MINLPs

MINLP stands for mixed-integer nonlinear programming, a class of optimization problems in which some decision variables are constrained to be integers and the objective function and/or the constraints are nonlinear. A typical formulation minimizes f(x, y) subject to g_i(x, y) ≤ 0 and h_j(x, y) = 0, with x in Z^p (integer variables) and y in R^q (continuous variables). If all nonlinearities are linear and all variables are integers, the problem reduces to a mixed-integer linear program (MILP); if all variables are continuous, it is a nonlinear program (NLP).

MINLPs are generally computationally challenging and many are NP-hard, especially when nonlinearities are non-convex. Global optimization seeks the best feasible solution across the entire search space, while many practical approaches obtain good local optima. The difficulty increases with problem size, nonlinearity strength, and the presence of discrete decisions.

Common solution strategies include exact methods such as branch-and-bound or branch-and-cut with embedded nonlinear solvers, outer-approximation, and generalized Benders decomposition. Decomposition techniques, linearization or convexification of nonlinear terms, and piecewise linear approximations are frequently used to improve tractability. Heuristics and hybrid methods may provide useful initial solutions or speed up convergence.

MINLPs are solved by both commercial and open-source solvers such as BARON, KNITRO, SCIP, BONMIN, and ANTIGONE, often through modeling environments like AMPL, GAMS, Pyomo, or JuMP. Applications span process and chemical engineering, energy systems, transportation and logistics, economics, and manufacturing, wherever discrete decisions interact with nonlinear relationships.