An extended cutting plane method for solving convex MINLP problems

Westerlund, Tapio; Pettersson, Frank

doi:10.1016/0098-1354(95)87027-x

Cited by 283 publications

(154 citation statements)

References 11 publications

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“…Rather, in the most rudimentary version, after each solution of the mixed-integer linear program (P K [l, u]), the most violated constraint (i.e, of f (x * , y * ) ≤ z and g(x * , y * ) ≤ 0) is linearized and appended to (P K [l, u]). This simple iteration is enough to easily establish convergence (see [135]). It should be noted that for the case in which there are no integer-constrained variables, then at each step (P K [l, u]) is just a continuous linear program and we exactly recover Kelley's Cutting-Plane Algorithm for convex continuous nonlinear programming.…”

Section: Gotomentioning

confidence: 99%

“…Substantially postdating the development of the OA Algorithm is the simpler and closely related Extended Cutting Plane (ECP) Algorithm introduced in [135]. The original ECP Algorithm is a straightforward generalization of Kelley's CuttingPlane Algorithm [77] for convex continuous nonlinear programming (which predates the development of the OA Algorithm).…”

Section: Gotomentioning

confidence: 99%

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Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

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Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems.It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

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Section: Gotomentioning

confidence: 99%

Section: Gotomentioning

confidence: 99%

Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

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“…Bonmin offers the possibility to choose one of five algorithms: a nonlinear programming based branch-and-bound [9], a pure outer approximation decomposition [10], a vanilla implementation of the Quessada-Grossmann branch-and-cut algorithm [20], a hybrid method [7] and a method based on extended cutting planes [24] (similar to the method proposed in [1]). Here we report results obtained with the hybrid method since it was consistently better than the others (with all four models) in preliminary experiments.…”

Section: Computational Experimentsmentioning

confidence: 99%

“…A special case of interest is that of convex MINLPs where the objective function to minimize is convex and the feasible region obtained by dropping the integrality requirements on the variables is a convex set. For such problems, several algorithms have been developed [10,20,24,7] and have been implemented in solvers such as FilMINT [1] or Bonmin [7] which are able to solve problems of medium size. A challenge is to solve larger problems.…”

Section: Introductionmentioning

confidence: 99%

Mixed-integer nonlinear programs featuring “on/off” constraints

et al. 2011

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In this paper, we study MINLPs featuring "on/off" constraints. An "on/off" constraint is a constraint f (x) ≤ 0 that is activated whenever a corresponding 0-1 variable is equal to 1. Our main result is an explicit characterization of the convex hull of the feasible region when the MINLP consists of simple bounds on the variables and one "on/off" constraint defined by an isotone function f . When extended to general convex MINLPs, we show that this result yields tight lower bounds compared to classical formulations. This allows us to introduce new models for the delayconstrained routing problem in telecommunications. Numerical results show gains in computing time of up to one order of magnitude compared to state-of-the-art approaches.

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“…In general, the continuous relaxation of these problems are nonconvex. As a result, the direct application of algorithms such as Generalized Benders Decomposition (GBD) [2] [3], Outer Approximation (OA) [4], Extended Cutting Plane [14], may fail to find the global optimum since the solution of the NLP subproblem may correspond to a local optimum and the cuts in the master problem may not be valid. Therefore, specialized algorithms should be used to find the global optimum [5] [6] [7].…”

Section: Introductionmentioning

confidence: 99%

Using redundancy to strengthen the relaxation for the global optimization of MINLP problems

Ruiz¹,

Grossmann²

2011

Computers & Chemical Engineering

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In this paper we present a strategy to improve the relaxation for the global optimization of nonconvex MINLPs. The main idea consists in recognizing that each constraint or set of constraints has a meaning that comes from the physical interpretation of the problem. When these constraints are relaxed part of this meaning is lost. Adding redundant constraints that recover that physical meaning strengthens the relaxation. We propose a methodology to find such redundant constraints based on engineering knowledge and physical insight.

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An extended cutting plane method for solving convex MINLP problems

Cited by 283 publications

References 11 publications

Nonlinear Integer Programming

Nonlinear Integer Programming

Mixed-integer nonlinear programs featuring “on/off” constraints

Using redundancy to strengthen the relaxation for the global optimization of MINLP problems

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