Project - University of Regina

cathamΤεχνίτη Νοημοσύνη και Ρομποτική

23 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

86 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


Determine best trade
-
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


Level Loading


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