A New multi-objective algorithm: Pareto archived dds Masoud asadzadeh Dr. Bryan Tolson Department of civil and environmental engineering Masoud Asadzadeh

Research goal Develop an efficient multi-objective optimization algorithm that has few parameters. Modify Dynamically Dimensioned Search (DDS), a simple efficient, parsimonious algorithm to solve unconstrained computationally expensive, multi-objective water resources problems. Set up the new tool so that it can easily scale to higher dimensional problems (not only problems with two objectives).

Dynamically Dimensioned Search (DDS) Background Dynamically Dimensioned Search (DDS) Tolson & Shoemaker [2007] Designed for: Single Objective Continuous Optimization Computationally Expensive Automatic Hydrologic Model Calibration Modified to solve problems with discrete decision variables, Tolson et al. [2008] Single-Solution Based algorithm (not population based) Simple & Fast Approximate Stochastic Global Optimization Algorithm Generate Good Results in Modeller's Time Frame Algorithm parameter tuning is unnecessary

Initialize starting solution Perturb the current best solution dds description Initialize starting solution Perturb the current best solution Y Globally search at the start of the search by perturbing all decision variables (DV) from their current best values Continue? STOP N Locally search at the end of the search by perturbing typically only one DV from its current best value Perturb each DV from a normal probability distribution centered on the current value of DV

F(x)=[f1(x),f2(x),…,fN(x)] Problem definition Minimize: F(x)=[f1(x),f2(x),…,fN(x)] Subject to: x=[x1,x2,…,xI]  RI f1 f2

PA-dds description Perturb the current ND solution Update the set of ND solutions if necessary Search for individual minima first Initialize starting solutions Create the non-dominated (ND) solutions set Pick a ND solution based on crowding distance Pick the New solution N New solution is ND? Y Y STOP N Continue?

PA-DDS on Bi-Objective Test Problems Results PA-DDS on Bi-Objective Test Problems Zitzler [1999] Actual Tradeoff Best Convergence Median Convergence Average Convergence Metric Y (Deb 2001) PA-DDS NSGA II* AMALGAM* ZDT4 0.049 0.052 0.002 ZDT6 0.050 0.001 * Vrugt and Robinson [2007]

Higher Dimensional Problem (25000 iterations) NEW RESULTS (TEST PROBLEMS DTLZ1) Higher Dimensional Problem (25000 iterations) DTLZ1, with 3 objectives Deb et. al [2002] 2D view f3 f1 Actual Tradeoff PA-DDS result f1 f3 f2

New York Tunnels Problem MORE NEW RESULTS 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

New York Tunnels Problem More New Results New York Tunnels Problem Perelman et al. [2008] 3,360,000

Conclusion 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

Thanks to our funding source NSERC Discovery grant Thank You

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*