Multi-Domain Diversity Preservation to Mitigate Particle Stagnation and Enable Better Pareto Coverage in Mixed-Discrete Particle Swarm Optimization
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The document presents advancements in multi-objective mixed-discrete particle swarm optimization (mo-mdpso) to address stagnation and improve Pareto coverage and robustness in optimization problems. Key innovations include updated diversity metrics based on the spread of 'follower' particles and a stochastic leader selection mechanism favoring less popular leaders. Performance evaluations demonstrate that mo-mdpso ii significantly outperforms the original algorithm, providing enhanced robustness and maintaining fast convergence rates in various test problems.
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Popsize = 100
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Modified Global Leader Selection Mechanism
• 𝑟 is a random number, where 𝑟 ∈ [0,1]
• if 𝐫 < 𝒑(𝑵𝒊
𝒈
) , Particle-i keeps
following its current global leader;
• otherwise, Particle-i will switch its
leader to a global leader from a
neighboring domain.
14
𝑝(𝑁𝑖
𝑔
) = 𝑒𝑥𝑝 −
80𝑁𝑖
𝑔
Γ(1 + 1 𝜌)
𝑁 𝑝 × 𝑁𝑎𝑣𝑔
𝜌
− 𝑵 𝒂𝒗𝒈 is the expected average number of followers per global leader (𝑁𝑎𝑣𝑔 = 𝑁 𝑝
/𝑁 𝑔
)
− 𝑵𝒊
𝒈
is the number of particles following the global leader of Particle-i
Hence, global leaders with number of followers significantly exceeding the
average number per leader will tend to lose followers – an evening out of
followers among the non-dominated solutions as the population converges.
A Stochastic global
leader selection
approach
Probability to
retain a follower](https://image.slidesharecdn.com/momdpso-mao2015-150626194745-lva1-app6891/85/Multi-Domain-Diversity-Preservation-to-Mitigate-Particle-Stagnation-and-Enable-Better-Pareto-Coverage-in-Mixed-Discrete-Particle-Swarm-Optimization-14-320.jpg)
Multi-Domain Diversity Preservation to Mitigate Particle Stagnation and Enable Better Pareto Coverage in Mixed-Discrete Particle Swarm Optimization
- 1. Multi-Domain Diversity Preservation to Mitigate Particle Stagnation and Enable Better Pareto Coverage in Mixed-Discrete Particle Swarm Optimization Weiyang Tong*, Souma Chowdhury#, and Achille Messac# * Syracuse University, Department of Mechanical and Aerospace Engineering # Mississippi State University, Department of Aerospace Engineering The AIAAAviation and Aeronautics Forum and Exposition June 22-26, 2015 Dallas, TX For citations, please refer to the journal version of this paper, by Tong et al., "A Multi-Objective Mixed-Discrete Particle Swarm Optimization with Multi-Domain Diversity Preservation", Structural and Multidisciplinary Optimization. DOI:10.1007/s00158-015-1319-8
- 2. Particle Swarm Optimization 2 Particle Swarm Optimization (PSO) • was introduced by Eberhart and Kennedy in 1995 • was inspired by the swarm behavior observed in nature • is a population based stochastic algorithm Advantages of basic PSO: • Fast convergence • Ease of implementation • Readily scalable to higher dimensions PSO often suffers from pre-mature stagnation, which is mainly attributed to the loss of population diversity during convergence. The leader-following scheme in the basic PSO dynamics is suited for single-objective optimization. Algorithm for single objective unconstrained continuous optimization
- 3. Outline • Motivation and Research Objectives • Search Strategy in MO-MDPSO • Introduction of MO-MDPSO • Scheme of Quantifying the Population Diversity • Numerical Experiments • Continuous Unconstrained Test Problems • Continuous Constrained Test Problems • Mixed-Discrete Test Problems • Concluding Remarks 3MO-MDPSO: Multi-objective Mixed-Discrete PSO
- 4. Single Objective Mixed-Discrete PSO* Position update: 𝒙𝑖 𝑡 + 1 = 𝒙𝑖 𝑡 + 𝒗𝑖 𝑡 + 1 Velocity update: 𝒗𝑖 𝑡 + 1 = 𝑤𝒗𝑖 𝑡 + 𝑟1 𝐶1 𝑃𝑖 𝑙 (𝑡) − 𝒙𝑖 + 𝑟2 𝐶2 𝑃 𝑔(𝑡) − 𝒙𝑖 + 𝑟3 𝛾𝑐 𝒗𝑖(𝑡) 𝑤 – inertia weight 𝒙𝑖 – position of a particle 𝒗𝑖 – velocity of a particle 𝐶1 – cognitive parameter 𝐶2 – social parameter 𝑡 – generation 𝑟 – random number between 0-1 𝑃𝑖 𝑙 – pbest of a particle 𝑃 𝑔 – gbest of the swarm 4 Diverging velocity Diversity preservation coefficient Inertia Local search Global search *Chowdhury et al (SMO, 2013) A mixed-discrete PSO was developed to solve nonlinear constrained problems with a mix of continuous and discrete/integer variables (in wind farm design and product family design).
