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.

本文研究了一种统一的框架方法用于解决一类混合整数优化问题，通过对其逻辑约束进行非线性方式的表达，结合规则化条件及基于混合精度算法，形成了凸二进制优化问题，并利用一种综合的数字策略方法解决问题。