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MITESD_77S10_lec10 (1)

# MITESD_77S10_lec10 (1) - Simulated Annealing A Basic...

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1 Simulated Annealing A Basic Introduction Olivier de Weck, Ph.D. Massachusetts Institute of Technology Lecture 10

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2 Outline • Heuristics • Background in Statistical Mechanics – Atom Configuration Problem – Metropolis Algorithm • Simulated Annealing Algorithm • Sample Problems and Applications • Summary
3 Learning Objectives • Review background in Statistical Mechanics : configuration, ensemble, entropy, heat capacity • Understand the basic assumptions and steps in Simulated Annealing (SA) • Be able to transform design problems into a combinatorial optimization problem suitable to SA • Understand strengths and weaknesses of SA

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4 Heuristics
5 What is a Heuristic? A Heuristic is simply a rule of thumb that hopefully will find a good answer. Why use a Heuristic? Heuristics are typically used to solve complex (large, nonlinear, non- convex (i.e. contain local minima)) multivariate combinatorial optimization problems that are difficult to solve to optimality. Unlike gradient-based methods in a convex design space, heuristics are NOT guaranteed to find the true global optimal solution in a single objective problem, but should find many good solutions (the mathematician 's answer vs. the engineer ’s answer) Heuristics are good at dealing with local optima without getting stuck in them while searching for the global optimum.

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6 Types of Heuristics Heuristics Often Incorporate Randomization Two Special Cases of Heuristics Construction Methods • Must first find a feasible solution and then improve it. Improvement Methods Start with a feasible solution and just try to improve it. 3 Most Common Heuristic Techniques Simulated Annealing – Genetic Algorithms – Tabu Search – New Methods: Particle Swarm Optimization, etc…
7 Origin of Simulated Annealing (SA) Definition : A heuristic technique that mathematically mirrors the cooling of a set of atoms to a state of minimum energy. Origin: Applying the field of Statistical Mechanics to the field of Combinatorial Optimization (1983) Draws an analogy between the cooling of a material (search for minimum energy state) and the solving of an optimization problem. Original Paper Introducing the Concept Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by Simulated Annealing,” Science , Volume 220, Number 4598, 13 May 1983, pp. 671- 680.

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8 MATLAB® “peaks” function • Difficult due to plateau at z=0, local maxima • SAdemo1 • x o =[-2,-2] • Optimum at x*=[ 0.012, 1.524] • z *= 8.0484 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x Peaks y Start Point Maximum
9 “peaks” convergence • Initially ~ nearly random search • Later ~ gradient search 0 50 100 150 200 250 300 -10 -8 -6 -4 -2 0 2 Iteration Number System Energy SA convergence history current configuration new best configuration

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10 Statistical Mechanics
11 The Analogy Statistical Mechanics: The behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature.

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MITESD_77S10_lec10 (1) - Simulated Annealing A Basic...

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