### Contents

Back to Integer Linear Programming or Nonlinear Programming

## Basic Concepts

*Mixed integer nonlinear programming* (MINLP) refers to optimization problems with continuous and discrete variables and nonlinear functions in the objective function and/or the constraints. MINLPs arise in applications in a wide range of fields, including chemical engineering, finance, and manufacturing. The general form of a MINLP is

\[\begin{array}{lllll}

\mbox{min} & f(x,y) & & & \\

\mbox{s.t.} & c_i(x,y) & = & 0 & \forall i \in E \\

& c_i(x,y) & \leq & 0 & \forall i \in I \\

& x & \in & X & \\

& y & \in & Y & \mbox{integer}

\end{array}

\]

where each \(c_i(x,y) \,\) is a mapping from \(R^n \,\) to \(R \,\), and \(E \,\) and \(I \,\) are index sets for equality and inequality constraints, respectively. Typically, the functions \(f\) and \(c_i\) have some smoothness properties, i.e., once or twice continuously differentiable.

Software developed for MINLP has generally followed two approaches:

**Outer Approximation/Generalized Bender's Decomposition**: These algorithms alternate between solving a mixed-integer LP master problem and nonlinear programming subproblems.**Branch-and-Bound**: Branch-and-bound methods for mixed-integer LP can be extended to MINLP with a number of tricks added to improve their performance.

For a recent survey of MINLP applications, models, and solution methods, see Belotti et al. (2013).

## Software Resources

The NEOS Server offers a number of MINLP solvers. See the list here.

## Test Problems

- MacMINLP, a collection of MINLP test problems in AMPL
- MINLPLib, a collection of MINLP test problems at MINLP World

## References

Optimization Online Integer Programming area (area covers both linear and nonlinear submissions)

- Belotti, P., C. Kirches, S. Leyffer, J. Linderoth, J. Luedtke, and A. Mahajan. 2013. Mixed-Integer Nonlinear Optimization.
*Acta Numerica***22**:1-131. DOI: http://dx.doi.org/10.1017/S0962492913000032 - Leyffer, S. and Mahajan, A. 2011. Software For Nonlinearly Constrained Optimization.
*Wiley Encyclopedia of Operations Research and Management Science*, John Wiley & Sons, New York.