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 the quality of solutions obtained from sample-average approximations to two-stage stochastic linear programs with recourse. We use a recently developed software tool executing on a computational grid to solve many large instances of these problems, allowing us to obtain high-quality solutions and to verify optimality and near-optimality of the computed solutions in various ways.

Abstract. This paper provides a survey of recent progress and software for solving mixed integer nonlinear programs (MINLP) wherein the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in very years. By exploiting analogies to the case of well-known techniques for solving mixed integer linear programs and incorporating these techniques into the software, significant improvements have been made in our ability to solve the problems.Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.1. Introduction. Mixed-Integer Nonlinear Programs (MINLP) are optimization problems where some of the variables are constrained to take integer values and the objective function and feasible region of the problem are described by nonlinear functions. Such optimization problems arise in many real world applications. Integer variables are often required to model logical relationships, fixed charges, piecewise linear functions, disjunctive constraints and the non-divisibility of resources. Nonlinear functions are required to accurately reflect physical properties, covariance, and economies of scale.In all its generality, MINLP forms a particularly broad class of challenging optimization problems as it combines the difficulty of optimizing over integer variables with handling of nonlinear functions. Even if we restrict our model to contain only linear functions, MINLP reduces to a Mixed-Integer Linear Program (MILP), which is an NP-Hard problem [56]. On the other hand, if we restrict our model to have no integer variable but allow for general nonlinear functions in the objective or the constraints, then MINLP reduces to a Nonlinear Program (NLP) which is also known to be NP-Hard [91]. Combining both integrality and nonlinearity can lead to examples of MINLP that are undecidable [68].

The branch and bound procedure for solving mixed integer programming (MIP) problems using linear programming relaxations has been used with great success for decades. Over the years, a variety o f r e s e a r c hers have studied ways of making the basic algorithm more e ective. Breakthroughs in the elds of computer hardware, computer software, and mathematics have led to increasing success at solving larger and larger MIP instances. The goal of this paper is to survey many of the results regarding branch a n d bound search strategies and evaluate them again in light of the other advances that have taken place over the years. In addition, novel search strategies are presented and shown to often perform better than those currently used in practice.

We describe a new solver for convex mixed-integer nonlinear programs (MINLPs) that implements a linearization-based algorithm. The solver is based on an algorithm of Quesada and Grossmann [Quesada, I., I. E. Grossmann. 1992. An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chemical Engrg. 16(10–11) 937–947] that avoids the complete re-solution of a master mixed-integer linear program (MILP) by adding new linearizations at open nodes of the branch-and-bound tree whenever an integer solution is found. The new solver, FilMINT, combines the MINTO branch-and-cut framework for MILP with filterSQP to solve the nonlinear programs that arise as subproblems in the algorithm. The MINTO framework allows us to easily employ cutting planes, primal heuristics, and other well-known MILP enhancements for MINLPs. We present detailed computational experiments that show the benefit of such advanced MILP techniques. We offer new suggestions for generating and managing linearizations that are shown to be efficient on a wide range of MINLPs. By carefully incorporating and tuning all these enhancements, an effective solver for convex MINLPs is constructed.