Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of … Read more
A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and $8$. Thus, for even $n\geq 10$, instances of this … Read more
Nonconvex quadratic programming (QP) is an NP-hard problem that optimizes a general quadratic function over linear constraints. This paper introduces a new global optimization algorithm for this problem, which combines two ideas from the literature—finite branching based on the first-order KKT conditions and polyhedral-semidefinite relaxations of completely positive (or copositive) programs. Through a series of … Read more
We recommend an implementation of the Markowitz problem to generate stable portfolios with respect to perturbations of the problem parameters. The stability is obtained proposing novel calibrations of the covariance matrix between the returns that can be cast as convex or quasiconvex optimization problems. A statistical study as well as a sensitivity analysis of the … Read more
Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In … Read more
Combinatorial and logic constraints arising in a number of challenging optimization applications can be formulated as vanishing constraints. Quadratic programs with vanishing constraints (QPVCs) then arise as subproblems during the numerical solution of such problems using algorithms of the Sequential Quadratic Programming type. QPVCs are nonconvex problems violating standard constraint qualifications. In this paper, we … Read more
Parametric Active Set Methods (PASM) are a relatively new class of methods to solve convex Quadratic Programming (QP) problems. They are based on tracing the solution along a linear homotopy between a QP with known solution and the QP to be solved. We explicitly identify numerical challenges in PASM and develop strategies to meet these … Read more
The quasi-Newton strategy presented in this paper preserves one of the most important features of the stabilized Sequential Quadratic Programming method, the local convergence without constraint qualifications assumptions. It is known that the primal-dual sequence converges quadratically assuming only the second-order sufficient condition. In this work, we show that if the matrices are updated by … Read more
Computational methods are considered for finding a point that satisfies the second-order necessary conditions for a general (possibly nonconvex) quadratic program (QP). The first part of the paper considers the formulation and analysis of an active-set method for a generic QP with both equality and inequality constraints. The method uses a search direction that is … Read more
Burer has shown that completely positive relaxations of nonconvex quadratic programs with nonnegative and binary variables are exact when the binary variables satisfy a so-called key assumption. Here we show that introducing binary variables to obtain an equivalent problem that satisfies the key assumption will not improve the semidefinite relaxation, and only marginally improve the … Read more