# Solving mixed integer nonlinear programs by outer approximation

@article{Fletcher1994SolvingMI, title={Solving mixed integer nonlinear programs by outer approximation}, author={Roger Fletcher and Sven Leyffer}, journal={Mathematical Programming}, year={1994}, volume={66}, pages={327-349} }

A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems.Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be…

## 608 Citations

### An Outer-Inner Approximation for Separable Mixed-Integer Nonlinear Programs

- Computer ScienceINFORMS J. Comput.
- 2014

A common structure in convex mixed-integer nonlinear programs (MINLPs) is separable nonlinear functions and three improvements to the outer approximation algorithms are proposed, including a simple extended formulation and a heuristic inner approximation of the feasible region.

### Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

- Computer Science, MathematicsMath. Program.
- 2004

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented and numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.

### Continuous optimization methods for onvex mixed-integer nonlinear programming

- Computer Science
- 2015

The topic of this dissertation is the design of fast branch-and-bound algorithms that use intelligently adapted approaches from continuous optimization for solving convex mixed-integer nonlinear…

### The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

- Computer ScienceJ. Glob. Optim.
- 2020

Preliminary numerical results show that due to the generated cuts in the LP phase, on average only 2–3 MIP problems have to be solved in the MIP phase.

### FilMINT: An Outer Approximation-Based Solver for Convex Mixed-Integer Nonlinear Programs

- Computer ScienceINFORMS J. Comput.
- 2010

A new solver for convex mixed-integer nonlinear programs (MINLPs) that implements a linearization-based algorithm 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.

### Polyhedral Approximation Strategies in Nonconvex Mixed-Integer Nonlinear Programming*

- Computer Science
- 2020

Some extensions to SHOT are discussed that significantly extend its applicability to nonconvex problems, including utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions.

### Using Interior-Point Methods within an Outer Approximation Framework for Mixed Integer Nonlinear Programming

- Computer Science
- 2012

This paper proposes the exact primal-dual penalty approach, and examines cases where the nonlinear subproblems take on special forms, namely those of second-order cone programming problems and semidefinite programming problems.

### FilMINT: An Outer-Approximation-Based Solver for Nonlinear Mixed Integer Programs

- Computer Science
- 2008

A new solver for mixed integer nonlinear programs (MINLPs) that implements a linearization-based algorithm, FilMINT, that avoids the complete solution of master mixed integer linear programs (MILPs) by adding new linearizations at open nodes of the branch-and-bound tree whenever an integer solution is found.

### AN IMPROVED DIRECT SEARCH APPROACH FOR SOLVING MIXED-INTEGER NONLINEAR PROGRAMMING PROBLEMS

- Mathematics
- 2012

The strategy of releasing non-basic variables from their bounds, combined with the "active constraint" method and the notion of super-basics, has been developed for efficiently tackling the special class of a nonlinear mathematical programming problem.

### Outer approximation for global optimization of mixed-integer quadratic bilevel problems

- Computer ScienceMathematical Programming
- 2021

It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches.

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