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 symmetry-based analyses and methods for Linear Programming, Integer Linear Programming, Mixed-Integer Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and Mixed-Integer 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.

The best known method to find exact or at least ε-approximate solutions to polynomial programming problems is the spatial Branch-and-Bound 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 k-linear 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 well-studied applications: the Molecular Distance Geometry Problem and the Hartree-Fock Problem.

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 pairwise-comparison of relaxation strength, which applies to some of our pairs of relaxations for quadrilinear monomials. Our results can be used to configure a spatial Branch-and-Bound 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-

Abstract. The search tree size of the spatial Branch-and-Bound algorithm for Mixed-Integer 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 Branch-and-Bound 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 Branch-and-Bound, range reduction, constraint programming. 1

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 Reformulation-Optimization Software Engine (ROSE) for performing (automatic) symbolic computation on mathematical programming formulations. Keywords: reformulation, MINLP. 1

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 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 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.

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.

networks reconstruction

In this article we will give a brief overview of the start-of-the-art on software for the solution of mixed integer nonlinear programs (MINLP). We establish several groupings with respect to various features and give concise individual descriptions for each solver.

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