Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation.Comment: Technical Report, School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, Scotland, United Kingdo

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Solution Techniques For Non-convex Optimization Problems

This thesis focuses on solution techniques for non-convex optimization problems. The first part of the dissertation presents a generalization of the completely positive reformulation of quadratically constrained quadratic programs (QCQPs) to polynomial optimization problems. We show that by explicitly handling the linear constraints in the formulation of the POP, one obtains a refinement of the condition introduced in Bai\u27s (2015) Thoerem on QCQPs, where the refined theorem only requires nonnegativity of polynomial constraints over the feasible set of the linear constraints. The second part of the thesis is concerned with globally solving non-convex quadratic programs (QPs) using integer programming techniques. More specifically, we reformulate non-convex QP as a mixed-integer linear problem (MILP) by incorporating the KKT condition of the QP to obtain a linear complementary problem, then use binary variables and big-M constraints to model the complementary constraints. We show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. The solution approach is implemented and labeled as quadprogIP, where computational results are presented in comparison with quadprogBB, BARON and CPLEX. The third part of the thesis involves the formulation and solution approach of a problem that arises from an on-demand aviation transportation network. A multi-commodity network flows (MCNF) model with side constraints is proposed to analyze and improve the efficiency of the on-demand aviation network, where the electric vertical-takeoff-and-landing (eVTOLs) transportation vehicles and passengers can be viewed as commodities, and routing them is equivalent to finding the optimal flow of each commodity through the network. The side constraints capture the decisions involved in the limited battery capacity for each eVTOL. We propose two heuristics that are efficient in generating integer feasible solutions that are feasible to the exponential number of battery side constraints. The last part of the thesis discusses a solution approach for copositive programs using linear semi-infinite optimization techniques. A copositive program can be reformulated as a linear semi-infinite program, which can be solved using the cutting plane approach, where each cutting plane is generated by solving a standard quadratic subproblem. Numerical results on QP-reformulated copositive programs are presented in comparison to the approximation hierarchy approach in Bundfuss (2009) and Yildirim (2012)

A clique algorithm for standard quadratic programming

A standard Quadratic Programming problem (StQP) consists in minimizing a (nonconvex) quadratic form over the standard simplex. For solving a SLQP we present an exact and a heuristic algorithm, that are based on new theoretical results for quadratic and convex optimization problems. With these results a StQP is reduced to a constrained nonlinear minimum weight clique problern in an associated graph. Such a Clique problem, which does not seem to have been Studied before, is then solved with all exact and a heuristic algorithm. Some computational experience shows that Our algorithms are able to solve StQP problems of at least one order of magnitude larger than those reported in the literature. (c) 2007 Elsevier B.V. All rights reserved