Advanced Spring School Thermal Measurements & Inverse Techniques Station Biologique de ROSCOFF June 13-18, 2011 Zero order optimization algorithms Tutorial 2 - Emmanuel RUFFIO*, Daniel PETIT, Didier SAURY, Manuel GIRAULT* emmanuel.ruffio@let.ensma.fr, manuel.girault@let.ensma.fr Institut P' (UPR CNRS 3346) CNRS, ENSMA, Université de Poitiers Département fluides, thermique, combustion ENSMA - BP. 40109 1 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011 86961 Futuroscope Chasseneuil
Outline 1) What is an optimization problem? 2) Traditional local search algorithms 3) Metaheuristics 3) Common nature-inspired metaheuristics 5) The Particle Swarm Optimization algorithm 6) Example 7) Conclusion 8) PSO and ES implementations METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011 2
Outline 1) What is an optimization problem? 2) Traditional local search algorithms 3) Metaheuristics 3) Common nature-inspired metaheuristics 5) The Particle Swarm Optimization algorithm 6) Example 7) Conclusion 8) PSO and ES implementations METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011 3
An optimization problem Continuous optimization Multiobjective N T ( ,..., ) R 1 N Multimodale Dynamic Single objective optimization Combinatorial ( J ) R Linear constraints Boundary constraints A . B i L U [ 1 ; N ] i i i Non-linear constraints f ( ) 0 Search space S [ L ; U ] [ L ; U ] Soft constraints 1 1 N N Our optimization problem ˆ arg[min J ( )] S 4 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
What kind of objective function? The simple case Ex: Least squares (Flash method for parameter estimation) A little bit more challenging? Ex: Least median of squares Capacity (J/m 3 /K) Diffusivity (m²/s) 5 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Outline 1) What is an optimization problem? 2) Traditional local search algorithms 3) Metaheuristics 3) Common nature-inspired metaheuristics 5) The Particle Swarm Optimization algorithm 6) Example 7) Conclusion 8) PSO and ES implementation METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011 6
Gradient-based methods (1/2) Objective function Background of gradient-based methods: J ( . D ) J ( ) k k k k Step size Descent direction Unknown parameters In which direction “ D k ” ? The name of a gradient-based algorithm depends on the way D k is computed. 1) First-order method: - Gradient : D J ( ) k k Hessian Matrix 2) Second-order method: 1 - Newton: D [ H ( )] . J ( ) 1 and k k k k 3) Pseudo second-order method: H ( ) - Quasi- Newton method: is approximated (DFP, SR1, BFGS,…) k exp 2 - Levenberg-Marquardt (least-squares): J ( ) ( y y ( ) ) y H i ( ) is approximated using the sensitivity matrix X ( ) ( ) k k k 7 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Gradient-based methods (2/2) Objective function Background of gradient-based methods: J ( . D ) J ( ) k k k k Step size Descent direction Unknown parameters How far in that direction ? What is the step size “α k ” ? Different ways to compute the step size α k : (1) Armijo , (2) Goldstein , (3) Wolfe method or (4) the traditional linear search : arg[min ( J ( . D ))] k 0 k k Function along the gradient direction Gradient 8 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Order and efficiency Iterative procedure: until a convergence criterion is satisfied . D k 1 k k k The Rosenbrock function N 1 2 2 2 J ( 1 ) 10 .( ) i i 1 i i 1 Starting point Iteration Solution First order Second order Pseudo second order Gradient method Newton’s method Quasi- Newton’s method 9 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Simplex methods (1/2) Definition : a simplex is a generalization of a triangle to arbitrary dimension. If k is the number of dimensions, then a simplex is defined by k+1 points in this space. The algorithm : (George Dantzig, 1947) -Deterministic -Derivative-free (zero-order) -Local search 0 Simple but efficient method 1 2 The worst is iteratively replaced by its symmetric with respect to the center of gravity 3 of the other points of the simplex -> Few rules to handle tricky situations A 2D-case J x , y 10 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Simplex methods (2/2) Many variants of the algorithm ! Simple but efficient The simplest one Trickier Sequential fixed size Fixed size Variable size 11 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
What about this one ? The simple case Ex: Least squares (Flash method for parameter estimation) A little bit more challenging? Ex: Least median of squares ? 12 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Outline 1) What is an optimization problem? 2) Traditional local search algorithms 3) Metaheuristics 3) Common nature-inspired metaheuristics 5) The Particle Swarm Optimization algorithm 6) Example 7) Conclusion 8) PSO and ES implementations METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011 13
What is a difficult problem ? A hard problem : The time required using any currently known algorithm increases exponentially with the size of the problem. Here we are… Then, what can we do ? three possibilities: 1. Use algorithms that compute all feasible solutions and keep best one, but it may take exponential time . 2. Use “approximation algorithms” that always run in polynomial time but they may not always produce the optimal solution 3. Find a new job 14 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
A metaheuristic ? (1/3) Heuristic (from the Greek heuriskein: " find " or " discover ” ) : A commonsense rule (or set of rules) based on experience that aids problem solving Examples: • If you are packing odd-shaped items into a box: start with the largest remaining items, fit the smaller items into the spaces left • If you are having difficulty understanding a problem, try drawing a picture . 15 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
A metaheuristic ? (2/3) Heuristic in computer science: An algorithm that generally produces acceptable solutions with a reasonable amount of time in many situations, but: -the solution can be bad -the algorithm can be horribly slow Example: Greedy heuristic: to maximize profit, choose the solution of the neighborhood that maximizes local profit (Hill climbing) 16 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
A metaheuristic ? (3/3) Meta (from Greek for “ beyond ” or “ higher ” ) : a prefix used to indicate a concept which is an abstraction from another concept. Meta-heuristic: No commonly agreed definition ! • Metaheuristics are strategies that guide the search process . • Algorithms that find acceptable solutions of problems by using several heuristics . • A metaheuristic is a heuristic method for solving a very general class of computational problems by combining heuristics in the hope of obtaining a more efficient or more robust procedure. 17 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
Fundamental properties of metaheuristics • Efficiently explore the search space . • May be approximation and stochastic algorithms. • Mechanisms to avoid getting trapped local optima . • Use search experience to be more efficient than random search. Three principles Exploration Exploitation (diversification, global search) (Intensification, local search) Finding areas of the search space Improving the current solutions found by performing generally small changes that are still not investigated Memory (Search experience) Store information 18 METTI V – Advanced Spring School: Thermal Measurements & Inverse Techniques, ROSCOFF, June 13-18, 2011
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