# Project - University of Regina

Τεχνίτη Νοημοσύνη και Ρομποτική

23 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

94 εμφανίσεις

1

Multi
-
Objective Non
-
linear
Optimization via
Parameterization and Inverse
Function Approximation

University of Regina

Industrial Systems Engineering

M.A.Sc. Thesis Defense

May 23, 2003

Mariano Arriaga Mar
í
n

2

Thesis Contributions

Novel technique for attaining the global
solution of nonlinear optimization
problems.

Novel technique for multi
-
objective
nonlinear optimization (MONLO).

Artificial Neural Networks (ANN)
Implementation

Methods tested in:

Highly nonlinear optimization problems,

MONLO problems, and

Practical scheduling problem.

3

Current Global Optimization
Techniques

Common Techniques

Multistart

Clustering Method

Genetic Algorithms

Simulated Annealing

Tabu Search

4

Multiple Objective Optimization

Current
MONLO

procedure:

Divide the problem in two parts

1.
Multi
-
Objective

Single Objective
Problem

2.
Solve with a Nonlinear Optimization
Technique

5

Multi
-
Objective

Single Objective

Common Techniques

Weighting Method

E
-
Constraint

Interactive Surrogate Worth Trade
-
Off Solution

Lexicographic Ordering

Goal Programming

Problems

Include extra parameters which might be
difficult to determine their value.

Determining their value gets more difficult as
the number of objective functions increases

6

Proposed Optimization Algorithm

Min

F
(
x
) = {
f
1
(
x
)
,…,f
m
(
x
)}

where

x

n

f
i
(x)

;

i

=

1
,

,
m

Optimization of:

Non
-
Linear functions

Multi
-
objective

Avoid local minima and inflection
points

7

General Idea

Set an initial value for
x

and calculate
f(x)

Decrease the value of the function via a
parameter

Calculate corresponding
f
-
1
(x)

Note: The algorithm does not necessarily follow the
function

x
0

8

General Idea

When the algorithm reaches a local minima

it looks for a lower value

if this value exists, the algorithm “jumps” to it and
continues the process

This process continues until the algorithm
reaches the global minimum.

x
0

x
f

9

Inverse Function Approximation

Inverse Function Approximation

Continuous functions

Full Theoretical Justification
1

1

Mayorga R.V. and Carrera J., (2002), “A Radial Basis Function Network Approach for the Computation of Inverse Time
Variant Functions”, IASTED Int. Conf. on Artificial Intelligence & Soft Computing, Banff, Canada. (To appear in the
International Journal of Neural Systems).

10

Global Optimization Example

Consider the function:

11

Initial Model

Part 1

Initial point in “front”
side of curve (1)

Gets out of two local
minima (2 & 3)

Converges to the
global minimum (4)

v=0

and

Z
-
1
=0

12

Initial Model

Part 2

Initial point in “back”
side of curve

Gets stuck in an
inflection point

Does not get to the
global minimum

v=0

and

Z
-
1
=0

13

Model with vector
v

and

Z
-
1

Initial point in
“back” side of curve

Goes around the
curve (null space
vector)

Converges to the
global minimum

14

Artificial Neural Networks Model

Initial point in “back”
side of curve

Calculate
J(x)

and
v

with ANNs

Follows almost the
same trajectory as
previous model

Converges to the
global minimum

15

The Griewank Function
-

Example

Consider the function:

16

Griewank Function Optimization

Initial Model

Z
-
1
=0

&
v=0

Model with

Z
-
1
&
v

Model Using
ANN

17

Multi
-
Objective Nonlinear Example

I
-
Beam Design Problem
2

-
off dimensions

Minimize conflicting
objectives

Cross
-
sectional Area

Static Deflection

2

Osyczka, A., (1984),
Multicriterion Optimization in Engineering with FORTRAN programs
. Ellis Horwood Limited.

18

What if both objectives are solved
separately?

Cross
-
Sectional Area

Static Deflection

Static Deflection

Cross
-
Sectional Area

19

I
-
Beam Results

Feasible Solutions

Strong Pareto

Solutions

Weak Pareto

Solutions

20

I
-
Beam Results

Result

Proposed approach achieves very similar results to
state
-
of
-
the
-
art Genetic Algorithms (GA)

Gives a diverse set of strong Pareto solutions

The result of the ANN implementations varies by 0.88%

Computational Time
3

If compared to a standard floating point GA
4
, the
computational time decreases in 83%

From 15.2 sec to 2.56 sec

3 Experiments performed in a Sun Ultra 4 Digital Computer. GA: 100 individuals and 50 generations.

4 Passino, K., (1998), Genetic Algorithms Code, September 21st,

http://eewww.eng.ohio
-
state.edu/~passino/ICbook/ic_code.html (accessed February 2003).

21

Multi
-
Objective Optimization:

Just
-
In
-
Time Scheduling Problem

Consider
5

products manufactured in
2

production
lines

Minimize:

Cost

Line Unbalance

Plant Unbalance

Variables:

Production Rate

Production Time

Production Constraints

22

Scheduling Problem

Optimization
Results

Minimize:

Cost

23

Multi
-
Objective Optimization
Example

Minimize:

Cost

Line unbalance

Production Rate
variance / line

24

Multi
-
Objective Optimization
Example

Minimize:

Cost

Line unbalance

Production Rate
variance / line

Plant unbalance

Distribute production
in both production
lines

25

Conclusion

Novel global optimization method

It avoids local minima and inflection
points

The algorithm leads to convexities
via a null space vector
v

It can also be used for constraint
nonlinear optimization

26

Conclusion (cont.)

Novel MONLO deterministic method

Starts from a single point instead of a
population

Computational Time

For the I
-
Beam example, the computational
time is 83% less than Genetic Algorithms

The implementation of ANN reduces the
number of calculations to compute the
Inverse Function

For the scheduling example, the ANN
implementation reduces computational time
by 70%

27

Publications

3
rd

ANIROB/IEEE
-
RAS International
Symposium of Robotics and Automation,
Toluca, Mexico. Sept 1
-
4, 2002

Three journal papers already submitted:

Journal of Engineering Applications of Artificial
Intelligence

International Journal of Neural Systems

Journal of Intelligent Manufacturing

One paper to be published as a Chapter in
a book on Intelligent Systems.
Editor: Dr. Alexander M. Meystel