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The ACO Metaheuristic

The ACO Metaheuristic. ACO 2.1 - 2.3 January 2008 C. Colson. Ants in Mythology. Zeus turned the hardworking ants of the uninhabited island of Aegina into the subjects of Aeacus. The people were called the Myrmidons. Pēleús (son of Aeacus) was the father of Achilles

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The ACO Metaheuristic

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  1. The ACO Metaheuristic ACO 2.1 - 2.3 January 2008 C. Colson

  2. Ants in Mythology • Zeus turned the hardworking ants of the uninhabited island of Aegina into the subjects of Aeacus. The people were called the Myrmidons. • Pēleús (son of Aeacus) was the father of Achilles • Although Achilles was from Thessaly, he leads his distant kinsmen, the Myrmidons in the Trojan War as told by Homer in the Iliad. • Of the Myrmidons: “Industry, thrift, endurance; they are eager for gain, and never easily relinquish, what they have won!"

  3. Metaheuristic? • Refers to a master strategy that modifies sub-heuristics. • A general purpose heuristic designed to guide an underlying problem-specific heuristic towards promising solutions. • NP-hard problems are believed to be unsolvable in polynomial time (tractable and intractable problems). • As a result, heuristic methods are applied to get near-optimal results in a reasonable time. • Metaheuristics have entered the picture to guide “heuristic methods” applicable to widely varying problems. •    ACO is a metaheuristic framework: applicable to many different applications.

  4. Combinational Optimization • Involve finding values for discrete variables such that an optimal solution with respect to a given objective function is found. • Traveling salesman • Supply-chain logistics planning • Asset allocation • Maximize or Minimize applications: П(S, f, Ω) • Problem: general question to be answered • Instance: a case of specified values for a problem • S: set of candidate solutions • f(s): objective function (s Є S) • Ω: constraints • Feasible solutions: Ŝ subset of S, that satisfies Ω • Globally optimal solution: s* Є Ŝ • Minimization example: f(s*) ≤ f(s) for all {s Є Ŝ} • Maximization example: f(s*) ≥ f(s) for all {s Є Ŝ}

  5. Computational Complexity • Straight-forward approach to solution: exhaustive search! • The possible solutions grows exponentially with instance size (n). • Time complexity: maximum time an algorithm needs to find a solution to an instance of n-size (aka worst-time complexity). • “Big-O” formal notation: Ŏ(function) • A function g(n) is said to be Ŏ( h(n) ), if two positive constants A and n0 exist such that g(n) ≤ Ah(n) for all n≥n0 . This is the asymptotic upper bound. • Polynomial time complex: if Ŏ( g(n) ) where g(n) is a polynomial. If k is the largest exponent of the polynomial g(n), then the problem is said to be solvable in Ŏ( nk ) time. • Exponential time complex: if Ŏ( g(n) ) cannot be bounded by a polynomial. • Intractable = not polynomial time complex.

  6. NP-hard??? NP-Completeness (Bottom of page 28) • P-class: an algorithm that outputs the correct answer (“yes” or “no) in poly-time. • NP-class (stands for Non-deterministic Polynomial): an algorithm that verifies every instance is indeed “yes” in poly-time. • P is a subset of NP • Poly-time reductions: the transformation of one problem into another one by a poly-time algorithm. • Key point: if the resultant problem is solvable in poly-time, then the original problem is likewise solvable in poly-time! NP Problems P = NP-complete ??? P Problems NP-complete Problems

  7. Two more classifications of algorithms: • Exact : • guaranteed to find optimal solution • prove optimality for every (finite-size) instance • runs within instance-dependent time (worst case scenario: exponential-time) • Approximate (trades optimality for speed/efficiency) heuristic methods; seeks near-optimal solutions but cannot guarantee optimality. • Further classified into constructive and local searches • Constructive (iteratively add solution components to the empty solution set until solution set is found): incremental solutions without backtracking (see NNH for TSP problem on pg. 30) • Local search (starts from a full solution set and tries to make improvements by local changes): iterative exploration that seeks to improve the solution with local changes (see best-improvement rule for neighborhood examination scheme on pg. 31) • Neighborhood structure: the set of possible solutions that the algorithm can “move” to from the current solution.

  8. The ACO Metaheuristic • A colony of artificial ants cooperate to find good solutions to difficult discrete optimization problems. • Good solutions are an emergent property of cooperation! • Static problems: all characteristics of the problem are defined once and do not change. • Dynamic problems: characteristics vary according to underlying functions and the optimization must adapt to the changing environment. • See problem definition on pgs. 34 & 35.

  9. ACO (continued)… • Ants build solutions by performing stochasic action on the construction graph which is made up of components (nodes) and connections (paths). • Sometimes the ants find feasible solutions, sometimes not (and that’s ok). • The pheromone trail is coded long-term memory. • Heuristic information: additional information that the ants have, a priori, from a source other than the environment (example: estimated path cost) • Although ants act concurrently, independently, and most times dumbly, good–quality solutions arise from collective interaction of the ants.

  10. ACO Components • ConstructAntsSolutions: • manages ants concurrently • stochastic engine resides here • evaluation function for ant performance resides here • UpdatePheromones: • deposits or evaporates pheromone trails. • DaemonActions: • centralized actions not resident in individual ants • optional functionality • example: pheromone bonuses for shortest path yet • See page 38 for pseudocode.

  11. ACO Applications • Hamiltonian circuit: a trip solution on an “undirected” graph which visits each city (node) exactly once and returns to the starting city. • Symmetric TSP: • only paths have “cost” • shortest Hamiltonian circuit trip length • “cost” from node i to j is identical to j to i. • Asymmetric traveling salesman problem • “cost” from node i to j is not identical to j to i. • Sequential Ordering Problem: similar to TSP • an asymmetric traveling salesman problem with additional constraints • doesn’t need to return to starting city • a precedence constraint must be considered • the precedence constraint requires that some node i has to be visited before some other node j • Note: pheromones play roughly the same role in these cases.

  12. ACO Applications (continued…) • Generalized Assignment Problem: • a number of agents and a number of tasks • any agent can be assigned to perform any task, but incurs some cost/profit that varies with the assignment • each agent has a budget and the sum of the costs of task assigned (cannot exceed its budget). • solution is an assignment in which all agents do not exceed their budget • good solution minimizes cost or maximizes “profit” • pheromones associated with either the next task to choose or which agent to assign the task to. • Multiple Knapsack Problem: • goal is to maximize valuable items that can fit into one bag to be carried on a trip • given a set of items, each with a cost and a value, determine the number of each item to include in a collection so that the total cost is less than a given limit and the total value is as large as possible • pheromones are associated only with the desirability of adding a particular item to a solution

  13. ACO Applications (continued…) • Network Routing Problem: • minimum path costs between pairs of nodes in the network • each “connection” (path) should have multiple pheromone trails associated for each different node destination. • Dynamic TSP: • time is a factor • cities can be added or removed from the graph • pheromones act similarly to standard TSP problem.

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