Mixed-integer Nonlinear Optimization: A Hatchery for Modern Mathematics (original) (raw)
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Optimization is a vibrant growing area of Applied Mathematics. Its many successful applications depend on efficient algorithms and this has pushed the development of theory and software. In recent years there has been a resurgence of interest to use "non-standard" techniques to estimate the complexity of computation and to guide algorithm design. New interactions with fields like algebraic geometry, representation theory, number theory, combinatorial topology, algebraic combinatorics, and convex analysis have contributed non-trivially to the foundations of computational optimization. In this expository survey we give three example areas of optimization where "algebraic-geometric thinking" has been successful. One key point is that these new tools are suitable for studying models that use non-linear constraints together with combinatorial conditions. Keywords Algorithms · Non-linear mixed-integer optimization · Linear optimization · Algebraic techniques in optimization · Graver bases · Generating functions · Complexity of the simplex method Mathematics Subject Classification 90CXX · 90C05 · 90C11 · 90C27 · 90C29 · 90C60
We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund ( LP/NLP-BB ) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance.
An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems
American Journal of Operations Research, 2011
We propose an exact penalty approach for solving mixed integer nonlinear programming (MINLP) problems by converting a general MINLP problem to a finite sequence of nonlinear programming (NLP) problems with only continuous variables. We express conditions of exactness for MINLP problems and show how the exact penalty approach can be extended to constrained problems.