Newtonian Law Inspired Optimization Techniques Based on Gravitational Search Algorithm
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This document presents a thesis seminar on optimization techniques inspired by Newtonian laws of gravity and gravitational search algorithms. It introduces the gravitational search algorithm and how it is based on Newton's law of universal gravitation. It then proposes some new hybrid algorithms that combine gravitational search with mutation operators and differential evolution. These proposed algorithms (GSA-m and GDE) are tested on benchmark functions and shown to perform better than the original GSA and other popular algorithms like genetic algorithms and particle swarm optimization. The document also extends the approach to multi-objective optimization problems and proposes a new multi-objective gravitational optimization algorithm.
Newtonian Law Inspired Optimization Techniques Based on Gravitational Search Algorithm
- 1. AThesis Seminar on Newtonian Law Inspired Optimization Techniques Based on Gravitational Search Algorithm Presented by- Rajdeep Chatterjee M.Tech, 2009-11 School of Computer Engineering KIIT University Under the guidance of- Prof. (Dr.) Madhabananda Das Dean, School of Computer Engineering KIIT University
- 2. Overview Gravitational Search Algorithm (GSA) PID Controller and GSA Simulation Results Mutation Differential Evolution (DE) Proposed Algorithm ◦ Gravitational Search Algorithm with mutation (GSA-m) ◦ Gravitational Differential Evolution (GDE) Benchmark Functions & Experimental Results Pareto optimality & Domination Proposed Multi-objective Gravitational Optimization (MOGO) Benchmark Functions & Simulations Observations Publications References
- 3. Gravitational Search Algorithm This algorithm is based on the Newtonian gravity: „„Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them”. The position of the mass corresponds to a solution of the problem, and its masses are determined using a fitness function.
- 5. Gravitational Search Algorithm Major equations of GSA: …(1) …(2) …(3) …(4) …(5) …(6)
- 6. Gravitational Search Algorithm Major equations of GSA: …(7) …(8) …(9) …(10) …(11) …(12) …(13)
- 8. PID Controller Fig. 5 PID Controller Transfer Function
- 9. GSA and PID Controller Simulation Result GSA PSO BFO KP 0.85 0.56 0.80 KI - - - KD 0.57 0.62 0.53 Cost 15.3026 19.1416 15.7906 Table 3
- 10. GSA and PID Controller Fig. 6 Closed Loop Response
- 11. Mutation Optimization algorithm often trapped into local optima. We cannot obtain the global optima and rather ended up with local optimal value. Mutation operator is used to lift the population from local optima to not yet explored search space.
- 12. Mutation Fig. 2
- 13. Differential Evolution It is a heuristic approach for minimizing possibly nonlinear and non differentiable continuous space. It was introduced by Rainer Storn and Kenneth Price in the year 1995.
- 14. Differential Evolution Concept behind the DE: …(15) Where and NP is the population size. …(16) …(17) …(18)
- 16. Proposed Algorithms GSA–m :: Gravitational Search Algorithm with mutation GDE :: Gravitational Differential Evolution DE is used as mutation operator to improve the convergence. DE-1 :: DE/best/1/exp DE-2 :: DE/best/2/exp DE-3 :: DE/best/2/bin
- 17. Proposed GSA-m Algorithm 1. Generate initial population 2. For I: 0 to max-iteration or stop criteria is reached do 3. Evaluate the fitness for each agent 4. Update the G, best and worst of the population 5. Calculate M , F and a for each agent 6. Update velocity and position i.e updated-agent 7. If last r iterations give same result or (I mod k) == 0 8. Create Difference-Offspring from updated-agent 9. Evaluate fitness; 10. If an offspring is better than updated-agent 11. Then replace the updated-agent by offspring in the next generation; 12. End If; 13. End If; 14. End For 15. Return approximate global optima
