Solving mixed integer nonlinear programs by outer approximation
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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 thel1 norm is also given.
KeywordsNonlinear integer programming Mixed integer nonlinear programming Decomposition Outer approximation
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- R. Breu and C.-A. Burdet, “Branch-and-bound experiments in zero—one programming,”Mathematical Programming Study 2 (1974) 1–50.Google Scholar
- M. Duran and I.E. Grossmann, “An outer-approximation algorithm for a class of Mixed Integer Nonlinear Programs,”Mathematical Programming 36 (1986) 307–339.Google Scholar
- R. Fletcher,Practical Methods of Optimization (John Wiley, Chichester, 1987).Google Scholar
- O.E. Flippo et al., “Duality and decomposition in general mathematical programming,” Econometric Institute, Report 8747/B, University of Rotterdam (1987).Google Scholar
- A.M. Geoffrion, “Generalized Benders decomposition,”Journal of Optimization Theory and Applications 10 (1972) 237–262.Google Scholar
- G.R. Kocis and I.E. Grossmann, “Global optimization of nonconvex MINLP in process synthesis,”Industrial and Engineering Chemistry Research 27 (1988) 1407–1421.Google Scholar
- F. Körner, “A new branching rule for the branch-and-bound algorithm for solving nonlinear integer programming problems,”BIT 28 (1988) 701–708.Google Scholar
- R. Lazimy, “Improved algorithm for mixed-integer quadratic programs and a computational study,”Mathematical Programming 32 (1985) 100–113.Google Scholar
- J. Viswanathan and I.E. Grossmann, “A combined penalty function and outer-approximation method for MINLP optimization,”Computers and Chemical Engineering 14 (1990) 769–782.Google Scholar
- 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.Google Scholar