- 5. Research Motivation 5 • A multi-objective version of the MDPSO algorithm was developed through fundamental modifications to the leader selection strategy, and was successfully tested on benchmark problems and practical engineering MOO problems *. • However, previous research showed that the robustness of MO-MDPSO needs to be further improved, where robustness refers to lower deviation in the final results over multiple runs (less sensitive to the random parameters). Constrained Multiobjective Non-convex Pareto frontier Nonlinear Mixed types of variables MDPSO • We hypothesized that a more coherent interaction of the diversity preservation technique and the leader selection mechanism can improve the robustness of the algorithm, while preserving its Pareto capture and fast convergence capabilities. *Tong et al. IDETC, 2014; SMO 2015.
- 6. Research Objective • Investigate diversity-preservation schemes in MO-MDPSO, which is also cognizant of the multi-leader-follower behavior of the particle population; here the aim is to: 1. Avoid stagnation and capture the complete Pareto frontier 2. Keep a desirably even distribution of Pareto solutions 3. Improve the robustness of the algorithm (i.e., demonstrate lower deviation over multiple runs, or higher likelihood to capture the actual Pareto front in any given run). 6
- 7. Search Strategies in Multi-Objective PSO 7 • MOPSO by Parsoupulos and Vrahatis (2002) • DNPSO by Hu and Eberhart (2002) • NSPSO by Li (2003) • MOPSO by Coello (2004) • To best retain the original dynamics, the Pareto based strategy is selected Aggregating function based Single objective based Hybrid with other techniques Pareto dominance based Basic PSO MOPSO Local leader The best solution is based on a particle’s own history Local set with non-dominated historical solutions are found within the local neighborhood Global leader The best solution among all local best solutions The global Pareto solutions are obtained using all local ones
- 8. 1 2 3 4 5 1 2 3 4 5 f1 f2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Infeasible R egion Actual Boundary Local Pareto set Dynamics of MO-MDPSO Position update: 𝒙𝑖 𝑡 + 1 = 𝒙𝑖 𝑡 + 𝒗𝑖 𝑡 + 1 Velocity update: 𝒗𝑖 𝑡 + 1 = 𝑤𝒗𝑖 𝑡 + 𝑟1 𝐶1 𝑷𝑖 𝑙 (𝑡) − 𝒙𝑖 + 𝑟2 𝐶2 𝑷𝑖 𝑔 (𝑡) − 𝒙𝑖 + 𝑟3 𝛾 𝑐,𝑖 𝒗𝑖(𝑡) 8 𝑷𝑖 𝑙 – local leader of particle-i, which is the selected from the local Pareto set 𝑷𝑖 𝑔 – global leader of particle-i that is determined by a stochastic process Crowding Distance – to manage the size of local/global Pareto set Multiple global leaders Inertia Local search Global search Applied w.r.t. each particle’s global leader Current particle Stored particle
