Simulated Annealing Methods Matthew Kelly April 12, 2011

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.

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

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

Simulated Annealing as Applied to Travelling Salesman 1.Number cities 1 to n 2.Build a path by either: 1.Removing a link between two cities and reverse their order of traversal 2.Remove a link between two cities and add a link between two and place it between two other randomly chosen cities 3.Assign where point n+1 is point 1 4.Generate random arrangements to get range of ΔE. 5.Choose initial T that is much larger than the largest ΔE. 6.Iteratively decrease T by 10% after holding it constant 100*N configurations or 10*N successful configurations, whichever comes first.

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

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

How Slow Should T Be Decreased? Various methods exist: 1. T i+1 = (1- ϵ) T after m moves. Determine ϵ / m experimentally 2. T i+1 = T 0 (1-k/K) α, k=# moves so far, α is a constant depending on data distribution of relative minima 3. After m moves, T =β(f 1 -f b ) where β is constant, f 1 smallest function in simplex, f b is best function ever. Never reduce T by more than some fraction of time γ

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.

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.