Solving mixed integer nonlinear programs by outer approximation
 Roger Fletcher,
 Sven Leyffer
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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.
 R. Breu and C.A. Burdet, “Branchandbound experiments in zero—one programming,”Mathematical Programming Study 2 (1974) 1–50.
 M. Duran and I.E. Grossmann, “An outerapproximation algorithm for a class of Mixed Integer Nonlinear Programs,”Mathematical Programming 36 (1986) 307–339.
 R. Fletcher,Practical Methods of Optimization (John Wiley, Chichester, 1987).
 O.E. Flippo et al., “Duality and decomposition in general mathematical programming,” Econometric Institute, Report 8747/B, University of Rotterdam (1987).
 A.M. Geoffrion, “Generalized Benders decomposition,”Journal of Optimization Theory and Applications 10 (1972) 237–262.
 G.R. Kocis and I.E. Grossmann, “Global optimization of nonconvex MINLP in process synthesis,”Industrial and Engineering Chemistry Research 27 (1988) 1407–1421.
 F. Körner, “A new branching rule for the branchandbound algorithm for solving nonlinear integer programming problems,”BIT 28 (1988) 701–708.
 R. Lazimy, “Improved algorithm for mixedinteger quadratic programs and a computational study,”Mathematical Programming 32 (1985) 100–113.
 J. Viswanathan and I.E. Grossmann, “A combined penalty function and outerapproximation method for MINLP optimization,”Computers and Chemical Engineering 14 (1990) 769–782.
 X. Yuan, S. Zhang, L. Pibouleau and S. Domenech, “Une méthode d'optimization non linéaire en variables mixtes pour la conception de procédés,”Operations Research 22/4 (1988) 331–346.
 Title
 Solving mixed integer nonlinear programs by outer approximation
 Journal

Mathematical Programming
Volume 66, Issue 13 , pp 327349
 Cover Date
 19940801
 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