Simulated Annealing Methods

1. Simulated Annealing Methods Matthew Kelly April 12, 2011

2. What is Annealing? • Slow cooling of a heated substance allows atoms to line themselves up, creating a stronger structure with minimum energy • Accelerating the cooling process produces a structure with more energy and fewer atoms optimally aligned.

3. How Does Simulated Annealing Related to Monte Carlo • If system is “cooled quickly”, we produce a local but not global minimum • Allowing system to find minimum takes more time but will result in global minimum, i.e. structure with minimum energy • Can be used to solve Travelling Salesman problem, which is NP-Hard • Has many local minima

4. Performing Simulated Annealing • Use Metropolis algorithm • f – objective function • x – system state • T – variable with annealing schedule (e.g. temperature), which is gradually reduced • ∆x – random step procedure • Should be consistently efficient, even if in a narrow valley or a minimum is being approached

5. Simulated Annealing as Applied to Travelling Salesman • Number cities 1 to n • Build a path by either: • Removing a link between two cities and reverse their order of traversal • Remove a link between two cities and add a link between two and place it between two other randomly chosen cities • Assign where point n+1 is point 1 • Generate random arrangements to get range of ΔE. • Choose initial T that is much larger than the largest ΔE. • Iteratively decrease T by 10% after holding it constant 100*N configurations or 10*N successful configurations, whichever comes first.

6. A Caveat • To show the robustness of using this procedure, add a penalty for “crossing the Mississippi”, i.e. a line to separate the points into two sets. • Each point has an attributed value μ with points left of line having μ = -1 and points right of line having μ = +1. • Modify the equation to enforce a penalty for crossing this line No Penalty Penalty Enforced Bonus For Crossing

7. Simulated Annealing for Continuous Minimization • Use Downhill Simplex Method • Store last D-1 vertices where D is the Dimensionality of the function • Move via reflections, expansions and contractions • First add a log. distributed random var prop. to T • Then subtract a log. distrib. rand var prop. to T • Determine region reachable at a large T • Perform stochastic tumbling Brownian (random and erratic) motion • Slowly reduce T and Repeat • Results in region exploration independent of narrowness and orientation • Like Metropolis, always accepts true downhill step and sometimes accepts uphill

8. How Slow Should T Be Decreased? • Various methods exist: • Ti+1= (1-ϵ)T after m moves. Determine ϵ /m experimentally • Ti+1=T0(1-k/K)α, k=# moves so far, α is a constant depending on data distribution of relative minima • After m moves, T=β(f1-fb) where β is constant, f1 smallest function in simplex, fbis best function ever. Never reduce T by more than some fraction of time γ

9. Advantages of this Method • Not easily fooled by a quick payoff from falling into an unfavorable local minima • Changes that cause large energy differences are sifted over when T is large. • e.g. In travelling salesman, river is cross minimum number of times (2) when T is large due to the large cost (λ) with neglect to correctness of the paths over the weight but are again considered when T is reduced. • Well-correlated with thermodynamics, so has practical use.

10. Disadvantages of this Method • Degree to which T should be reduced is not clearly defined, 3 potential method are given, all of which don’t work 100% of time • Claim of “effectively solving the traveling salesman problem” does not change the fact that it is an NP problem that would take a long time to resolve. Removing the time characteristic from the method makes attribution to NP (solvable in nondeterministic polynomial time) meaningless. • Scheme of restarting back to a best ever point appears to violate the Metropolis trait of always being able to reach the global minimum regardless of the starting point.