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

4

– 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

5

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