Invited Session Mon.1.MA 005

Monday, 10:30 - 12:00 h, Room: MA 005

Cluster 14: Mixed-integer nonlinear programming [...]

Global mixed-integer nonlinear optimization I


Chair: Ignacio E. Grossmann



Monday, 10:30 - 10:55 h, Room: MA 005, Talk 1

Ignacio E. Grossmann
Using convex nonlinear relaxations in the global optimization of nonconvex generalized disjunctive programs

Coauthor: Juan Pablo Ruiz


In this paper we address the global optimization of GDP problems that in addition to bilinear and concave terms, involve other terms such as linear fractional terms for which nonlinear convex relaxations have shown to provide rigorous convex envelopes that are magnitude much tighter than linear relaxations. The use of nonlinear convex relaxations leads to a nonlinear convex GDP which relaxation can be strengthen by using recently results from the recent work of Ruiz and Grossmann.
We first define the general nonconvex GDP problem that we aim at solving and review the use of the hull relaxation, the traditional method to find relaxations. Second, we show how we can strengthen the relaxation of the traditional approach by presenting a systematic procedure to generate a hierarchy of relaxations based on the application of basic steps to nonlinear convex sets in disjunctive programming. We outline a set of rules that avoids the exponential transformation to the Disjunctive Normal Form leading to a more efficient implementation of the method. Finally we assess the performance of the method by solving to global optimality engineering design test problems.



Monday, 11:00 - 11:25 h, Room: MA 005, Talk 2

MiloŇ° Bogataj
A multilevel approach to global optimization of MINLP problems

Coauthor: Zdravko Kravanja


In this work, we present an approach for global optimization of nonconvex mixed-integer nonlinear programs (MINLPs) containing bilinear and linear fractional terms. These terms are replaced by piecewise convex under-/ overestimators defined over domains of one or both complicating variables. The domains are partitioned over at least two levels with increasing grid density. The densest grid is dense enough to ensure the gap between the upper and lower bound falls below the predefined convergence criterion. The derived multilevel convex MINLP is then solved using a modified outer approximation/equality relaxation (OA/ER) algorithm. The key idea of the approach is progressive tightening of convex relaxation, whilst keeping low combinatorial complexity of the convexified MINLP throughout the solution procedure. After each major OA/ER iteration, tighter under-/overestimators are activated, however, only over the domain partitions containing currently optimal solution. Hence, only the most promising alternatives are being explored from the start on. The multilevel approach was tested on illustrative examples to show its advantage over a single level approach.



Monday, 11:30 - 11:55 h, Room: MA 005, Talk 3

Tapio Westerlund
A reformulation framework for global optimization

Coauthor: Andreas Lundell


In some previous papers we have published results connected to an optimization framework for solving non-convex mixed integer nonlinear programming problems, including signomial functions. In the framework the global optimal solution of such non-convex problems can be obtained by solving a converging sequence of convex relaxed MINLP problems. The relaxed convex problems are obtained by replacing the non-convex constraint functions with convex underestimators. The signomial functions are first convexified by single-variable power and exponential transformations. The non-convexities are then moved to the transformations. However, when replacing the transformations with piecewise linear approximations the problem will be both convexified and relaxed.
The scope of this paper is to show how any twice-differentiable function can be handled in an
extended version of the global optimization framework. For C2-functions it is shown how a spline
version of the so-called αBB-underestimator can be applied in a slightly similar way as the
approach utilized for signomial functions. It is, further, shown how this underestimator can easily
be integrated in the actual reformulation framework.


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