A unified approach to mixed-integer optimization: Nonlinear formulations and scalable algorithms

by   Dimitris Bertsimas, et al.

We propose a unified framework to address a family of classical mixed-integer optimization problems, including network design, facility location, unit commitment, sparse portfolio selection, binary quadratic optimization 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.



There are no comments yet.


page 1

page 2

page 3

page 4


A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

We introduce a unified framework for the study of multilevel mixed integ...

On Polyhedral and Second-Order-Cone Decompositions of Semidefinite Optimization Problems

We study a cutting-plane method for semidefinite optimization problems (...

Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem

The article demonstrates rather general approach to problems of discrete...

Cardinality Minimization, Constraints, and Regularization: A Survey

We survey optimization problems that involve the cardinality of variable...

Application and Assessment of the Divide-and-Conquer method on some Integer Optimization Problems

In this paper the Divide-and-Conquer method is applied and assessed to t...

The CCP Selector: Scalable Algorithms for Sparse Ridge Regression from Chance-Constrained Programming

Sparse regression and variable selection for large-scale data have been ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.