- 9. Scheme 1: Original Diversity Metrics (MO-MDPSO I) Measured based on the spread of the entire swarm in design space for all continuous variables: 𝐷𝑐 = 𝑗=1 𝑛 𝑥 𝑚𝑎𝑥,𝑗(𝑡) − 𝑥 𝑚𝑖𝑛,𝑗(𝑡) 𝑋 𝑚𝑎𝑥,𝑗 − 𝑋 𝑚𝑖𝑛,𝑗 1 𝑛 for each discrete variable: 𝐷 𝑑 𝑗 = 𝑥 𝑚𝑎𝑥,𝑗(𝑡) − 𝑥 𝑚𝑖𝑛,𝑗(𝑡) 𝑋 𝑚𝑎𝑥,𝑗 − 𝑋 𝑚𝑖𝑛,𝑗 9 Considering the impact of outlier solutions, the diversity metrics are modified as 𝐷𝑐 = 𝐹𝜆 𝐷𝑐, and 𝐷 𝑑,𝑖 𝑗 = 𝐹𝜆𝑖 𝐷 𝑑 𝑗 where 𝐹𝜆𝑖 = 𝜆 𝑁 𝑝 + 1 𝑁𝑖 + 1 1 𝑚+𝑛 𝜆 is used to define the fractional domain w.r.t. the global leader of particle-i The spread along the jth dimension The upper & lower bounds along the jth dimension Number of candidate solutions per global leader Number of candidate solutions enclosed by the 𝜆-fractional domain
- 10. Hypercube enclosing 72 candidate solutions X1 X2 0 1 2 3 0 1 2 3 Particles Global leaders Original Multiple λ-Fractional Domains 10 • 7 global leaders present. • With a total of 72 particles, ideally, each fractional domain should enclose around 10 particles. • Particles located in the overlapping regions often tend to be re- allocated between the neighboring domains. • Outlier particles are allowed to have marginal impact on diversity. 13 10 14 𝜆 = 0.25 Design Variable Space
- 11. Determining the Location of the Multi-domain The boundaries of the 𝜆𝑖 - fractional domain are defined as 𝑥𝑖 𝑚𝑎𝑥,𝑗 = 𝑚𝑎𝑥 𝑥 𝑚𝑖𝑛,𝑗 + 𝜆𝑖 𝛿𝑥 𝑗, 𝑚𝑖𝑛 𝑷𝑖 𝑔,𝑗 + 1 2 𝜆𝑖 𝛿𝑥 𝑗, 𝑥 𝑚𝑎𝑥,𝑗 𝑥𝑖 𝑚𝑖𝑛,𝑗 = 𝑚𝑖𝑛 𝑥 𝑚𝑎𝑥,𝑗 − 𝜆𝑖 𝛿𝑥 𝑗, 𝑚𝑎𝑥 𝑷𝑖 𝑔,𝑗 − 1 2 𝜆𝑖 𝛿𝑥 𝑗, 𝑥 𝑚𝑖𝑛,𝑗 11 • Continuous variables 𝛾 𝑐,𝑖 = 𝛾 𝑐0 𝑒𝑥𝑝 − 1 2 𝐷𝑐,𝑖 𝜎𝑐 2 , 𝜎𝑐 = 1 2 ln 1 𝛾 𝑚𝑖𝑛 • Discrete variables 𝛾𝑑,𝑖 𝑗 = 𝛾 𝑑0 𝑒𝑥𝑝 − 1 2 𝐷 𝑑,𝑖 𝑗 𝜎 𝑑 2 , 𝜎 𝑑 = 1 2 ln 1 𝑀 𝑗
- 12. Scheme 2: Modified Diversity Metrics Measured based on the spread of “followers” in design space, i.e., based on the design vector of particles following the global leader of Particle-i: 𝐷𝑐,𝑖 = 𝑗=1 𝑁𝑖 𝑔 𝑥 𝑚𝑎𝑥,𝑗(𝑡) − 𝑥 𝑚𝑖𝑛,𝑗(𝑡) (𝑋 𝑚𝑎𝑥,𝑗 − 𝑋 𝑚𝑖𝑛,𝑗) 1 𝑁𝑖 𝑔 12 The spread along the jth dimension The upper & lower bounds along the jth dimension Number of particles following the global leader of Particle-i • The scheme for handling discrete variables is the same as in MO-MDPSO. • Outlier particles could improve the Pareto coverage in multi-objective problems, and hence are accounted for in the diversity metric. • Greater the spread of the followers (in the variable space) of a global leader, greater is its diversity footprint (it has a more diverse team).
