An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Durán, Marco A; Grossmann, Ignacio E.

doi:10.1007/bf02592081

Cited by 106 publications

(153 citation statements)

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“…Similar to the mixed integer linear case, such branch and bound algorithms are often combined with cutting plane techniques (e.g., see [14]). Further approaches originating from Kelly's cutting plane method are the Generalized Bender Decomposition method [6] and the outer-approximation algorithms introduced in [4] and later refined and improved, e.g., cf. [3,5].…”

Section: Related Literature and Main Resultsmentioning

confidence: 99%

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Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes

et al. 2012

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This paper is about the minimization of Lipschitz-continuous and strongly convex functions over integer points in polytopes. Our results are related to the rate of convergence of a black-box algorithm that iteratively solves special quadratic integer problems with a constant approximation factor. Despite the generality of the underlying problem, we prove that we can find efficiently, with respect to our assumptions regarding the encoding of the problem, a feasible solution whose objective function value is close to the optimal value. We also show that this proximity result is the best possible up to a factor polynomial in the encoding length of the problem.

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Section: Related Literature and Main Resultsmentioning

confidence: 99%

“…Kelly's cutting plane method has been adapted and extended to general mixed integer convex optimization problems of the form (6) introduced below, see [4,15].…”

Section: Related Literature and Main Resultsmentioning

confidence: 99%

Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes

et al. 2012

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“…Now suppose that QP (1) is unbounded below. Let {ρ k } be a sequence of increasing scalars tending to ∞ such that with x k being an optimal solution of the truncated QP (8) …”

Section: Theorem 1 Suppose the Qp (1) Is Feasible And Thatmentioning

confidence: 99%

“…Beginning with the seminal work of Egon Balas [3], there is a vast literature on disjunctive programming, with the recent work of Grossman et al [8,14,24] treating the nonlinear case. Nevertheless much of this literature, including the cited references, assumes boundedness of the problems; in contrast, a main contribution of our work is on problems where such boundedness is not assumed.…”

Section: Introductionmentioning

confidence: 99%

An LPCC approach to nonconvex quadratic programs

Mitchell

Pang

2010

Math. Program.

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Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can be accomplished by a parameter-free, mixed integer-programming based, finitely terminating algorithm developed recently by the authors, which can be enhanced in this context by a new kind of valid cut derived from the second-order conditions of the QP and by exploiting the special structure of the LPCCs. Throughout, our treatment makes no boundedness assumption 123 244 J. Hu et al.of the QP; this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption. The general theory is illustrated by 3 classes of indefinite problems: QPs with simple upper and lower bounds (existence of optimal solutions is guaranteed); same QPs with an additional inequality constraint (extending the case of simple bound constraints); and nonnegatively constrained copositive QPs (no guarantee of the existence of an optimal solution). We also present numerical results to support the special cuts obtained due to the QP connection.

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“…• Outer-Approximation (OA): Proposed by Duran and Grossmann (1986), it exploits the outer approximation linearization technique. Namely, given a point (x * , y * ), we can derive the OA cut…”

Section: Convex Minlpmentioning

confidence: 99%

Mixed integer nonlinear programming tools: a practical overview

d'Ambrosio

Lodi

2011

4OR-Q J Oper Res

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An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Abstract: [Mathematical Programming 36 (1986) The solution that was reported in Table 4A

Cited by 106 publications

References 0 publications

Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes

Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes

An LPCC approach to nonconvex quadratic programs

Mixed integer nonlinear programming tools: a practical overview

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