Interactive Multiobjective
Optimization via the Internet
Providing Distributed Interface
and Centralized Computing
Kaisa Miettinen & Marko M. Mäkelä
miettine@mit.jyu.fi
Laboratory of Scientific Computing
Department of Mathematics
University of Jyväskylä
P.O. Box 35
FIN-40351 Jyväskylä
FINLAND
1. Overview
Method: Nondifferentiable Interactive
Multiobjective BUndle-based optimization System = NIMBUS
Interface: World-Wide Web (WWW), Internet
Product: WWW-NIMBUS
Example: Using WWW-NIMBUS to solve a test problem
2. The Problem
We consider a problem
We denote the feasible region by S.
A point is Pareto optimal if there does not
exist another point such that for all and for at least one j.
A point is weakly Pareto optimal if there does not
exist another point such that for all .
3. NIMBUS Method
Motivation:
Solving general single and multiobjective
optimization problems.
Easiness of use.
Assumptions:
All the objective and constraint functions are locally
Lipschitzian (less demanding than continuous differentiability).
The feasible region S is convex.
Less is preferred to more to the DM.
The DM is asked to divide the
objective functions into up to five classes such that
: whose values
() should be decreased,
() should be decreased down till
some aspiration level,
are satisfactory at the moment,
are allowed to
increase up till some upper bound, and
() are allowed to change freely.
Functions in are to be minimized as far
as possible but the functions in
( aspiration functions) only till the aspiration
level. Also weighting coefficients can be given to the functions in
these two classes.
Then we form a new problem
with weighting coefficients , ,
and . In addition, for and
for .
MPB Routine
The MPB routine is an extension of the proximal bundle method into a
multiobjective case. The basic
idea is to move into a direction where the values of all the current objective
functions improve.
NIMBUS Algorithm
Checking the Pareto optimality:
4. NIMBUS Implementations
Mainframe version
+ Applicable for even large-scale problems.
- No graphical interface
difficult to use.
- Trouble in delivering updates.
Windows version
+ Good graphical interface with illustrations.
- Not suitable for large-scale problems.
- Computing capacity required.
- Trouble in delivering updates.
Need for WWW-NIMBUS
5. World-Wide Web = WWW
Graphical user interface to Internet.
Information can be made available easily.
Accessible for many.
Graphical illustrations and interaction possible (to
some extent).
6. WWW-NIMBUS Implementation
Computation takes place in a server computer (HP9000/835).
Information is carried between server and local WWW browser via the Internet.
The scripts have been coded in C and the underlying
calculation involves Fortran subroutines.
7. WWW-NIMBUS
The problem to be handled may have one or several objective functions.
Linear and nonlinear constraints and bounds for the variables
are treated separately, each type in the most efficient way.
+ NIMBUS is a flexible and versatile method for solving general
(even nondifferentiable) optimization problems.
+ The user can select to what extent to
exploit the versatile possibilities of the method. The DM is free to
move around and also to change her or his mind. Previous acts do not
limit the movements. The DM can also extract undesirable solutions
from further consideration.
+ WWW provides necessary graphical user interface.
+ The latest version is always available for the user.
+ The browsing computer does not need excessive computational
capacities, nor appropriate complilers.
All you need is a connection to the Internet and a WWW browser.
+ WWW-NIMBUS can be used free of charge.
+ WWW-NIMBUS provides a user interface to the well-tested
nondifferentiable optimization system NSOLIB.
9. Under Development
Automatic (sub)differentiation.
Graphics.
Aids for selecting the most preferred alternative.
Graphical support for single objective optimization.
is Pareto optimal if there does not
exist another point 
such that 
for all 
and 
for at least one j.
is weakly Pareto optimal if there does not
exist another point 
such that 
for all 
.
: 
whose values
) should be decreased,
) should be decreased down till
some aspiration level,
are satisfactory at the moment,
are allowed to
increase up till some upper bound, and
) are allowed to change freely.
are to be minimized as far
as possible but the functions in 
( aspiration functions) only till the aspiration
level. Also weighting coefficients can be given to the functions in
these two classes.
, 
,
and 
. In addition, 
for 
and
for 
.
difficult to use.
Need for WWW-NIMBUS
