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 Gröbner basis) of linear subproblems that are used in a dual search strategy.

Abstract. The redundancy allocation problem is formulated minimizing the design cost for a series-parallel system with multiple component choices while ensuring a given system reliability level. The obtained model is a nonlinear integer programming problem with a nonlinear, nonseparable constraint. We propose a method based on the construction of a test set of an integer linear problem, which allows us to obtain an exact solution of the problem. It is compared to other approaches in the literature and standard nonlinear solvers.

Abstract. Many problems in engineering design involve the use of nonlinearities and some integer variables. Methods based on test sets have been proposed to solve some particular problems with integer variables, but they have not been frequently applied because of computation costs. The walk-back procedure based on a test set gives an exact method to obtain an optimal point of an integer programming problem with linear and nonlinear constraints, but the calculation of this test set and the identification of an optimal solution using the test set directions are usually computationally intensive.In problems for which obtaining the test set is reasonably fast, we show how the effectiveness can still be substantially improved. This methodology is presented in its full generality and illustrated on two specific problems: (1) minimizing cost in the problem of scheduling jobs on parallel machines given restrictions on demands and capacity, and (2) minimizing cost in the series parallel redundancy allocation problem, given a target reliability. Our computational results are promising and suggest the applicability of this approach to deal with other problems with similar characteristics or to combine it with mainstream solvers to certify optimality.Non-linear Integer Programming and test set and Gröbner basis and chance constrained programming

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