Application-oriented mixed integer non-linear programming

d'Ambrosio, Claudia

doi:10.1007/s10288-010-0118-8

Cited by 16 publications

(6 citation statements)

References 87 publications

(168 reference statements)

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“…In particular, FP should be effective on the instances in which the nonlinear aspects of the problems play a crucial role, thus suggesting its fruitful integration within COUENNE or any other GO solver (as happened for FP algorithms in MILP). Indeed, the current trunk version of COUENNE is more sophisticated in terms of heuristics also due to our investigation preliminary reported in [15,16] and some results at the of Section 5.4 seem promising in this concern.…”

Section: Fp Variants and Preliminary Resultsmentioning

confidence: 80%

A storm of feasibility pumps for nonconvex MINLP

et al. 2012

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One of the foremost difficulties in solving Mixed-Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed-Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed-Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pump algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pump algorithms is that, unlike most previous successive projection methods from the literature, projection is "naturally" taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performance of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.

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Section: Fp Variants and Preliminary Resultsmentioning

confidence: 80%

A storm of feasibility pumps for nonconvex MINLP

et al. 2012

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“…While for the 0-1 case the equivalence between the FP heuristic and the Frank-Wolfe method has been pointed out first in [14] and then further investigated and exploited in [12]; the relationship between FP and the Frank-Wolfe method in the general integer case is by far less clear [9,10]. The aim of this section is that of clarifying this equivalence showing that the ) -FP heuristic without any perturbation (i.e.…”

Section: The Fp Heuristic and The Frank-wolfe Methodsmentioning

confidence: 99%

Feasibility Pump-like heuristics for mixed integer problems

Santis

Lucidi

Rinaldi

2014

Discrete Applied Mathematics

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Available online xxxx Keywords:Mixed integer programming Concave penalty functions Frank-Wolfe algorithm Feasibility problem a b s t r a c t Finding a feasible solution to a MIP problem is a tough task that has received much attention in the last decades. The Feasibility Pump (FP) is a heuristic for finding feasible solutions to MIP problems that has encountered a lot of success as it is very efficient also when dealing with very difficult instances. In this work, we show that the FP heuristic for general MIP problems can be seen as the Frank-Wolfe method applied to a concave nonsmooth problem. Starting from this equivalence, we propose concave non-differentiable penalty functions for measuring solution integrality that can be integrated in the FP approach.

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“…An earlier version of this work was published in [22]. Very recently, a Feasibility Pump for nonconvex MINLPs was also proposed [8,Chapter 2;9].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Set R ← R ∩ N G(x I ) 11: if x * feasible then 12: return x * 13: else 14: return failure Some of the ideas that led to devising our IR scheme have also been employed by the recently developed Feasibility Pump (FP) for nonconvex MINLPs [8,Chapter 2;9], even though the aim of the FP is to find any feasible solution, whereas in our case, we want to find a good solution. We briefly highlight the main differences between the two approaches: step, the point generated in one trajectory tries to get closer to the other trajectory.…”

Section: Main Algorithmic Ideasmentioning

confidence: 99%

“…It was also shown to be effective within the MINLP world [19,23]. Again, we face the problem of choosing k; for the binary case, we can follow the suggestion of [13] and use a small k ∈ [10,20]; for general integer variables, we will choose a k based on the distance between the bounds involved in (8), as in Sect. 4.1.…”

Section: Improvement-based Iterative Roundingmentioning

confidence: 99%

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Rounding-based heuristics for nonconvex MINLPs

Nannicini

Belotti

2011

Math. Prog. Comp.

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We propose two primal heuristics for nonconvex mixed-integer nonlinear programs. Both are based on the idea of rounding the solution of a continuous nonlinear program subject to linear constraints. Each rounding step is accomplished through the solution of a mixed-integer linear program. Our heuristics use the same algorithmic scheme, but they differ in the choice of the point to be rounded (which is feasible for nonlinear constraints but possibly fractional) and in the linear constraints. We propose a feasibility heuristic, that aims at finding an initial feasible solution, and an improvement heuristic, whose purpose is to search for an improved solution within the neighborhood of a given point. The neighborhood is defined through local branching cuts or box constraints. Computational results show the effectiveness in practice of these simple ideas, implemented within an open-source solver for nonconvex mixedinteger nonlinear programs. Mathematics Subject Classification (2000)

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Application-oriented mixed integer non-linear programming

Cited by 16 publications

References 87 publications

A storm of feasibility pumps for nonconvex MINLP

A storm of feasibility pumps for nonconvex MINLP

Feasibility Pump-like heuristics for mixed integer problems

Rounding-based heuristics for nonconvex MINLPs

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