Book ChapterDOI
A Suggested Extension of Special Ordered Sets to Non-Separable Non-Convex Programming Problems*
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In this article, a branch and bound method for solving non-separable non-convex programming problems where the nonlinearities are piecewise linearly approximated using the standard simplicial subdivision of the hypercube is proposed.Abstract:
This paper suggests a branch and bound method for solving non-separable non-convex programming problems where the nonlinearities are piecewise linearly approximated using the standard simplicial subdivision of the hypercube. The method is based on the algorithm for Special Ordered Sets, used with separable problems, but involves using two different types of branches to achieve valid approximations.read more
Citations
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Journal ArticleDOI
Mixed-integer nonlinear optimization
TL;DR: An emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP is described and a range of approaches for tackling this challenging class of problems are discussed, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non- Convex structures to obtain improved convex Relaxations.
Journal ArticleDOI
Non-convex mixed-integer nonlinear programming: A survey
Samuel Burer,Adam N. Letchford +1 more
TL;DR: In this paper, the authors survey the literature on non-convex mixed-integer nonlinear programs, discussing applications, algorithms, and software, and special attention is paid to the case in which the objective and constraint functions are quadratic.
Journal ArticleDOI
Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions
TL;DR: This work reviews several new and existing MIP formulations for continuous piecewise-linear functions with special attention paid to multivariate nonseparable functions.
Journal ArticleDOI
Mixed integer models for the stationary case of gas network optimization
TL;DR: This work describes techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program and shows that the number of vertices is computationally tractable yielding exact separation algorithms.
Journal ArticleDOI
Modeling disjunctive constraints with a logarithmic number of binary variables and constraints
TL;DR: It is proved that the new formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions have favorable tightness properties and can significantly outperform other mixed integer binary formulations.
References
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Journal ArticleDOI
Large-scale linearly constrained optimization
TL;DR: An algorithm for solving large-scale nonlinear programs with linear constraints is presented, which combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities.
Journal ArticleDOI
A Nonlinear Programming Technique for the Optimization of Continuous Processing Systems
R. E. Griffith,R. A. Stewart +1 more
TL;DR: A numerical example, a model construction example, and a description of a particular existing computer system are included in order to clarify the mode of operation of the method.
Journal ArticleDOI
Homotopies for computation of fixed points
TL;DR: Given a point to set mapf on a simplex with certain conditions, an algorithm for computing fixed points is described, which operates by following the fixed point as an initially affine function is deformed towardsf.