- 13. Modified Multiple Domains 13 • Ideally, each of the 7 global leaders should have 10 followers. • Outlier particles are not neglected 𝛾𝑐,𝑖 = 𝛾 𝑐0 𝑒𝑥𝑝 − 1 2 𝐷𝑐,𝑖 𝜎𝑐 2 , 𝜎𝑐 = 1 2 ln 1 𝛾 𝑚𝑖𝑛 • The population diversity is measured based on the spread of “followers” w.r.t each global leader Design Variable Space 6 12
- 14. 0 20 40 60 80 100 120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ng i Probability Popsize = 100 Navg = 20 Navg = 30 Navg = 40 Modified Global Leader Selection Mechanism • 𝑟 is a random number, where 𝑟 ∈ [0,1] • if 𝐫 < 𝒑(𝑵𝒊 𝒈 ) , Particle-i keeps following its current global leader; • otherwise, Particle-i will switch its leader to a global leader from a neighboring domain. 14 𝑝(𝑁𝑖 𝑔 ) = 𝑒𝑥𝑝 − 80𝑁𝑖 𝑔 Γ(1 + 1 𝜌) 𝑁 𝑝 × 𝑁𝑎𝑣𝑔 𝜌 − 𝑵 𝒂𝒗𝒈 is the expected average number of followers per global leader (𝑁𝑎𝑣𝑔 = 𝑁 𝑝 /𝑁 𝑔 ) − 𝑵𝒊 𝒈 is the number of particles following the global leader of Particle-i Hence, global leaders with number of followers significantly exceeding the average number per leader will tend to lose followers – an evening out of followers among the non-dominated solutions as the population converges. A Stochastic global leader selection approach Probability to retain a follower
- 15. • Three classes of test problems are used to evaluate the performance of the new MO-MDPSO algorithm (called MO-MDPSO II) • Each test problems is run 30 times to compensate for the impact of random parameters • Sobol’s quasirandom sequence generator is used for the initial population User-defined parameters in MO-MDPSO Numerical Experiment 15 Parameter Continuous uncon- strained problems Continuous con- strained problems Mixed-discrete constrained problems 𝑤 0.5 0.5 0.5 𝐶1 1.5 1.5 1.5 𝐶𝑐0 1.5 1.5 1.5 𝛾 𝑐0 1.0 1.0 1.0 𝛾 𝑚𝑖𝑛 1e-4 1e-6 1e-8 𝛾 𝑑0 NA NA 0.5,1.0 𝜆 0.2 0.1 0.1 Local set size 5 6 10 Global set size 50 50 Up to 100 Population size 100 100 min(5n, 100)
- 16. Performance Metric for Continuous Unconstrained Problems Results: Continuous Unconstrained Problems 16 Function name Accuracy 𝜸 Uniformity and Spread ∆ Mean 𝝁 𝜸 Std. deviation 𝝈 𝜸 Mean 𝝁∆ Std. deviation 𝝈∆ MO-MDPSO I MO-MDPSO II MO-MDPSO I MO-MDPSO II MO-MDPSO I MO-MDPSO II MO-MDPSO I MO-MDPSO II Fonseca 2 4.4e-3 4.2e-3 4.6e-4 1.2e-4 0.58 0.53 1.5e-2 5.4e-4 Coello 8.2e-3 8.4e-3 1.2e-3 3.2e-4 0.57 0.57 2.9e-2 3.2e-4 