An algebraic approach to Integer Portfolio problems

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 Matemáticas de la Universidad de Sevilla MTM2007-67433-C02-0

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

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

Software tools for stochastic programming: A Stochastic Programming Integrated Environment (SPInE)

SP models combine the paradigm of dynamic linear programming with modelling of random parameters, providing optimal decisions which hedge against future uncertainties. Advances in hardware as well as software techniques and solution methods have made SP a viable optimisation tool. We identify a growing need for modelling systems which support the creation and investigation of SP problems. Our SPInE system integrates a number of components which include a flexible modelling tool (based on stochastic extensions of the algebraic modelling languages AMPL and MPL), stochastic solvers, as well as special purpose scenario generators and database tools. We introduce an asset/liability management model and illustrate how SPInE can be used to create and process this model as a multistage SP application

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

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 $N$ convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size $N$ 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 $N$. 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 $S$ of $\mathbb R^d$. The key elements are generalizations of Helly's theorem where the convex sets are required to intersect $S \subset \mathbb R^d$. The size of samples in both algorithms will be directly determined by the $S$-Helly numbers. Motivated by the first half of the paper, for any subset $S \subset \mathbb R^d$, we introduce the notion of an $S$-optimization problem, where the variables take on values over $S$. It generalizes continuous, integer, and mixed-integer optimization. We illustrate with examples the expressive power of $S$-optimization to capture sophisticated combinatorial optimization problems with difficult modular constraints. We reinforce the evidence that $S$-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 $S$.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

A review of portfolio planning: Models and systems

In this chapter, we first provide an overview of a number of portfolio planning models which have been proposed and investigated over the last forty years. We revisit the mean-variance (M-V) model of Markowitz and the construction of the risk-return efficient frontier. A piecewise linear approximation of the problem through a reformulation involving diagonalisation of the quadratic form into a variable separable function is also considered. A few other models, such as, the Mean Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the Minimax (MM) model which use alternative metrics for risk are also introduced, compared and contrasted. Recently asymmetric measures of risk have gained in importance; we consider a generic representation and a number of alternative symmetric and asymmetric measures of risk which find use in the evaluation of portfolios. There are a number of modelling and computational considerations which have been introduced into practical portfolio planning problems. These include: (a) buy-in thresholds for assets, (b) restriction on the number of assets (cardinality constraints), (c) transaction roundlot restrictions. Practical portfolio models may also include (d) dedication of cashflow streams, and, (e) immunization which involves duration matching and convexity constraints. The modelling issues in respect of these features are discussed. Many of these features lead to discrete restrictions involving zero-one and general integer variables which make the resulting model a quadratic mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the algorithms and solution methods for this class of problems are also discussed. The issues of preparing the analytic data (financial datamarts) for this family of portfolio planning problems are examined. We finally present computational results which provide some indication of the state-of-the-art in the solution of portfolio optimisation problems