

Advantages and disadvantages

1) Simplicity and ease of use

2) It is better than weighted sum method because the designer specify two different values of weights for each objective on the two sides of the target value

1) Specifying weights for the designer preference is not easy

2) What about…?

Testing for Pareto

Why the solution was not a Pareto optimal?

Because the designer set a pessimistic target value

Larbani & Aounini method

Goal programming method: (Program 1)

The output of Program 1 is X1

Pareto Method: (Program 2)

The output of Program 2 is X2

If X1 is a solution of program 2, therefore it is Pareto optimal solution and vice versa

Multiobjective Optimization

using Pareto Genetic

Algorithm

Genetic Algorithms (GAs) :

Adaptive heuristic search algorithm based on the

evolutionary ideas of natural selection.

Darvin’s Theory:

The individuals who best adapt to the environment are

the ones who will most likely survive.

Important Concepts in GAs

1. Fitness:

Each nondominated point in a model should be equally

important and considered an optimal goal.

Nondominated rank procedure.

2. Reproduction

a. Crossover:

Produces new individuals in

combining the information

contained in two or more parents.

b. Mutation:

Altering individuals with low

probability of survival.

3. Niche

Maintaining variety

Genetic Drift

4. Pareto Set Filter

Reproduction cannot guarantee that best characteristics of

the parents are inherited by their next generation.

Some of them maybe Pareto optimal points

Filter pools nondominated points ranked 1 at each

generation and drops dominated points.

NNP: no. of

nondominated

points

PFS: Pareto set

Filter Size

Detailed Algorithm

Pn= Population Size

Constrained Multiobjective Optimization via GAs

Transform a constrained optimization problem into an

unconstrained one via penalty function method.

Minimize F(x)

subject to,

g(x) <= 0

h(x) = 0

Transform to,

Minimize Φ(x) = F(x) + rp P(x)

A penalty term is added to the fitness of an infeasible

point so that its fitness never attains that of a feasible

point.

Fuzzy Logic (FL) Penalty Function Method

Derives from the fact that classes and concepts for natural phenomenon tend to be fuzzy instead of crisp.

Fuzzy Set

A point is identified with its degree of membership in that set.

A fuzzy set A in X( a collection of objects) is defined as,

μA : mapping from X to unit interval [0,1] called as membership function

0: worst possible case

1: best possible case

When treating a points violated amount for

constraints, a fuzzy quantity- such as the points

relationship to feasible zone as very close, close , far,

very far- can provide the information required for GA

ranking.

Fuzzy penalty function

For any point k,

KD value depends on membership function.

Entire search space is divided into zones.

Penalty value increases

from zone to zone.

Same penalty for points

in the same zone.

Advantages of GAs

Doesn’t require gradient information

Only input information required from the given problem is

fitness of each point in present model population.

Produce multiple optima rather than single local optima.

Disadvantages

Not good when Function evaluation is expensive.

Large computations required.