# Mixed-Integer Nonlinear Optimization

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 inmethods 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. Nonconvex MINLPs pose additional challenges, because they contain nonconvex functions in the objective or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally nonconvex, resulting in many local minima. We discuss a range of approaches to tackle this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations nonconvex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of nonconvex 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.

## Citation

Argonne National Laboratory, Mathematics and Computer Science Division, Preprint ANL/MCS-P3060-1121