- 18. Proposed GDE Algorithm 1. Generate initial population 2. For I: 0 to max-iteration or stop criteria is reached do 3. Evaluate the fitness for each agent 4. Update the G, best and worst of the population 5. Calculate M , F and a for each agent 6. Update velocity and position i.e updated-agent 7. Create Difference-Offspring from updated-agent 8. Evaluate fitness; 9. If an offspring is better than updated-agent 10. Then replace the updated-agent by offspring in the next generation; 11. End If; 12. End For 13. Return approximate global optima
- 19. Benchmark Functions Unimodal & Multimodal Benchmark Functions Table. 1
- 20. Experimental Results Simulation Results on six Benchmark Functions Table. 2 Test Func tions GSA GSA-m GDE DE-1 DE-2 DE-3 DE-1 DE-2 DE-3 F1(x) 7.3E-11 2.06 E-17 2.07 E-17 1.99 E-17 1.76 E-17 1.92 E-17 1.83 E-17 F2(x) 25.16 5.3E-2 6.5E-2 4.9E-2 2.53E-4 1.75E-4 2.35E-4 F3(x) -2.8E+3 -2.78 E+3 -2.82 E+3 -2.77 E+3 -1.045 E+4 -1.045 E+4 -9.5 E+3 F4(x) 6.9E-6 3.68E-9 3.65E-9 3.55E-9 3.2E-9 3.4E-9 3.5E-9 F5(x) 8.0E-3 5.4E-3 5.6E-3 5.8E-3 3.6E-3 3.2E-3 3.4E-3 F6(x) -9.33 -7.39 -7.39 -7.45 -10.04 -9.162 -9.162
- 21. Performance Graphs Fig. 4 Fig. 5
- 22. Comparison Results Test Functions GA PSO GSA GSA-m GDE F1(x) 23.13 1.8E-3 7.3E-11 1.99E-17 1.76E-17 F2(x) 1.1E+3 3.6E+4 25.16 4.9E-2 1.75E-4 F3(x) -1.2E+4 -9.8E+3 -2.8E+3 -2.82E+3 -1.045E+4 F4(x) 2.14 9.0E-3 6.9E-6 3.55E-9 3.2E-9 F5(x) 4.0E-3 2.8E-3 8.0E-3 5.4E-3 3.2E-3 F6(x) -7.3421 -9.1120 -9.33 -7.45 -10.04 Table. 3
- 23. Pareto Optimality A well formed Multi-objective problem, there should not be a single solution that simultaneously minimizes each objective to its fullest. In each case we are looking for a solution for which each objective has been optimized to the extent that if we try to optimize it any further, then the other objective(s) will suffer as a result. Finding such a solution, and quantifying how much better this solution is compared to other such solutions (there will generally be many) is the goal when setting up and solving a Multi-objective optimization problem.
- 24. Types of Domination Given two decision or solution vectors x and y, we say that decision vector x weakly dominates (or simply dominates) the decision vector y (denoted by x y) if and only if fi(x) fi(y)∀ i = 1, ...,M (i.e., the solution x is no worse than y in all objectives) and fi(x) ≺ fi(y) for at least one i ∈ 1, 2, ...,M (i.e., the solution x is strictly better than y in at least one objective). A solution x strongly dominates a solution y (denoted by x ≺ y ), if solution x is strictly better than solution y in all M objectives.
- 25. Domination Fig. 7 Strict & weakly domination
- 26. Pareto Front Fig. 8 Non-dominated solutions
- 27. Multi-objective Gravitational Optimization (MOGO) Equation (8) is modified to (14) …(8) …(14) Where m is the number of objectives; bestk and abestk are the maximum and minimum fitness value among the solutions for kth objective. Mass of an agent is the summation of the masses in all dimensions of the objective space.
- 28. MOGO 1. Generate initial population and set the parameters 2. Evaluate fitness and add solutions to the archive 3. For I: 0 to max-iteration or stop criteria is reached do 4. Update the G, best and worst of the population 5. Calculate M , F and a for each agent 6. Update velocity and position 7. Evaluate fitness 8. Domination check for the new set of solutions with the solutions in the archive 9. End For 10. Return set of non-dominated set of solutions
- 30. Simulation Plots
- 31. Observations GSA is implemented to optimize gain parameters in PID Controller. GSA provides better gain values than other popular algorithms – PSO and BFO. Proposed algorithm GSA-m produces better results than Classical GSA except F6. Again, proposed algorithm GDE generates better results than Classical GSA as well as GSA-m for all the test functions.
