Ordinal optimization for a class of deterministic and stochastic discrete resource allocation problems

Abstract

The authors consider a class of discrete resource allocation problems which are hard due to the combinatorial explosion of the feasible allocation search space. In addition, if no closed-form expressions are available for the cost function of interest, one needs to evaluate or (for stochastic environments) estimate the cost function through direct online observation or through simulation. For the deterministic version of this class of problems, the authors derive necessary and sufficient conditions for a globally optimal solution and present an online algorithm which they show to yield a global optimum. For the stochastic version, they show that an appropriately modified algorithm, analyzed as a Markov process, converges in probability to the global optimum. An important feature of this algorithm is that it is driven by ordinal estimates of a cost function, i.e., simple comparisons of estimates, rather than their cardinal values. They can therefore exploit the fast convergence properties of ordinal comparisons, as well as eliminate the need for "step size" parameters whose selection is always difficult in optimization schemes. An application to a stochastic discrete resource allocation problem is included, illustrating the main features of their approach.

A survey of recent advances in discrete input parameter discrete-event simulation optimization

“…This goal softening produces a considerable reduction in the computational cost, but the user must be willing to accept some loss of precision. Cassandras et al (1998) examine a class of resource allocation problems that can be formulated as discrete optimization problems. They address the stochastic version of the problem with the use of ordinal estimates to derive an efficient solution scheme.…”

Stochastic approximation on discrete sets using simultaneous difference approximations

“…The problem is to minimize L using only the measurements . The constrained version of this problem assumes that , where the subset Θ of Similar to [6], we restrict our attention to cost functions that satisfy a certain integer convexity condition. For the case p = 1, the function satisfies the inequality :…”

“…An ordinal optimization method for finding the minimum was introduced in [6]. The method discussed here, which was introduced in [7], relies on simultaneous perturbation difference approximations.…”

A new approach to discrete stochastic optimization problems

“…Some of them are specific to a certain subclass of stochastic optimization problems, e.g. approach to solve stochastic resource allocation problems [2] while others are designed for more general problems.…”