Optimality Conditions For A Nonlinear Stochastic Knapsack Problem

We investigate a nonlinear stochastic knapsack problem with application in Call Admission Control (CAC) with two classes of users, preliminary studied in [1,2]. Among possible stochastic nonlinear generalizations [3.4] of the NP-hard 0/1 knapsack problem, we consider the following model. One has a knapsack of capacity C and K classes of objects. The objects belonging to each class become available randomly. The inter-arrival times are exponentially-distributed with means depending on the class and on the state of the knapsack. The sojourn time of each object is independent of the others and described by a class-dependent distribution. When included in the knapsack, an object from class k generates revenue at a positive rate rk. The occupied portion of knapsack is given by a nonlinear function bk(nk), where, for k=1, . . . K, nk is the number of objects of class k currently inside. The objects can be inserted as long as the sum of their sizes does not exceed the capacity C. The stochastic nonlinear 0,1-programming problem consists in deciding either acceptance or rejection of the arriving objects in dependence of the current state of the knapsack, in such a way to maximize the average revenue. The functions used to generate such decisions are called \policies". We focus on coordinate-convex policies. We provide an algorithm which generates all coordinate-convex policies satisfying three different necessary conditions for optimality. Then we derive exact expressions of the cardinalities of such three sets of policies. Finally, we give conditions under which these cardinalities are significantly smaller than the cardinality of the set of all coordinate-convex policies