Solving 0-1 Integer Programs with Unknown Knapsack Constraints Using Membership Oracles

TL;DR

This paper introduces a novel framework for solving 0-1 integer programs with unknown knapsack constraints by learning surrogate linear constraints through active learning, improving sampling strategies and separation methods.

Contribution

It proposes an innovative active learning approach with new sampling and separation techniques to efficiently solve problems with unknown constraints.

Findings

Mixed-integer quadratic programming improves sampling.

Linear separation inspired by convex optimization enhances learning.

Experimental results show better solution quality and efficiency.

Abstract

We consider solving a combinatorial optimization problem with unknown knapsack constraints using a membership oracle for each unknown constraint such that, given a solution, the oracle determines whether the constraint is satisfied or not with absolute certainty. The goal of the decision maker is to find the best possible solution subject to a budget on the number of oracle calls. Inspired by active learning for binary classification based on Support Vector Machines (SVMs), we devise a framework to solve the problem by learning and exploiting surrogate linear constraints. The framework includes training linear separators on the labeled points and selecting new points to be labeled, which is achieved by applying a sampling strategy and solving a 0-1 integer linear program. Following the active learning literature, a natural choice would be SVM as a linear classifier and the information-based sampling strategy known as simple margin, for each unknown constraint. We improve on both sides: we propose an alternative sampling strategy based on mixed-integer quadratic programming and a linear separation method inspired by an algorithm for convex optimization in the oracle model. We conduct experiments on classical problems and variants inspired by realistic applications to show how different linear separation methods and sampling strategies influence the quality of the results in terms of several metrics including objective value, dual bound and running time.