, Volume 66, Issue 1, pp 327349
First online:
Solving mixed integer nonlinear programs by outer approximation
 Roger FletcherAffiliated withDepartment of Mathematics and Computer Science, University of Dundee
 , Sven LeyfferAffiliated withDepartment of Mathematics and Computer Science, University of Dundee
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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 tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I.
The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving rise to a new quadratic outer approximation algorithm.
An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems. It is possible to develop the theory in an elegant way for a large class of nonsmooth MINLPs based on the use of convex composite functions and subdifferentials, although an interpretation for thel _{1} norm is also given.
Keywords
Nonlinear integer programming Mixed integer nonlinear programming Decomposition Outer approximation Title
 Solving mixed integer nonlinear programs by outer approximation
 Journal

Mathematical Programming
Volume 66, Issue 13 , pp 327349
 Cover Date
 199408
 DOI
 10.1007/BF01581153
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Nonlinear integer programming
 Mixed integer nonlinear programming
 Decomposition
 Outer approximation
 Industry Sectors
 Authors

 Roger Fletcher ^{(1)}
 Sven Leyffer ^{(1)}
 Author Affiliations

 1. Department of Mathematics and Computer Science, University of Dundee, DD1 4HN, Dundee, Scotland, UK