# A unified approach to mixed-integer optimization problems with logical constraints

## TL;DR

This paper introduces a unified, non-linear formulation for mixed-integer optimization with logical constraints, enabling faster and larger-scale solutions than traditional linear methods across various applications.

## Contribution

It presents a novel non-linear reformulation and regularization approach that transforms complex logical mixed-integer problems into convex binary problems, improving scalability and solution quality.

## Key findings

- Solves network design problems with hundreds of nodes faster and better.
- Handles sparse portfolio selection with up to 3,200 securities.
- Addresses sparse regression with up to 100,000 covariates.

## Abstract

We propose a unified framework to address a family of classical mixed-integer optimization problems with logically constrained decision variables, including network design, facility location, unit commitment, sparse portfolio selection, binary quadratic optimization, sparse principal analysis and sparse learning problems. These problems exhibit logical relationships between continuous and discrete variables, which are usually reformulated linearly using a big-M formulation. In this work, we challenge this longstanding modeling practice and express the logical constraints in a non-linear way. By imposing a regularization condition, we reformulate these problems as convex binary optimization problems, which are solvable using an outer-approximation procedure. In numerical experiments, we establish that a general-purpose numerical strategy, which combines cutting-plane, first-order and local search methods, solves these problems faster and at a larger scale than state-of-the-art mixed-integer linear or second-order cone methods. Our approach successfully solves network design problems with 100s of nodes and provides solutions up to 40\% better than the state-of-the-art; sparse portfolio selection problems with up to 3,200 securities compared with 400 securities for previous attempts; and sparse regression problems with up to 100,000 covariates.