Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of black-box functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1

The performance of Branch-and-Bound algorithms is severely impaired by the presence of symmetric optima in a given problem. We describe a method for the automatic detection of formulation symmetries in MINLP instances. A software implementation of this method is used to conjecture the group structure of the problem symmetries of packing equal circles in a square. We provide a proof of the conjecture and compare the performance of spatial Branch-and-Bound on the original problem with the performance on a reformulation that cuts away symmetric optima. Keywords: MINLP, spatial Branch-and-Bound, Global Optimization, group, reformulation.

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.

Abstract. Solution symmetries in integer linear programs often yield long Branch-and-Bound based solution processes. We propose a method for finding elements of the permutation group of solution symmetries, and two different types of symmetry-breaking constraints to eliminate these symmetries at the modelling level. We discuss some preliminary computational results.

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

Summary. We discuss a mixed-integer nonlinear programming formulation for the problem of covering a set of points with a given number of slabs of minimum width, known as the bottleneck variant of the hyperplane clustering problem. We derive several linear approximations, which we solve using a standard mixed-integer linear programming solver. A computational comparison of the performance of the different linearizations is provided. Key words: MINLP, k-line center problem, reformulation, linearization. 1

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We discuss a problem arising in the strategic management of IT enterprises: that of replacing some existing services with new services without impairing operations. We formalize the problem by means of a Mathematical Programming formulation of the Mixed-Integer Nonlinear Programming class and show it can be solved to a satisfactory optimality approximation guarantee by means of existing off-the-shelf software tools.