A nonlinear minimax allocation problem with multiple knapsack constraints

Abstract

We consider a nonlinear minimax allocation problem with multiple knapsack-type resource constraints. Each term in the objective function is a nonlinear, strictly decreasing and continuous function of a single variable. All variables are continuous and nonnegative. A previous algorithm for such problems repeatedly solves relaxed problems without the nonnegativity constraints. That algorithm is particularly efficient for certain nonlinear functions for which there are closed-form solutions for the relaxed problems; for other functions, however, the algorithm must employ search methods. We present a new algorithm that uses at each iteration simple-to-compute algebraic expressions to check optimality conditions, instead of solving the relaxed minimax problems. The new algorithm is therefore significantly more efficient for more general nonlinear functions.