Optimization Problems Greg Stitt ECE Department University of

Optimization Problems Greg Stitt ECE Department University of Florida

Introduction n n Why are we studying optimization problems? Just about every RC-tool related issue is an optimization problem n This lecture should provide abstract solutions to many RC tool problems

Time Complexity n Time Complexity - Informal definition: n n Defines how execution time increases as input size increases We are interested in worst case execution time n Big O Notation n O(n) n Execution time increases linearly with input size O(n 2) n Quadratic growth O(2 n) n Exponential growth

Problem Classes n P n A problem is “in P” if it has a polynomial time solution n n Undecidable n Provably impossible to solve n n n Does not mean “Not-Polynomial”!!!! A NP problem has a non-deterministic polynomial time solution NP-hard n A problem is NP-hard if every problem in NP is reducible to it n n Example: halting problem NP n n O(n), O(n 2), O(n 3), etc. Basically means that NP-hard problems are at least as hard as the hardest problems in NP NP-complete n n NP + NP-hard Most interesting problems are NP-complete!!!! n n Traveling salesman, Subset sum, 0 -1 knapsack, graph coloring, vertex cover Place and route, logic minimization, minimum resource scheduling, hw/sw partitioning, etc.

The Big Question n Does P = NP? n n n Been studied for a long time Never been proven either true or false Why do we care? n Currently, best known solutions for NP-complete problems typically have complexity of O(2 n), O(n!) n n If one NP-complete problem can be solved in polynomial time (is in P), then all NP-complete problems can be solved in polynomial time n n Known as intractable – takes more than your lifetime to solve Remember, many interesting problems are NP-complete If you can find a polynomial time solution to an NPcomplete RC problem: n I’ll give you an A And, I’ll get a million dollars (www. claymath. org/millennium/P_vs_NP/ )

Problem Classes

Optimization Problems n Informal definition n Problem of finding the best solution from all possible solutions n n Possible solution: exhaustive search n n n Check every possible solution, save best one Works, but Many optimization problems are NP-complete n n Not feasible to generate all possible solutions Example: O(2 n) n n Typically involves finding best solution for millions, billions, or even more possibilities n = 5 => 32, n = 10 => 1024, n = 100 => 1. 3 * 1030 Problem: If most optimization problems are intractable, how do we solve them? n Could use solution to any NP-complete problem! n n But, this doesn’t really help n n Remember: any NP-complete problem can be reduced to any other NPcomplete problem There is no known polynomial solution to any NP-complete problem So, what can we do?

Branch and Bound n Another solution: n Branch and bound n n Idea: Try to eliminate many possibilities without evaluating them How it works: n n Imagine a tree representing all possible solutions Algorithm progressively builds tree n n When considering a branch in the tree, determines best possible solution represented by branch n n n Maintains best found solution so far If branch can’t possibly be better than current best, don’t bother to explore possible solutions “Prunes” solution space Result n + Very often, can eliminate large percentage of possible solution n + Usually, much faster than exhaustive search - But, still has exponential complexity n n + Still finds optimal solution O(2 n) - Still not feasible for large input sizes

Heuristics n Another solution: n n Forget about trying to find the best solution Focus on finding a “good” solution quickly n n Known as heuristics Good candidates for heuristics n Map problem to other NP-complete problem, use heuristic for that problem n n Source of way too many publications Use generalized optimization problem heuristic n n Due to large number of interesting optimization problems, research has introduced general heuristics that apply to all optimization problems Examples: n n n Hill Climbing Simulated Annealing Genetic Algorithms

Hill Climbing n Background: Graph of solution space n n x axis = possible solutions y axis = quality of solution n Informal description: n n n 1) Choose a solution s (usually randomly) 2) Find neighboring solution n (i. e. change 1 thing) 3) If n better than s (climbs the hill) n n s = n, Repeat from 2) 4) If all neighboring solutions worse than s n n Height of solution represents goodness Peak represents best solution s is answer (s is a peak) Example: Traveling Salesman n 1) Find a solution 2) Swap 2 cities, if improves solution keep, if not reject 3) repeat 2) until no improvement can be found

Hill Climbing n Advantages: n n Very fast, works well for certain problems Disadvantages: n What if there are multiple peaks? n n n Hill climbing gets stuck at all peaks, known as local maxima Optimal solution is highest peak – global maximum May result in extremely suboptimal solution if many peaks

Simulated Annealing n Problem: Hill climbing suffered from local peaks n n n Need way of escaping local peaks => Have to accept some bad moves Solution: Simulated annealing n n Annealing is process of heating and cooling metals in order to improve strength Idea: Controlled heating and cooling of metal n n When hot, atoms move around When cooled, atoms find configuration with lower internal energy n n i. e. makes metal stronger Analogy: n Temperature = probability of accepting worse neighboring solution n n When temperature is high, likely to accept worse neighboring solutions (but may lead to better overall solution) n Analogous to atoms wandering around Cooling represents shrinking probability of accepting worse solutions

Simulated Annealing n Informal description n n 1) Set initial temperature and probability p 2) Find initial solution s 3) Find neighboring solution n 4) If n better than s n n 5) If n worse than s n n s = n (always accept better solutions) Accept n with probability p 6) Reduce “temperature” and p by some amount 7) If final temperature not reached, repeat from 3) 8) If final temperature reached, report best solution found

Simulated Annealing n Advantages n n Can find near optimal solutions – escapes local maxima Disadvantages n Takes a long time to find near optimal solution n n But, still fast compared to algorithms Very sensitive to input parameters – must be configured well to get good results n How long to run? n n n Initial temperature/final temperature What should initial probability be? Cooling schedule n How much to reduce temperature and probability at each step?

Genetic Algorithms n Another possible solution: n n Imitate evolution – survival of the fittest Assumption: evolution produces “better” humans/animals n n Implies genetic processes must work well Genetic Algorithms n n Generates a “population” of solutions Selection chooses members of population (solutions) to survive by probability based on fitness function (i. e. natural selection) n n Reproduction combines attributes of selected population n Bad solutions less likely to “survive” Crossover/Inheritence – Combines traits for each parent (solution) Mutation – Random change to characteristic Repeat until solution has “evolved” to acceptable lavel

Genetic Algorithms n Advantages n n Can find near optimal solutions Disadvantages n Takes a long time to find near optimal solution n n Very sensitive to input parameters – must be configured well to get good results n n n But, still fast compared to algorithms How large should population be? How does reproduction occur? How many generations should be considered? How much of each generations should be killed off? Same advantages/disadvantages as simulated annealing

Other Heuristics n n Ant colony optimization Tabu search Stochastic optimization Many others

Summary n n Optimizations problems search huge solution space for best solution Many optimization problems are NP-complete n n n Heuristics must be used n n Find a “good” solution in reasonable time General optimization problem heuristics n n No known efficient solution Branch and bound helps a little Hill climbing – fast, but gets stuck at local maxima Simulated annealing – works well but sensitive to parameters Genetic algorithms - work well but sensitive to parameters Why should you care about any of this? n n Most RC tool problems are NP-complete optimization problems You now know how to solve almost all tool problems n You can use branch and bound, hill climbing, simulated annealing, genetic algorithms, ant colony optimization, tabu search, etc.