Solving Constrained Optimization Problems Using Probability Collectives and a Penalty Function Approach

The best option to deal with a complex system that is too cumbersome to be treated in a centralized way is to decompose it into a number of sub-systems and optimize them in a distributed and decentralized way to reach the desired system objective. These sub-systems can be viewed as a multi-agent system (MAS) with self-learning agents. Furthermore, another challenge is to handle the constraints involved in real world optimization problems. This paper demonstrates the theory of probability collectives (PC) in the collective intelligence (COIN) framework, supplemented with a penalty function approach for constraint handling. The method of deterministic annealing in statistical physics, game theory and Nash equilibrium are at the core of the PC optimization methodology. Three benchmark problems have been solved with the optimum results obtained at reasonable computational cost. The evident strengths and weaknesses are also discussed to determine the future direction of research.

[1]  David Mautner Himmelblau,et al.  Applied Nonlinear Programming , 1972 .

[2]  Sascha Ossowski,et al.  Collective-Based Multiagent Coordination: A Case Study , 2008, ESAW.

[3]  Zbigniew Michalewicz,et al.  Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization , 1999, Evolutionary Computation.

[4]  Zbigniew Michalewicz,et al.  Handling Constraints in Genetic Algorithms , 1991, ICGA.

[5]  Yoav Shoham,et al.  Multi-Agent Reinforcement Learning:a critical survey , 2003 .

[6]  Zbigniew Michalewicz,et al.  Numerical optimization: Handling nonlinear constraints , 1997 .

[7]  Jonathan A. Wright,et al.  Self-adaptive fitness formulation for constrained optimization , 2003, IEEE Trans. Evol. Comput..

[8]  Erwie Zahara,et al.  Solving constrained optimization problems with hybrid particle swarm optimization , 2008 .

[9]  Robert Czabanski,et al.  Deterministic Annealing Integrated with epsilon-Insensitive Learning in Neuro-fuzzy Systems , 2006, ICAISC.

[10]  M.H.A. Mehr,et al.  A Distributed Probability Collectives Optimization Method for Multicast in CDMA Wireless Data Networks , 2007, 2007 4th International Symposium on Wireless Communication Systems.

[11]  Bart De Schutter,et al.  A Comprehensive Survey of Multiagent Reinforcement Learning , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[12]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[13]  Dar Yun Chiang,et al.  Improved Simulated Annealing Search for Structural Optimization , 2000 .

[14]  D. Jeya Mala,et al.  On the Use of Intelligent Agents to Guide Test Sequence Selection and Optimization , 2009, Int. J. Comput. Intell. Appl..

[15]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[16]  Y. Dong,et al.  An application of swarm optimization to nonlinear programming , 2005 .

[17]  Jasbir S. Arora,et al.  12 – Introduction to Optimum Design with MATLAB , 2004 .

[18]  Masao Fukushima,et al.  Derivative-Free Filter Simulated Annealing Method for Constrained Continuous Global Optimization , 2006, J. Glob. Optim..

[19]  Kang Tai,et al.  Probability Collectives: A distributed optimization approach for constrained problems , 2010, IEEE Congress on Evolutionary Computation.

[20]  David H. Wolpert,et al.  Discrete, Continuous, and Constrained Optimization Using Collectives , 2004 .

[21]  Selmer M. Johnson,et al.  Chemical Equilibrium in Complex Mixtures , 1958 .

[22]  Brian M. Autry.  University course timetabling with probability collectives , 2008 .

[23]  K. C. Seow,et al.  MULTIOBJECTIVE DESIGN OPTIMIZATION BY AN EVOLUTIONARY ALGORITHM , 2001 .

[24]  P. Gupta,et al.  Min-cost Selfish Multicast with Network Coding , 2006, 2006 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks.

[25]  E. Hinton,et al.  Optimization of trusses using genetic algorithms for discrete and continuous variables , 1999 .

[26]  Xin Yao,et al.  Search biases in constrained evolutionary optimization , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[27]  K. Deb.  An Efficient Constraint Handling Method for Genetic Algorithms , 2000 .

[28]  Jasbir S. Arora,et al.  Introduction to Optimum Design , 1988 .

[29]  Russell C. Eberhart,et al.  Solving Constrained Nonlinear Optimization Problems with Particle Swarm Optimization , 2002 .

[30]  Kang Tai,et al.  Probability Collectives: A multi-agent approach for solving combinatorial optimization problems , 2010, Appl. Soft Comput..

[31]  Kang Tai,et al.  Target-matching test problem for multiobjective topology optimization using genetic algorithms , 2007 .

[32]  Tapabrata Ray,et al.  An Evolutionary Algorithm for Constrained Optimization , 2000, GECCO.