- 32. Observations Results obtained from GSA, new algorithms GSA-m and GDE has been compared with existing popular optimization techniques GA and PSO. In F3, GA and in F5 PSO outperform all the three physics inspired algorithms. But GDE outclasses GA and PSO in all other test functions. Also, results of GDE not far from these popular algorithms. Hence, our new Hybrid Algorithm GDE is very much competitive with the GA and PSO.
- 33. Observations Distribution of non-dominated points is not so uniform in nature in all the cases. As far as the spreads of the Pareto fronts for the benchmark test functions are concerned, our results are well suited except for Deb benchmark function. Unlike in MOPSO approaches, we have no leader selection strategy in our proposed MOGO. This in turn has reduced the computational complexity to a great extent as compared to MOPSO approaches. Hence, proposed MOGO is a novel algorithm and it serves the purpose quite well. It could lead us to a complete new arena with very high possibilities.
- 34. Publications R. Chatterjee and M. N. DAS, “Physics Inspired Optimization Algorithms: Introducing New Hybrid Gravitational Evolution & Gravitational Search Algorithm with mutation”, International Symposium on Devices MEMS Intelligence System Communication 2011, SMU, Sikkim, India, APR 2011. https://www.researchgate.net/publication/259193474_Physics_Inspi red_Optimization_Algorithms_Introducing_New_Hybrid_Gravitati onal_Differential_Evolution_and_Gravitational_Search_Algorithm_ with_mutation?ev=prf_pub
- 35. References Darrell Whitley. A Genetic Algorithm Tutorial. Computer Science Department, Colorado State University, page www.cs.colostate.edu/ genitor/MiscPubs/tutorial.pdf. David E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning.Addison-Wesley Longman Publishing Co. Inc, 1 edition, 1989. E. Rashedi. Gravitational Search Algorithm. M.Sc Thesis, Shahid Bahonar University of Kerman, 2007. Esmat Rashedi, Hossein Nezamabadi-pour and Saeid Saryazdi. GSA: A Gravitational Search Algorithm. Information Sciences, 179:2232–2248,2009. H. Nezamabadi-pour, S. Saryazdi and E. Rashedi. Edge detection using Ant Algorithm, . Soft Computing, 10:623–628,2006. Hai Shen, Yunlong Zhu, Xiaoming Zhou, Haifenf Gho and Chuanguang Chang. Bacterial Foraging Optimization Algorithm with Particle Swarm Optimization Strategy for Global Numerical Optimization. In Proceeding GEC '09 Proceedings of the rst ACM/SIGEVO Summit on Genetic and Evolutionary Computation.
- 36. References Hui Liu, Zixing Cai and Yong Wang. Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization.Applied Soft Computing, 10. James Kennedy and Russell Eberhart . Particle Swarm Optimization. In proceedings of the IEEE, International Conference on Neural Network, pages 1942–1948,1995. Wael Mansour Korani. Bacterial foraging oriented by particle swarm optimization strategy for PID tuning. GECCO '08 Conference on Genetic and evolutionary computation,ACM , 2008. Xiaohui Hu, Russell C. Eberhert and Yuhui Shi. Particle Swarm with Extended Memory for Multiobjective Optimization. In IEEE Swarm Intelligence Symposium, pages 193–197,2003. Yuhui Shi and Russell C. Eberhert. Empirical study of Particle Swarm Optimization. Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress , 2009.
- 37. References Liping Xie, Jianchao Zeng and Zhihua Cui. General framework of Artificial Physics Optimization Algorithm. Nature & Biologically Inspired Computing (NaBIC 2009), pages pp 1321–1326,2009. N. C. Jagan . Control System. B S Publication , 2 edition, 2008. R. A. Formato. Central Force Optimization: A New Metaheuristic with Applications in Applied Electromagnetic. Progress In Electromagnetic Research, PIER,77:425–491,2007. Rainer Storn and Kenneth Price. Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization archive, 11 Issue 4, 1997.
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