Overview. Optimization. Easy optimization problems. Monte Carlo for Optimization. 1. Survey MC ideas for optimization: (a) Multistart
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1 Monte Carlo for Optimization Overview 1 Survey MC ideas for optimization: (a) Multistart Art Owen, Lingyu Chen, Jorge Picazo (b) Stochastic approximation (c) Simulated annealing Stanford University Intel Corporation (e) Genetic algorithms (f) Ant colony methods (d) Tabu search Describe theses of Chen and Picazo 1 Optimization Find best value of some Best means: minimize or maximize a computable cost or utility is a (possibly long) list of variables under our control Predicted profit credit policy Easy optimization problems Lift/Drag Investment return Travel distance variables describing wing portfolio order in which cities visited Easy to optimize smooth differentiable unimodal functions Repeated quadratic approximations (Newton and variants) Extends to high dimensional (easy) problems 3 4
2 Newton s results on easy function e e e e e e e e e multimodal not quite smooth Hard problems 3 measured with noise 4 is where for missing random 5 combinatorial, eg all circuits through cities 5 6 A harder optimization Several or many local optima 15 0 Minima can be twisty passages in high dimensions End point depends on start point Another hard optimization Derivatives may be unreliable a nearest neighbor selection, a secondary iteration, an adaptive grid, etc Maybe contains:
3 10 0 $ 3 Multistart Repeat Newton-like algorithm times:! "! "! # # # " $ %! &! ' Keep best $ %! &! 1 Start with random, or, a grid if low dimensional Domains of attraction Many! converge to same $ %! &! Need a good $ %! &! with a large domain of attraction ( ) Seen best * + * is prob of starting in best domain of attraction Pure random search Multistart, with no Newton-like optimization Pick! at random, record!, keep the best one Can beat multistart on rough functions Non Monte Carlo: Hooke and Jeeves Nelder-Mead simplex Low discrepancy search Pattern search 9 10 Another hard optimization might be measurable only with noise Eg: controlling temp with noisy sensors, or missing random with criterion, Optimizing expectations Often we can simulate- (make our own noise) Optimize $, / $ for very large0 (by eg Newton s) " " " # # # Wasteful: too much work at suboptimal s 0 Alternatives: use more iterations, smaller 11 1
4 / K Kiefer-Wolfowitz (195) For and random- minimize, - Suppose* : 7 $ 7 / $ $ 8 $ - $ 9 ; $ 8 $ - $ : 8 $ Estimate slope at $ for random- Better yet use common- $ 8 $ - $ ; $ 8 $ - $ 8 $ Step in downhill direction $ 9 $ < $ 7 / $ Kiefer-Wolfowitz, ctd Search distance: 8 $ Move distance: < $ 7 / $ < $ > < $ "? 8 $ > 8 $ " $ < < $ 8 $ $ < $ B 8 $ Eg 8 $ 8 B 0 C < $ < B 0 $ " Local optimum D $ D E 0 : C F for examples above Note: search distance8 $ move distance= < $ Simulated annealing Emulating nature Some natural systems approximately solve hard optimizations: 1 Cooling metals approach minimum energy Plants and animals evolve 3 Ants communicate to find food These provide fruitful paradigms, Starting as proposed panaceas, Evolving to niches Slowly cooling metals anneal, finding near minimum energy arrangement of many atoms, eg F atoms Boltzman distribution ( ) = G H I J K L M N Energy of system (Hamiltonian) M Temperature L M Boltzmann s constant M Low O small K strongly favored High O small K weakly favored 15 16
5 K Simulated annealing 1 Pick an energy K to match (a) K P, or, (b) K, or etc Sample M $ Q G H I K B L M $ 3 Where $ decreases slowly to zero (a) eg cooling every M samples (b) or continously like $ = B 0 At step : Metropolis-Hastings At $ propose random $ $ 9 ST R $ with probu $ " $ $ with prob U $ " $ U $ " $ V WX G H I $ K $ B L M Accept energy reductions, and some energy increases (to get out of local minima) Travelling salesman problem Find the shortest (or a short) round trip passing through each of0 cities Skill is in proposing a good $ for $ Often $ is a neighbor of $ If $ isu " U " # # # " U $ " U Then $ may be: 1 U,U, # # #,U! :, then U Y,U Y :, # # #,U! (running backwards), then 3 U Y 9,U Y 9, # # #,U $,U Tabu search Some random searches tend to cycle Tabu list contains forbidden moves to inhibit cycling 1 Pick, set L, start tabu list M [ Randomly pick\ neighbors of : ] M 3 Disqualify any neighbors in tabu list 4 Find best non-tabu neighbor 5 If L A then 9 6 L M, update, goto Customization details: M Termination rules, picking neighbors, describing Tabu ctd Suppose is Extend credit if variable has been for is predicted value using the last F days, and# # # F # # # Bank Balance # # # Bank Balance over # # # Card debt under # # # F Bank Balance over ; # # # ] Bank Balance over # # # Tabu lists 1 Recently tried s are tabu (eg last^ ), or, Recent changes can t be undone (eg forbid: " ; " ), or, 3 Recently changed variables can t be changed, or, 4 unless tabu violation yields really good
6 9 9 Genetic algorithms 1 Encode in binary ^ " " Define fitness_ related to 3 Initial random L population ] 4 Generation : (a) Sample ` with prob= _ ` (b) Randomly marry pairs of (c) Randomly crossover D D " D D (d) Randomly mutate each bit (with very small probability) Genetic variations 1 Non-binary encodings Replace single strand by: (a) paired strands, with dominant and recessive traits (b) multiple (paired) strands, analogs of chromosomes 3 Replace pairing by analogs of natural and agricultural practices 4 Dynamically vary the fitness function Strength of GAs Works in parallel Population evolves to higher fitness Survivors have solved some problem Offspring may inherit solutions from both parents 1 Ant colony heuristic Model from nature: Ants work together to find food source communicate through pheremone left on paths For TSP: 1 Place\ ants at random in graph Each ant chooses a random next city higher probability on nearby cities 3 Ants add to pheremone level on each arc they use 4 At end of tour they add more pheremone to the path they took, inversely proportional to the length of their tour Features Still very new Parallel, like GA s, but communicating a few parameters to tweak artificial ants can use lookahead, memory, calculus competitive results reported for TSP Starting points Stochastic approximation: Kushner and Yin Stochastic Approximation Algorithms and Applications, 1997 Simulated Annealing: Numerical Recipes, Laarhoven and Aarts: Simulated Annealing: Theory and Applications Tabu Search: F Glover Annals of Operations Research vol 41, 1993 Genetic algorithms: Goldberg Genetic Algorithms in Search, Optimization, and Machine Learning, 1989 Falkenauer Genetic Algorithms and Grouping Problems,1998 Ant colony heuristic: Dorigo, Caro, Sambardella: Ant Algorithms for Discrete Optimization, Artificial Life, Vol 5, No 3,
7 Lingyu Chen s work Optimize, a - for a with random- By stochastic optimization Applied to portfolio allocation allowing for fund loads, taxes Citing: Spall, Kushner, Yin, Breiman, Cover Jorge Picazo s work Simulate multivariable American options Like Longstaff and Schwartz But using classification instead of regression By boosted stumps Also citing: Freund, Shapire, Friedman, Hastie, Tibshirani, Broadie, Glasserman, Fu, Laprise, Madan, Su, Wu Contact Art Owen owen@statstanfordedu Lingyu Chen LingyuChen@intelcom Jorge Picazo jpicazo@statstanfordedu 5 6
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