On Optimal Allocation of Indivisibles Under Uncertainty

Abstract

The optimal allocation of indivisible resources is formalized as a stochastic optimization problem involving discrete decision variables. A general stochastic search procedure is proposed, which develops the concept of the branch and bound method. The main idea is to process large collections of possible solutions and to devote more attention to the most promising groups. By gathering more information to reduce the uncertainty and by narrowing the search area, the optimal solution can be found with probability one. Special techniques for calculating stochastic lower and upper bounds are discussed. The results are illustrated by a computational experiment. iii iv Contents 1 Models of discrete stochastic optimization 2 The stochastic branch and bound method 2.1 Outline of the method : :

S-ACO: An Ant-Based Approach to Combinatorial Optimization Under Uncertainty

“…It should be mentioned that, contrary to its deterministic counterpart, problem (1) is typically nontrivial already for a very small number |S| of feasible solutions: Even for |S| = 2, except when F (x) can be computed directly, a nontrivial statistical hypothesis testing problem is obtained (see [18]). …”

“…Traditional methods for optimization under uncertainty such as Stochastic Approximation or the Response Surface Method are not well-suited for an application in the context of combinatorial optimization. Problems that are both stochastic and combinatorial can, however, be treated by Sample Average Approximation [16], Variable-Sample Random Search Methods [14], the Stochastic Branch-and-Bound Method [18], the Stochastic Ruler Method [2], or the Nested Partition Method [19]. As approaches drawing from metaheuristic algorithms ideas, we mention Stochastic Simulated Annealing ( [12], [1]) or Genetic Algorithm for Noisy Functions [9].…”

A Lagrangean decomposition approach for oil supply chain investment planning under uncertainty with risk considerations

“…These approaches either try to adapt the L-Shaped algorithm into the context of stochastic integer programming problems through the use of convexification techniques for the second-stage problem (see, for example Laporte and Louveaux [17], Sherali and Fraticelli [30], Zheng et al [34]), enumerative branch-and-bound strategies (see for example Carøe and Schultz [7], Norkin et al [23], Ahmed et al [2]), or else apply dual decomposition methods by means of Lagrangean decomposition approaches [7]. In this case, the problem is decomposed into scenario subproblems through the relaxation of non-anticipativity constraints (NAC) and the solution strategy relies on finding the optimal dual multipliers.…”

Convex approximations for a class of mixed-integer recourse models

“…1 Structural properties of mixed-integer recourse models were studied in Schultz (1993Schultz ( , 1995, and first algorithms are described in Carøe and Tind (1997), Laporte and Louveaux (1993), Norkin et al (1998). Surveys of properties, algorithms, and applications for (mixed-)integer recourse models can be found in Klein Haneveld and Van der Vlerk (1999), Louveaux and Schultz (2003), Van der Vlerk (1997, 2003).…”