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15
On convex relaxations of quadrilinear terms
, 2009
"... The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are o ..."
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The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we establish that, any relaxation for klinear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear convex envelope). We present a computational analysis which helps establish which relaxations are strictly tighter, and we apply our findings to two wellstudied applications: the Molecular Distance Geometry Problem and the HartreeFock Problem.
Reformulations in mathematical programming: Automatic symmetry detection and exploitation
 Mathematical Programming
"... symmetrydetection andexploitation ..."
Symmetry in mathematical programming
 Mixed Integer Nonlinear Programming. Volume IMA
"... Abstract. Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric ..."
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Abstract. Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; (c) solving the modified problem. Sometimes (b) and (c) are performed concurrently: the solution algorithm generates a sequence of subproblems, some of which are recognized to be symmetrically equivalent and either discarded or treated differently. We review symmetrybased analyses and methods for Linear Programming, Integer Linear Programming, MixedInteger Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and MixedInteger Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the original formulation. We finally present a new theoretical and computational study of the formulation symmetries of the Kissing Number Problem.
Experiments with a Feasibility Pump approach for nonconvex MINLPs
 SYMPOSIUM ON EXPERIMENTAL AND EFFICIENT ALGORITHMS
, 2010
"... We present a new Feasibility Pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programming problems. Differences with the previously proposed Feasibility Pump algorithms and difficulties arising from nonconvexities in the models are extensively discussed. The main methodological innovati ..."
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Cited by 4 (2 self)
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We present a new Feasibility Pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programming problems. Differences with the previously proposed Feasibility Pump algorithms and difficulties arising from nonconvexities in the models are extensively discussed. The main methodological innovations of this variant are: (a) the first subproblem is a nonconvex continuous Nonlinear Program, which is solved using global optimization techniques; (b) the solution method for the second subproblem is complemented by a tabu list. We exhibit computational results showing the good performance of the algorithm on instances taken from the MINLPLib.
Feasibilitybased bounds tightening via fixed points
"... Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, whi ..."
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Cited by 3 (1 self)
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Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole BranchandBound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints. Keywords: global optimization, MINLP, spatial BranchandBound, range reduction, constraint programming. 1
Comparison of convex relaxations of quadrilinear terms
"... In this paper we compare four different ways to compute a convex linear relaxation of a quadrilinear monomial on a box, analyzing their relative tightness. We computationally compare the quality of the relaxations, and we provide a general theorem on pairwisecomparison of relaxation strength, which ..."
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Cited by 2 (1 self)
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In this paper we compare four different ways to compute a convex linear relaxation of a quadrilinear monomial on a box, analyzing their relative tightness. We computationally compare the quality of the relaxations, and we provide a general theorem on pairwisecomparison of relaxation strength, which applies to some of our pairs of relaxations for quadrilinear monomials. Our results can be used to configure a spatial BranchandBound global optimization algorithm. We apply our results to the Molecular Distance Geometry Problem, demonstrating the usefulness of the present study. quadrilinear; convex relaxation; reformulation; global optimization, spatial Branch
A Storm of Feasibility Pumps for Nonconvex MINLP
, 2010
"... 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 successive pro ..."
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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 successive projection algorithms that iterate between solving a continuous relaxation and a mixedinteger relaxation of the original problems; such approaches currently exist in the literature for MixedInteger Linear Programs and convex MixedInteger 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 the main innovations in this paper are tailored algorithmic methods to overcome such a difficulty. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.
The ReformulationOptimization Software Engine ⋆
"... Abstract. Most optimization software performs numerical computation, in the sense that the main interest is to find numerical values to assign to the decision variables, e.g. a solution to an optimization problem. In mathematical programming, however, a considerable amount of symbolic transformation ..."
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Abstract. Most optimization software performs numerical computation, in the sense that the main interest is to find numerical values to assign to the decision variables, e.g. a solution to an optimization problem. In mathematical programming, however, a considerable amount of symbolic transformation is essential to solving difficult optimization problems, e.g. relaxation or decomposition techniques. This step is usually carried out by hand, involves human ingenuity, and often constitutes the “theoretical contribution ” of some research papers. We describe a ReformulationOptimization Software Engine (ROSE) for performing (automatic) symbolic computation on mathematical programming formulations. Keywords: reformulation, MINLP. 1