Mirror-Descent Methods in Mixed-Integer Convex Optimization

TL;DR

This paper explores mirror-descent methods for mixed-integer convex optimization, proposing an oracle-based approach that can solve certain instances efficiently and find multiple optimal solutions.

Contribution

It introduces an oracle-based algorithmic framework for mixed-integer convex optimization, with efficient implementation for problems with two integer variables and methods to find multiple optima.

Findings

Oracle can be implemented efficiently for two integer variables

Algorithm finds second and k-th best solutions in integer convex problems

Finite-time algorithm for mixed-integer convex optimization

Abstract

In this paper, we address the problem of minimizing a convex function f over a convex set, with the extra constraint that some variables must be integer. This problem, even when f is a piecewise linear function, is NP-hard. We study an algorithmic approach to this problem, postponing its hardness to the realization of an oracle. If this oracle can be realized in polynomial time, then the problem can be solved in polynomial time as well. For problems with two integer variables, we show that the oracle can be implemented efficiently, that is, in O(ln(B)) approximate minimizations of f over the continuous variables, where B is a known bound on the absolute value of the integer variables.Our algorithm can be adapted to find the second best point of a purely integer convex optimization problem in two dimensions, and more generally its k-th best point. This observation allows us to formulate a finite-time algorithm for mixed-integer convex optimization.