Metaheuristics The idea: search the solution space directly. No math models, only a set of algorithmic steps, iterative method. Find a feasible solution and improve it. A greedy solution may be a good starting point. Goal: Find a near optimal solution in a given bounded time. Applied to combinatorial and constraint optimization problems Diversification and intensification of the search are the two strategies for search in Metaheuristics. One must strike a balance between them. Too much of either one will yield poor solutions. Remember that you have only a limited amount of time to search and you are also looking for a good quality solution. Quality vs Time tradeoff. – For applications such as design decisions focus on high quality solutions (take more time) Ex. high cost of investment, and for control/operational decisions where quick and frequent decisions are taken look for good enough solutions (in very limited time) Ex: scheduling – Trajectory based methods are heavy on intensification, while population based methods are heavy on diversification. 1

Metaheuristics The idea: search the solution space directly. No math models, only a set of algorithmic steps, iterative method. Trajectory based methods (single solution based methods) – search process is characterized by a trajectory in the search space – It can be viewed as an evolution of a solution in (discrete) time of a dynamical system Tabu Search, Simulated Annealing, Iterated Local search, variable neighborhood search, guided local search Population based methods – Every step of the search process has a population – a set of- solutions – It can be viewed as an evolution of a set of solutions in (discrete) time of a dynamical system Genetic algorithms, swarm intelligence - ant colony optimization, bee colony optimization, scatter search Hybrid methods Parallel metaheuristics: parallel and distributed computing- independent and cooperative search You will learn these techniques through several examples 2

S-Metaheuristics – single solution based Idea: Improve a single solution These are viewed as walks through neighborhoods in the search space (solution space) The walks are performed via iterative procedures that move from the current solution to the next one Iterative procedure consist of generation and replacement from a single current solution. – Generation phase: A set of candidate solutions C(s) are generated from the current solution s by local transformation. – Replacement phase: Selection of s’ is performed from C(s) such that the obj function f(s’) is better than f(s). The iterative process continues until a stopping criteria is reached The generation and replacement phases may be memoryless. Otherwise some history of the search is stored for further generation of candidate solutions. Key elements: define the neighborhood structure and the initial solution. 3

Neighborhood Representation of solutions – Vector of Binary values – 0/1 Knapsack, 0/1 IP problems – Vector of discrete values- Location, and assignment problems – Vector of continuous values on a real line – continuous, parameter optimization – Permutation – sequencing, scheduling, TSP k-distance – For discrete values distance d(s,s’)<  for continuous values: sphere of radius d around s – For binary vector of size n, 1-distance neighborhood of s will have n neighbors (flip one bit at a time) Ex: hypercube, neighbors of 000 are 100, 010, 001. For permutation based representations – k-exchange (swapping) or k-opt operator (for TSP) Ex: 2-opt: for permutations of size n, the size of the neighborhood is n(n-1)/2. The neighbors of 231 are 321, 213 and 132 – for scheduling problems Insertion operator Exchange operator Inversion operator

Initial solution and objective function Random or greedy Or hybrid In most cases starting with greedy will reduce computational time and yield better solutions, but not always Sometimes random solutions may be infeasible Sometimes expertise is used to generate initial solutions For population-based metaheuristics a combination of greedy and random solutions is a good strategy. Complete vs incremental evaluation of the obj. function 5

Distance and Landscape For binary and flip move operators – For a problem of size n, the search space size is 2 n and the maximum distance of the search space is n (the maximum distance is by flipping all n values). For permutation and exchange move operators – For a problem of size n, the search space size is n! and the maximum distance between 2 permutations is n-1 Landscape – Flat, plain; basin, valley; rugged, plain; rugged, valley 6

Local search Hill climbing (descent), iterative improvement Select an initial solution Selection of the neighbor that improves the solution (obj func) – Best improvement (steepest ascent/descent). Exhaustive exploration of the neighborhood (all possible moves). Pick the best one with the largest improvement. – First improvement (partial exploration of the neighborhood) – Random selection – evaluate a few randomly selected neighbors and select the best among them. Great method if there are not too many local optimas. Issues: search time depends on initial solution and not good if there are many local optimas. 7

Local search Maximize f(x)= x 3 -60x x Use binary search Starting solution = = f (17) = 2774 Find the neighbors for Solution : local optima is = f(16) = 3136 But global optima is = f(10)=

Escaping local optimas Accept nonimproving neighbors – Tabu search and simulated annealing Iterating with different initial solutions – Multistart local search, greedy randomized adaptive search procedure (GRASP), iterative local search Changing the neighborhood – Variable neighborhood search Changing the objective function or the input to the problem in a effort to solve the original problem more effectively. – Guided local search 9

Tabu search – Job-shop Scheduling problems Single machine, n jobs, minimize total weighted tardiness, a job when started must be completed, N-P hard problem, n! solutions Completion time C j Due date d j Processing time p j Weight w j Release date r j Tardiness T j = max (C j -d j, 0) Total weighted tardiness = ∑ w j. T j The value of the best schedule is also called aspiration criterion Tabu list = list the swaps for a fixed number of previous moves (usually between 5 and 9 swaps for large problems), too few will result in cycling and too many may be unduly constrained. Tabu tenure of a move= number of iterations for which a move is forbidden. 10

Tabu Search - Job shop scheduling example Single machine, 4 jobs, minimize total weighted tardiness, a job when started must be completed, N-P hard problem, 4! Solutions Use adjacent pair-wise flip operator Tabu list – last two moves cannot be swapped again. Jobs j 1234 p j d j w j Initial solution 2143 weighted tardiness = 500 Stopping criterion: Stop after a fixed number of moves or after a fixed number of moves show no improvements over the current best solution. 11

Tabu search Static and dynamic size of tabu list Short term memory – stores recent history of solutions to prevent cycling. Not very popular because of high data storage requirements. Medium term memory – Intensification of search around the best found solutions. Only as effectives a the landscape structure. Ex: intensification around a basin is useless. Long term memory – Encourages diversification. Keeps an account of the frequency of solutions (often from the start of the search) and encourages search around the most frequent ones. 12