Schaffer 1 9.0e-3 5.2e-3 1.1e-3 4.3e-5 0.21 0.24 1.2e-2 8.5e-5 Schaffer 2 1.3e-2 1.2e-2 1.1e-3 4.1e-4 0.96 0.98 2.9e-3 9.1e-4 ZDT 1 8.9e-4 4.7e-4 1.2e-4 9.2e-5 0.20 0.19 4.0e-2 3.9e-5 ZDT 2 7.5e-4 8.4e-4 7.5e-3 1.2e-4 0.20 0.21 4.0e-2 1.5e-4 ZDT 3 4.2e-3 5.1e-3 4.0e-4 6.1e-4 0.54 0.45 4.0e-2 2.0e-5 ZDT 4 1.4 9.2e-1 2.0 1.3e-2 0.53 0.75 1.6e-2 6.9e-3 ZDT 6 4.6e-2 7.3e-3 5.3e-2 1.6e-4 0.60 0.62 2.4e-2 8.0e-4 γ: Measures the closeness of the computed Pareto solutions to actual Pareto front Δ: Measures the uniformity of the distribution of the computed Pareto solutions (along the front) and overall spread of the computed Pareto front. The better values are marked in bold blue font in this Table. Performance Metrics: Deb et al., IEEE Trans. on Evo. Comp., 2002; Okabe et al, CEC, 2003
- 17. 0 0.0005 0.001 0.0015 0 0.0002 0.0004 0.0006 0.0008 0.001 0 0.002 0.004 0.006 0.008 0.01 0 0.5 1 1.5 2 2.5 3 3.5 0 0.05 0.1 0.15 Comparison of Performance Indicators 17 Accuracy Uniformity and Spread The lower the better MO- MDPSOII MO- MDPSOI 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 MO- MDPSOI MO- MDPSOII ZDT 1 ZDT 2 ZDT 3 ZDT 4 ZDT 6 ZDT 1 ZDT 2 ZDT 3 ZDT 4 ZDT 6 MO-MDPSO II consistently offers performance that is 1-2 orders of magnitude more robust than the original algorithm (MO-MDPSO I)
- 18. Plots: Continuous Constrained Problems 18 SRN 2 design variables TNK 2 design variables Number of function evaluations used by MO-MDPSO is 10,000 (20,000 was used by NSGA-II) BNH 2 design variables CONSTR 2 design variables KITA 2 design variables
- 19. • The MINLP problem adopted from Dimkou and Papalexandri* 19 No. of design variables 6 No. of discrete variables 3 (binary) Function evaluations 10,000 Population size 100 Elite size of MO-MDPSO 100 *: Dimkou and Papalexandri (1998) Mixed-Discrete Constrained Test Problem 1 f1 f2 -60 -50 -40 -30 -20 -10 0 -20 0 20 40 60 80 100 by MO-MDPSO I by MO-MDPSO II
- 20. 20 • The design of disc brake problem adopted from Osyczka and Kundu* No. of design variables 4 No. of discrete variables 1 (integer) Function evaluations 10,000 Population size 100 Elite size of MO-MDPSO 100 *: Osyczka and Kundu (1998) Mixed-Discrete Constrained Test Problem 2 MO-MDPSO II retains its superior ability to better capture extreme solutions (anchor points) compared to NSGA II.
