Continuous Equality Knapsack with Probit-Style Objectives

TL;DR

This paper investigates a class of continuous equality knapsack problems with non-convex, antisymmetric objectives, providing structural insights and two efficient algorithms for their optimization.

Contribution

It introduces a novel model with non-convex objectives, proves structural properties, and develops two algorithms for fast optimization under general assumptions.

Findings

Structural properties of the model are characterized.

Two algorithms for efficient optimization are proposed.

Algorithms achieve linear time and constant operations with preprocessing.

Abstract

We study continuous, equality knapsack problems with uniform separable, non-convex objective functions that are continuous, antisymmetric about a point, and have concave and convex regions. For example, this model captures a simple allocation problem with the goal of optimizing an expected value where the objective is a sum of cumulative distribution functions of identically distributed normal distributions (i.e., a sum of inverse probit functions). We prove structural results of this model under general assumptions and provide two algorithms for efficient optimization: (1) running in linear time and (2) running in a constant number of operations given preprocessing of the objective function.