MINLP

MINLP stands for mixed-integer nonlinear programming. It refers to optimization problems in which the objective function and constraints can be nonlinear, and some decision variables are restricted to integer values, while others remain continuous. The problem generalizes both mixed-integer linear programming (MILP) and nonlinear programming (NLP).

A typical MINLP can be written as minimizing f(x, y) subject to g_i(x, y) ≤ 0 for i = 1,...,m and h_j(x, y) = 0 for j = 1,...,p, with x representing continuous variables and y representing integer variables (often binary). The nonlinearities may appear in the objective and in the constraints, and the integer variables can capture yes/no decisions, quantities, or assignments.

MINLP problems are generally NP-hard due to the combination of nonlinearities and combinatorial decisions. If the nonlinear parts are convex and the integer variables are fixed, the problem reduces to a convex NLP; conversely, fixing the continuous variables can yield an easier discrete problem. In practice, a variety of exact and heuristic methods are used to obtain global or near-global solutions.

Common exact methods include branch-and-bound with convex relaxations, outer-approximation, generalized Benders decomposition, and spatial branch-and-bound. Heuristic and metaheuristic approaches, such as genetic algorithms, tabu search, and local search, are employed when exact methods are too costly or impractical for large instances.

Applications span process design, chemical engineering, energy systems, power generation, facility location, logistics, scheduling, and engineering design problems that involve discrete decisions and nonlinear physics or economics. Software tools for MINLP include commercial and open-source solvers such as BARON, BONMIN, COUENNE, SCIP, KNITRO, and others.