An Algorithm for Separable Nonconvex Programming Problems

Falk, James E.; Soland, Richard M.

doi:10.1287/mnsc.15.9.550

Cited by 365 publications

(172 citation statements)

References 3 publications

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“…Initially conceived as an algorithm to solve combinatorial optimization problems [21,9], branch-and-bound has evolved to a method for solving more general multi-extremal problems like P [11,19]. To solve P, branch-and-bound computes lower and upper bounds on the optimal objective function value over successively refined partitions of the search space.…”

Section: Branch-and-bound In Continuous Spacesmentioning

confidence: 99%

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Tawarmalani

Sahinidis²

2004

Mathematical Programming

473

285

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This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm.In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems.In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0 − 1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

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Section: Branch-and-bound In Continuous Spacesmentioning

confidence: 99%

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Tawarmalani

Sahinidis²

2004

Mathematical Programming

473

285

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“…The function 4~(f) is discussed (in other terms) in [22], where it is called the convex envelope of f, and it provides a way of turning a general minimisation problem into a convex problem, since f and its convex evelope have the same minimum point and value.…”

Section: Vx Su(x) Vy F(y) -F(x) ->-Omentioning

confidence: 99%

Projections in minimax algebra

Cuninghame‐Green

1976

Mathematical Programming

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An axiomatic theory of linear operators can be constructed for abstract spaces defined over (R, *, e), that is over the (extended) real numbers R with the binary operations x + y = max (x,y) and x ~ y = x + y. Many of the features of conventional linear operator theory can be reproduced in this theory, although the proof techniques are quite different. Specialisation of the theory to spaces of n-tuples provides techniques for analysing a number of well-known operational research problems, whilst specialisation to function spaces provides a natural formal framework for certain familiar problems of approximation, optimisation and duality. A motivating exampleThe object of the present article is to give an overview of a system of ideas which, because of the extensive nature of the research program of which they form a part (and the long refereeing procedures so implied), will not all be published in detailed form for some time. The accent in what follows is on explanation rather than on proof.In the past 20 years (see references at the end) several different authors, often apparently unaware of one another's work, have discovered that an attractive formulation language is provided for a surprisingly wide class of problems by setting up an algebra of real numbers (perhaps extended by symbols such as -~, etc.) in which, however, the usual operations of multiplication and addition of two numbers are replaced by the operations:(i) arithmetical addition, and (ii) selection of the greater (or dually, the smaller) of the two numbers, respectively. Thus, if R is the set of (suitably extended) real numbers, we define for 111

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“…(12) In this formulation, the y variables have the same interpretation as in the multiple choice model.…”

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confidence: 99%

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

2003

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We study a generic minimization problem with separable non-convex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.Key words: piecewise-linear, integer programming, linear relaxation, Lagrangian relaxation. Resum6Nous considerons un probleme de minimisation gen6rique dans lequel l'objectif consiste d'une somme separable de fonctions lineaires par morceaux non convexes. Nous montrons que les relaxations lineaires de trois modeles classiques de programmation en nombres entiers sont equivalentes puisqu'elles fournissent comme approximation de l'objectif son enveloppe convexe inf6rieure. Nous etablissons galement une relation entre ce rsultat et la th6orie de la dualite lagrangienne classique.Mots-cls : lineaire par morceaux, progrmmation en nombres entiers, relaxation lin6aire, relaxation lagrangienne.ii

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An Algorithm for Separable Nonconvex Programming Problems

Cited by 365 publications

References 3 publications

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Projections in minimax algebra

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

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