On the Fastest Vickrey Algorithm

We investigate the algorithmic performance of Vickrey-Clarke-Groves mechanisms in the single item case. We provide a formal definition of a Vickrey algorithm for this framework, and give a number of examples of Vickrey algorithms. We consider three performance criteria, one corresponding to a Pareto criterion, one to worst-case analysis, and one related to first-order stochastic dominance. We show that Pareto best Vickrey algorithms do not exist and that worst-case analysis is of no use in discriminating between Vickrey algorithms. For the case of two bidders, we show that the bisection auction stochastically dominates all Vickrey algorithms. We extend our analysis to the study of weak Vickrey algorithms and winner determina-tion algorithms. For the case of two bidders, we show that the One-Search algorithm stochastically dominates all column monotonic weak Vickrey algorithms and that a suitably adjusted version of the bisection algorithm, the WD bisection algorithm

Valid Inequalities and Restrictions for Stochastic Programming Problems with First Order Stochastic Dominance Constraints

Stochastic dominance relations are well-studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance (SSD) constraints can be solved by linear programming (LP). However, problems involving first order stochastic dominance (FSD) constraints are potentially hard due to the non-convexity of the associated feasible regions. We consider a mixed 0--1 linear programming formulation of a discrete FSD-constrained optimization model and present an LP relaxation based on SSD constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional SSD constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem

Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

Stochastic dominance relations are well-studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance (SSD) constraints can be solved by linear programming (LP). However, problems involving first order stochastic dominance (FSD) constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 01 linear programming formulation of a discrete FSD-constrained optimization model and present an LP relaxation based on SSD constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional SSD-constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a portfolio optimization problem