Global Optimization Simulated Annealing and Tabu Search. Doron Pearl

Transcription

1 Global Optimization Simulated Annealing and Tabu Search Doron Pearl 1

2 A Greedy Approach Iteratively minimize current state, by replacing it by successor state that has lower value. When successor is higher return. 2

3 A Greedy Approach Problem Most Minimization strategies find the nearest local minimum. starting point descend direction local minimum barrier to local search 3 global minimum

4 Bouncing Ball Goal: Find the lowest valley in a terrain. Approach: A bouncing ball. Process 1. At the beginning allow the ball to make high bounces. 2. Slowly decreases the maximum bounce. starting point Will jump barrier descend direction local minimum 4 global minimum

5 Physical Annealing Process Gradual cooling of liquid At high temperatures, molecules move freely At low temperatures, molecules are "stuck if cooling is slow Low energy, organized crystal lattice formed. 5

6 Simulated Annealing A stochastic global optimization method that distinguishes between different local optima. Derived its name from the annealing process used to re-crystallize metals. 6 Simulated annealing is summarized with the following idea: When optimizing a very large and complex system (i.e., a system with many degrees of freedom), instead of always going downhill, try to go downhill most of the times [Haykin, 1999]

7 Algorithm s Sketch 7 Step 1: Initialize Start with a random initial placement. Initialize a very high temperature. Step 2: Move Perturb the placement through a defined move. Step 3: Calculate score calculate the change in the score due to the move made. Step 4: Choose Depending on the change in score, accept or reject the move. The probability of acceptance depending on the current temperature. Step 5: Update and repeat Update the temperature value by lowering the temperature. Go back to Step 2. The process is done until Freezing Point is reached.

8 Basic Algorithm 1. Choose a random X i, select the initial system temperature, and outline the cooling (ie. annealing) schedule. 2. Evaluate E(X i ) using a simulation model 3. Perturb X i to obtain a neighboring Design Vector (X i+1 ) 4. Evaluate E(X i+1 ) using a simulation model 5. If E(X i+1 )< E(X i ), X i+1 is the new current solution 6. If E(X i+1 )> E(X i ), then accept X i+1 as the new current solution with a probability e (-Δ /T) where Δ = E(X i+1 ) -E(X i ). 7. Reduce T according to the cooling schedule. 8. Terminate the algorithm. 8 T = System Temperature, E = Objective function

9 Rosenbrock Function SA s performance on the Rosenbrock function (2D): f = [(1 x) x x ] Search Pattern Objective reduction

10 General SA Software Overview Initial configuration X 0 Options (e.g. annealing schedule) Simulated Annealing Algorithm X best X history Evaluation function e(x) Perturbation function p(x) 10

11 Software: ASA Adaptive Simulated Annealing: C code that statistically finds the best global fit of a nonlinear constrained non-convex cost-function. Has over 100 options to provide robust tuning over many classes of nonlinear stochastic systems. Reannealing - periodically rescaling the annealing-time. ASAMIN Matlab Interface for ASA Reference: Lester Ingber, California Institute of Technology 11

12 Software: parsa parsa Parralel Simulated Annealing Code in C/C++ University of Paderborn, Germany Public, Using MPI, Platform independent 12

13 More Software JSimul Simulated Annealing Code in Java Taygeta Scientific Inc. GOFFE Fortan implementation William Goffe, Barkely California 13 simanneal Simulated Annealing Code in C, C++ and ADA Taygeta Scientific Inc.

14 Tabu Search (Giordano 96, based on Glover 89) General idea: search a space by choosing a point, and going to its best neighbor that is not in the tabu list. Tabu search is stochastic by nature: not guaranteed to always find an optimal solution even if one exists. 14 The Tabu restriction adds memory to local search - prevent cycles / revisiting - increase diversity of exploration - escape from local optima

15 Tabu Restrictions When we would tag a move as a Tabu? Enforce tabu restrictions. Example 1: After a move that changes the value of x i from 0 to 1, we would like to prevent x i from taking the value of 0 in the next p iterations. Example 2: After a move that exchanges the positions of element i and j in a sequence, we would like to prevent elements i and j from exchanging positions in the next p iterations Tabu tenure - 15

16 Basic Tabu Search Algorithm 1. initialize current-solution c to some random solution 2. for x 1 to num-iterations 1. generate and estimate neighbors of c 2. prune neighbors that are in the tabu-list 3. c best of remaining neighbors 4. update tabu-list 3. return c 16 Best neighbor Tabu

17 Tabu Move 17 Best neighbor Tabu

18 Software No off the shelf codes because generally each problem requires particular implementation. Some code framework are available - OpenTS Java Tabu Search. CPL License Metslib meta heuristics framework in C++ - GNU, OO 18 You would still need to add your own implementation (neighborhood generator, solution estimator etc )

19 Probabilistic Tabu Search 1. Instead of evaluating each neighbor consider only a random sample. 2. When 2 or more neighbors evaluated as best move choose by random. Benefit: Reduce significantly the computational burden. Acts as an anti-cycling mechanism. Allows usage of shorter tabu lists. 19

20 TS vs. SA TS gives a better result for average costs 20 In 10 out 15 problems TS found a better (smaller) cost function

21 TS vs. SA For the problem : Cost Value Number of Moves Iterations per move for SA 21 TS always finds better solutions SA needs more iterations as it proceeds

22 TS vs. SA CPU time in seconds (averaged over 10 runs) required by TS and SA to carry out 200,000 moves It takes about half the time for TS 22 Move=A better solution was found SA won t find a move in most iterations

23 TS vs. SA For the problem : Influence of the size of TS list on the cost function: Bad results when the size is too small or too big Influence of the initial temp. of SA on the cost function: 23 SA is sensitive to initial temp.

24 Bibliography Randomized Feature Selection, PRISM Texas A&M University Simulated Annealing, Statistical Computing, University of Michigan Artificial Intelligence Seminar, Adele Howe, Colarado State University Tabu Search for Military Analysis, Ray Hill, Department of Operational Sciences Air Force Institute of Technology 24 Simulated Annealing and Tabu Search for Constraint Solving, Jin- Kao-Hao and Jerome Pannier (1998), Parc Scientifique Georges Besse Nimes, France