A Simple Multi-objective Optimization Problem
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Transcript of A Simple Multi-objective Optimization Problem
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A simple multi-objective
optimization problem
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Introduction
Lets introduce a geometrical optimization problem, named cones problem, with thefollowing characteristics:
multi-objective problem (two objective functions): the solution is not a single optimum
design, but instead it is represented by the set of designs belonging to the Pareto frontie
simple mathematical formulation: easy and quick implementation from scratch of the
relevant modeFRONTIER project
constrained problem: objectives space and designs space present feasible and
unfeasible regions
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Problem definition
Right circular cone:
r= base radius
h = height
s= slant height
V= volume
B = base area
S= lateral surface area
T= total area
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Cones problem
two input variables: r, h
two objectives:
one constraint:
The cone shape (i.e. the
design) is defined univocallywhen both rand h are given.
We want to minimize both
the lateral surface area andthe total surface area
A constraint for the cone volume is
given, in order to guarantee aminimum volume.
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Project building
Lets build from scratch the pertinent modeFRONTIERproject:
1. Work Flow setup: fill the work canvas with the projects
building blocks
2. Script Node setup: use your favourite math tool
Jython script
Matlab node
Excel Workbook node OpenOffice Spreadsheet node
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Work Flow setup
two input variables
three output variablestwo objectivesone constraintscript node
logic flow
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Work Flow setup
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Script node: Jython
Jython (Python) script case:
Load math module
Note the syntax of mathematical functions and constants
Write down the formulae
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Script node: MatlabMatlab case:
Write down the formulae
Check Matlab version
Load the matlab file
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Script node: ExcelExcel Workbook case:
Insert the formulae
Build the spreadsheet
Load the xls file
Link variables to cells
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Script node: OpenOfficeOpenOffice Spreadsheet case:
Insert the formulae
Build the spreadsheet
Link variables to cells
Load the sxc file
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Runs examples
Lets see some examples of runs with different DOEsand/or schedulers:
Full Factorial DOE
random samplings: Random Sequence and Sobol
DOEs
genetic algorithms: MOGA-II, NSGA-II
MOSA
NBI-NLPQLP
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Full Factorial
Full Factorial DOE
10 levels per variable
100 eval. designs
feasible designsunfeasible designs
Pareto designs
objectives space
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Random Sequence
Random Sequence DOE
1000 eval. designs
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Sobol
Sobol DOE
1000 eval. designs
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MOGA-II
MOGA-II
20 individuals (Sobol)
50 generations
1000 eval. designs
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MOGA-II
MOGA-II
20 individuals (Sobol)
50 generations
1000 eval. designs
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NSGA-II
NSGA-II
20 individuals (Sobol)
50 generations
1000 eval. designs
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MOSA
MOSA
10 points (Sobol)
100 iterations
1000 eval. designs
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NBI-NLPQLP
NBI-NLPQLP
(DOE: 10 Sobol)
20 NBI-subproblems
346 eval. designs
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Final considerations
Lets consider the difference between single-objective problem solutions: two different minima
multi-objective problem solutions: the Pareto frontier
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Single-objectives minimaEach design represents
the optimum solution for
its corresponding single-objective problem.
...but what about the in
between designs?
?
...we would like to get a
compromise solution. A
trade-off of the twoobjectives...
What we want is the Pareto frontier!
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The Pareto frontier
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