m asoud a sadzadeh dr. bryan tolson
DESCRIPTION
A New multi-objective algorithm: Pareto archived dds. D epartment of c ivil and e nvironmental e ngineering. M asoud a sadzadeh Dr. Bryan Tolson. Research goal. Develop an efficient multi-objective optimization algorithm that has few parameters. - PowerPoint PPT PresentationTRANSCRIPT
Masoud AsadzadehDr. Bryan Tolson
Department of Civil and Environmental Engineering
A NEW MULTI-OBJECTIVE ALGORITHM: PARETO ARCHIVED DDS
RESEARCH GOAL
• Modify Dynamically Dimensioned Search (DDS), a simple
efficient, parsimonious algorithm to solve unconstrained
computationally expensive, multi-objective water resources
problems.
• Develop an efficient multi-objective optimization algorithm that
has few parameters.
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• Set up the new tool so that it can easily scale to higher
dimensional problems (not only problems with two objectives).
BACKGROUND
• Simple & Fast Approximate Stochastic Global Optimization Algorithm Generate Good Results in Modeller's Time Frame
Algorithm parameter tuning is unnecessary
3
• Single-Solution Based algorithm (not population based)
• Designed for:
Single Objective Continuous Optimization
Computationally Expensive Automatic Hydrologic Model Calibration
Modified to solve problems with discrete decision variables, Tolson et al. [2008]
Tolson & Shoemaker [2007]
Dynamically Dimensioned Search (DDS)
DDS DESCRIPTION
Perturb the current best solution
Initialize starting solution
Continue?STOP
– Globally search at the start of the search by perturbing all decision variables (DV) from their current best values
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– Perturb each DV from a normal probability distribution centered on the current value of DV
– Locally search at the end of the search by perturbing typically only one DV from its current best value
N
Y
PROBLEM DEFINITION
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F(x)=[f1(x),f2(x),…,fN(x)]
Subject to: x=[x1,x2,…,xI] RI
Minimize:
f1
f2
PA-DDS DESCRIPTION
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Perturb the current ND solution
Update the set of ND solutions if necessary
Search for individual
minima first
Continue?STOP
New solution is ND?
Pick the New solution
Pick a ND solution based on
crowding distance
Initialize starting
solutions
YN
Create the non-dominated (ND)
solutions set
YN
RESULTS
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 ZDT6 (15000 iterations)
f1
f2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1ZDT4 (15000 iterations)
f1
f2
Average Convergence Metric Y (Deb 2001)
PA-DDS NSGA II* AMALGAM*
ZDT4 0.049 0.052 0.002
ZDT6 0.002 0.050 0.001
Actual Tradeoff Best Convergence Median Convergence
PA-DDS on Bi-Objective Test Problems Zitzler [1999]
* Vrugt and Robinson [2007]
0 0.1 0.2 0.3 0.4 0.5 0.60 0.1 0.2 0.3 0.4 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
f2
DTLZ1 Pareto FrontIteration number is:30000
f1
f 3
f3
f1
Actual Tradeoff
PA-DDS result
NEW RESULTS (TEST PROBLEMS DTLZ1)
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f1
f3
f2
Higher Dimensional Problem (25000 iterations)DTLZ1, with 3 objectives Deb et. al [2002]
2D view
MORE NEW RESULTS
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New York Tunnels Problem
• Water Distribution Network (WDN) Rehabilitation of an existing WDN
21 pipes (decision variables)
15 standard pipe sizes for each pipe
1 more option - no change in the pipe
1621 size of the discrete decision space
Minimum cost in the single objective version of the problem is $38.638 million
Objectives: Cost and Hydraulic deficit
3,360,000
PADDS
3,360,000
100,000
MORE NEW RESULTS
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New York Tunnels ProblemPerelman et al. [2008]
CONCLUSION
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• PA-DDS inherits simplicity and parsimonious characteristics of
DDS
Generating good approximation of tradeoff in the modeller's time frame
Reducing the need to fine tune the algorithm parameters
Solving both continuous and discrete problems
• PA-DDS can scale to higher dimensional problems
Research for the efficiency assessment is ongoing
Thank You
Thanks to our funding sourceNSERC Discovery grant
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only modification is to discretize the DV perturbation distribution
Discrete probability distribution of candidate solution option numbers for a single decision variable with 16 possible values and a current best solution of xbest=8. Default
DDDS-v1 r-parameter of 0.2*
0.00
0.05
0.10
0.15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Option # for Decision Variable x
Prob
abili
tyxbest = 8