Iterative improvement algorithmsIterative improvement algorithms = iterative refinement = local searchHill-climbing SearchHill-climbing Search…Slide 5E.g. 2-swap in TSPSimulated AnnealingSimulated Annealing…Heuristic RepairHeuristic Repair (cont.)Iterative improvement algorithmsProf. Tuomas SandholmCarnegie Mellon UniversityComputer Science DepartmentIterative improvement algorithms= iterative refinement = local searchUsable when the solution are states, not paths.Start with a complete configuration and make modifications to improve its quality.Hill-climbing SearchIterative improvement algorithms try to find peaks on a surface of states where height is defined by the evaluation functionHill-climbing Search…function HILL-CLIMBING(problem) returns a solution state inputs: problem, a problem static: current, a node next, a node current MAKE-NODE(INITIAL-STATE[problem]) loop do next a highest-valued successor of current if VALUE[next] < VALUE[current] then return current current next endBest-swap vs. first-swapHill-climbing Search…Problems:1. Local maxima•No progress2. Plateaux (essentially flat evaluation fn)•Random walk3. Ridges•Search may oscillate from side to side, making little progressPotential solutions: random restarts•Eventually finds the optimal solution•On NP-complete problems it is likely that there are exponentially many local optimaUsually good solutions can be found quickly.Performance depends on the “state-space surface”.How to find feasible neighbors?E.g. 2-swap in TSP3-swaps…Simulated AnnealingSimulated Annealing…function SIMULATED-ANNEALING(problem,schedule) returns a solution state inputs: problem, a problem schedule, a mapping from time to “temperature” static: current, a node next, a node T, a “temperature” controlling the probability of downward steps current MAKE-NODE(INITIAL-STATE[problem]) for t 1 to do T schedule[t] if T=0 then return current next a randomly selected successor of current E VALUE[next] – VALUE[current] if E > 0 then current next else current next only with probability e E/TDoes not always find an optimal solution, and does not know whether it has found an optimal solution.[Theoretical results show that if T is lowered slowly enough (extremely slowly), the optimum will be found]Heuristic RepairIterative improvement in CSPs called heuristic repair.Min-conflicts heuristic: choose a value that results in a minimum number of conflicts with other variables.Heuristic Repair (cont.)A two-step solution for an 8-queen problem using min-conflicts. At each stage, a queen is chosen for reassignment in its column. The number of conflicts (in this case, the number of attacking queens) is shown in each square. The algorithm moves the queen to the min-conflict square, breaking ties randomly.Surprisingly effective- 106 queens in 50 steps on average- Hubble space telescope scheduling (3 weeks 10 minutes for scheduling a week of
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