[33]  Garret N. Vanderplaats,et al.  Numerical optimization techniques for engineering design , 1999 .

[34]  Anthony Chen,et al.  Constraint handling in genetic algorithms using a gradient-based repair method , 2006, Comput. Oper. Res..

[35]  Ilan Kroo,et al.  Distributed optimization and flight control using collectives , 2005 .

[36]  Carlos A. Coello Coello,et al.  Constraint-handling in genetic algorithms through the use of dominance-based tournament selection , 2002, Adv. Eng. Informatics.

[37]  Kenneth Rose,et al.  A Deterministic Annealing Approach for Parsimonious Design of Piecewise Regression Models , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  Ilan Kroo,et al.  Fleet Assignment Using Collective Intelligence , 2004 .

[39]  Ling Wang,et al.  A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization , 2007, Appl. Math. Comput..

[40]  David E. Goldberg,et al.  ENGINEERING OPTIMIZATION VIA GENETIC ALGORITHM, IN WILL , 1986 .

[41]  David H. Wolpert,et al.  Information Theory - The Bridge Connecting Bounded Rational Game Theory and Statistical Physics , 2004, ArXiv.

[42]  Garret N. Vanderplaats,et al.  Numerical Optimization Techniques for Engineering Design: With Applications , 1984 .

[43]  Edmund M. Yeh,et al.  Distributed Algorithms for Minimum Cost Multicast with Network Coding in Wireless Networks , 2006 .

[44]  U. Rajendra Acharya,et al.  Distributed Diagnosis and Home Healthcare , 2010 .

[45]  Geoffrey S. Ryder,et al.  A probability collectives approach to weighted clustering algorithms for ad hoc networks , 2005, Communications and Computer Networks.

[46]  Sajal K. Das,et al.  An on-demand weighted clustering algorithm (WCA) for ad hoc networks , 2000, Globecom '00 - IEEE. Global Telecommunications Conference. Conference Record (Cat. No.00CH37137).

[47]  Xin Yao,et al.  Stochastic ranking for constrained evolutionary optimization , 2000, IEEE Trans. Evol. Comput..

[48]  Robert Czabanski.  Deterministic Annealing Integrated with ε-Insensitive Learning in Neuro-fuzzy Systems , 2006 .

[49]  Kang Tai,et al.  Probability Collectives for decentralized, distributed optimization: A Collective Intelligence Approach , 2008, 2008 IEEE International Conference on Systems, Man and Cybernetics.

[50]  C. Coello,et al.  Cultured differential evolution for constrained optimization , 2006 .

[51]  Tarek Helmy,et al.  Hybrid Computational Intelligence Models for Porosity and Permeability Prediction of Petroleum reservoirs , 2010, Int. J. Comput. Intell. Appl..

[52]  Manuela M. Veloso,et al.  Multiagent learning using a variable learning rate , 2002, Artif. Intell..

[53]  Manuela M. Veloso,et al.  Rational and Convergent Learning in Stochastic Games , 2001, IJCAI.

[54]  David S. Leslie,et al.  Sequentially updated Probability Collectives , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[55]  Anand J. Kulkarni,et al.  Probability Collectives: A Decentralized, Distributed Optimization for Multi-Agent Systems , 2009 .

[56]  David H. Wolpert,et al.  A comparative study of probability collectives based multi-agent systems and genetic algorithms , 2005, GECCO '05.

[57]  Makoto Yokoo,et al.  Adopt: asynchronous distributed constraint optimization with quality guarantees , 2005, Artif. Intell..

[58]  Kenneth Rose,et al.  Mixture of experts regression modeling by deterministic annealing , 1997, IEEE Trans. Signal Process..

[59]  Ricardo Landa Becerra,et al.  Efficient evolutionary optimization through the use of a cultural algorithm , 2004 .

[60]  Abdollah Homaifar,et al.  Constrained Optimization Via Genetic Algorithms , 1994, Simul..

[61]  Stefan J. Johansson.  No Free Lunches in Multi-Agent Systems , 2002, Int. J. Comput. Intell. Appl..

[62]  S. Sathiya Keerthi,et al.  Deterministic annealing for semi-supervised kernel machines , 2006, ICML.

[63]  Wei Yu,et al.  A cross-layer optimization framework for multicast in multi-hop wireless networks , 2005, First International Conference on Wireless Internet (WICON'05).

[64]  Carlos A. Coello Coello,et al.  Use of a self-adaptive penalty approach for engineering optimization problems , 2000 .

[65]  Babak Hossein Khalaj,et al.  A Distributed Cross-Layer Optimization Method for Multicast in Interference-Limited Multihop Wireless Networks , 2008, EURASIP J. Wirel. Commun. Netw..

[66]  Kagan Tumer,et al.  An Introduction to Collective Intelligence , 1999, ArXiv.