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Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

We investigate a network design problem under traffic uncertainty that arises when provisioning Virtual Private Networks (VPNs): given a set of terminals that must communicate with one another, and a set of possible traffic matrices, sufficient capacity has to be reserved on the links of the large underlying public network to support all possible traffic matrices while minimizing the total reservation cost. The problem admits several versions depending on the desired topology of the reserved links, and the nature of the traffic data uncertainty. We present compact linear mixed-integer programming formulations for the problem with the classical hose traffic model and for a less conservative robust variant relying on the traffic statistics that are often available. These flow-based formulations allow us to solve optimally medium-to-large instances with commercial MIP solvers. We also propose a combined branch-and-price and cutting-plane algorithm to tackle larger instances. Computational results obtained for several classes of instances are reported and discussed.

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Mixed Integer Linear Programming (MILP) is commonly used to model indicator constraints, i.e., constraints that either hold or are relaxed depending on the value of a binary variable. Unfortunately, those models tend to lead to weak continuous relaxations and turn out to be unsolvable in practice, like in the case of Classification problems with Ramp Loss functions that represent an important application in this context.In this paper we show the computational evidence that a relevant class of these Classification instances can be solved far more efficiently if a nonlinear, nonconvex reformulation of the indicator constraints is used instead of the linear one. Inspired by this empirical and surprising observation, we show that aggressive bound tightening is the crucial ingredient for solving this class of instances, and we devise a pair of computationally effective algorithmic approaches that exploit it within MILP.More generally, we argue that aggressive bound tightening is often overlooked in MILP, while it represents a significant building block for enhancing MILP technology when indicator constraints and disjunctive terms are present.