5,565 research outputs found

    An algebraic approach to Integer Portfolio problems

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    Integer variables allow the treatment of some portfolio optimization problems in a more realistic way and introduce the possibility of adding some natural features to the model. We propose an algebraic approach to maximize the expected return under a given admissible level of risk measured by the covariance matrix. To reach an optimal portfolio it is an essential ingredient the computation of different test sets (via Gr\"obner basis) of linear subproblems that are used in a dual search strategy.Universidad de Sevilla P06-FQM-01366Junta de Andalucía (Plan Andaluz de Investigación) FQM-333Ministerio de Ciencia e Innovación (España) MTM2007-64509Instituto de Matemåticas de la Universidad de Sevilla MTM2007-67433-C02-0

    Finding Multiple Solutions in Nonlinear Integer Programming with Algebraic Test-Sets

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    We explain how to compute all the solutions of a nonlinear integer problem using the algebraic test-sets associated to a suitable linear subproblem. These test-sets are obtained using Gröbner bases. The main advantage of this method, compared to other available alternatives, is its exactness within a quite good efficiency.Ministerio de Economía y Competitividad MTM2016-75024-PMinisterio de Economía y Competitividad MTM2016-74983-C2- 1-RJunta de Andalucía P12-FQM-269

    Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of SS-optimization

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    The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs utilizes random sampling on the uncertainty parameter to substitute the original problem with a representative continuous convex optimization with NN convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size NN of the sampling relaxation to yield high-likelihood feasible solutions of the chance-constrained problem. They measured the probability of the original constraints to be violated by the random optimal solution from the relaxation of size NN. This paper has two main contributions. First, we present a generalization of the Calafiore-Campi results to both integer and mixed-integer variables. In fact, we demonstrate that their sampling estimates work naturally for variables restricted to some subset SS of Rd\mathbb R^d. The key elements are generalizations of Helly's theorem where the convex sets are required to intersect S⊂RdS \subset \mathbb R^d. The size of samples in both algorithms will be directly determined by the SS-Helly numbers. Motivated by the first half of the paper, for any subset S⊂RdS \subset \mathbb R^d, we introduce the notion of an SS-optimization problem, where the variables take on values over SS. It generalizes continuous, integer, and mixed-integer optimization. We illustrate with examples the expressive power of SS-optimization to capture sophisticated combinatorial optimization problems with difficult modular constraints. We reinforce the evidence that SS-optimization is "the right concept" by showing that the well-known randomized sampling algorithm of K. Clarkson for low-dimensional convex optimization problems can be extended to work with variables taking values over SS.Comment: 16 pages, 0 figures. This paper has been revised and split into two parts. This version is the second part of the original paper. The first part of the original paper is arXiv:1508.02380 (the original article contained 24 pages, 3 figures
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