- 21. Concluding Remarks A new multiobjective implementation of the Mixed-Discrete PSO algorithm (called MO-MDPSO II) was developed and tested. In seeking to improve the algorithm robustness, the following two advancements were made to the original MO-MDPSO: The diversity metric is defined based on the spread of particles (followers) following each global leader (instead of the spread of the entire particle population). A stochastic leader selection is adopted that favors global leaders with fewer followers (greater the follower-team size, greater the likelihood of its followers to switch team). For unconstrained problems, the new MO-MDPSO is observed to provide significantly better robustness (lower std-dev over multiple runs). For constrained MOO and MO-MINLP problems, the new MO-MDPSO algorithm retains its fast convergence and Pareto coverage capabilities. Owing to the involvement of a relatively high number of prescribed parameters future work should pursue a parametric analysis of this algorithm. 21
- 22. Acknowledgement • I would like to thank the co-authors, Dr. Weiyang Tong and Prof. Achille Messac for their substantial contributions to this research. • Support from the NSF Award CMMI 1437746 is also acknowledged. 22
- 24. PDF of Global Leader Selection Mechanism • Starting from the generalized normal distribution, which is given by 𝑝 = 𝜌 2𝛼Γ(1 𝜌) 𝑒𝑥𝑝 − 𝑥 − 𝜇 𝛼 𝜌 where 𝛼 is the scale parameter; 𝜌 is the shape parameter; 𝜇 is the mean value; and Γ denotes the Gamma function, which is defined as Γ 𝑡 = 0 ∞ 𝑥 𝑡−1 𝑒−𝑡 𝑑𝑥. • Let 𝑝 be ranged between 0 and 1, then 𝛼 becomes 2𝜌 Γ 1 𝜌 • By adjusting the slope and the probability of 𝑝(𝑁𝑎𝑣𝑔) , we have 𝑝(𝑁𝑖 𝑔 ) = 𝑒𝑥𝑝 − 80𝑁𝑖 𝑔 Γ(1 + 1 𝜌) 𝑁 𝑝 × 𝑁𝑎𝑣𝑔 𝜌 24 𝜌 𝜌 𝜌 𝜌 𝜌 𝜌 Nadarajah, Journal of Applied Statistics, 2011
- 25. Solutions Comparison if both solu-x and solu-y are infeasible choose the one with the smaller constraint violation; else if solu-x is feasible and solu-y is infeasible choose solu-x; else if solu-x is infeasible and solu-y is feasible choose solu-y; else both solu-x and solu-y are feasible apply non-dominance comparison: solu-x strongly dominates solu-y if and only if 𝑓𝑘 𝑥 < 𝑓𝑘 𝑦 , ∀k = 1,2,…, N solu-x weakly dominates solu-y if 𝑓𝑘 𝑥 ≤ 𝑓𝑘 𝑦 for at least one k solu-x is non-dominated with solu-y when 𝑓𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑘: 𝑓𝑘 𝑥 > 𝑓𝑘 𝑦 , for the rest: 𝑓𝑘 𝑥 < 𝑓𝑘 𝑦 25
- 26. Multi-directional Diversity Preservation (cont’d) • Continuous variables 𝛾 𝑐,𝑖 = 𝛾 𝑐0 𝑒𝑥𝑝 − 1 2 𝐷𝑐,𝑖 𝜎𝑐 2 , 𝜎𝑐 = 1 2 ln 1 𝛾 𝑚𝑖𝑛 • The diversity coefficient for continuous variables is to apply a repulsion away from each of the global leaders • Discrete variables 𝛾𝑑,𝑖 𝑗 = 𝛾 𝑑0 𝑒𝑥𝑝 − 1 2 𝐷 𝑑,𝑖 𝑗 𝜎 𝑑 2 , 𝜎 𝑑 = 1 2 ln 1 𝑀 𝑗 • A stochastic update process is applied to help particles jump out of the local hypercude: if 𝑟 > 𝛾𝑑,𝑖 𝑗 , use the nearest vertex approach (NVA) else, update randomly to the upper or lower bound of the local hypercube 26 The diversity coefficient is expressed as a monotonically decreasing function of the current diversity metric Size of feasible values of the jth variable
- 27. Discrete Variables and Constraints Handling • The Nearest Vertex Approach (NVA) is used to deal with discrete variables, where a local hypercube is defined as 𝑯 = 𝑥 𝐿1, 𝑥 𝑈1 , 𝑥 𝐿2, 𝑥 𝑈2 , … , 𝑥 𝐿 𝑚, 𝑥 𝑈 𝑚 and 𝑥 𝐿 𝑗 < 𝑥 𝑗 < 𝑥 𝑈 𝑗, ∀𝑗 = 1,2, … , 𝑚 • The net constraint is used to handle constraints, as given by 𝑓𝑐 = 𝑝=1 𝑃 max 𝑔 𝑞, 0 + 𝑞=1 𝑄 max ℎ 𝑞 − 𝜖, 0 27 Nearest Vertex Approach Normalized inequality constraints Normalized equality constraints Tolerance