In this paper, we study a class of biquadratic optimization problems. We first relax the original problem to its semidefinite programming (SDP) problem and discuss the approximation ratio between them. Under some conditions, we show that the relaxed problem is tight. Then we consider how to approximately solve the problems in polynomial time. Under several … Read more
Recently, researchers have been interested in studying the semidefinite programming (SDP) relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained quadratic programming (QCQP). Comparing to the basic SDP relaxation, this doubly-positive SDP model possesses additional O(n2) constraints, which makes the SDP solution complexity substantially higher than that for the … Read more
In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods … Read more
In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved … Read more
We present two strategies for choosing a “hot” starting-point in the context of an infeasible Potential Reduction (PR) method for convex Quadratic Programming. The basic idea of both strategies is to select a preliminary point and to suitably scale it in order to obtain a starting point such that its nonnegative entries are sufficiently bounded … Read more
We consider a quadratic programming (QP) problem ($\Pi$) of the form $\min x^T C x$ subject to $Ax \ge b$ where $C\in {\mathbb R}^{n\mbox{\tiny\texttimes} n}_+, rank(C)=1$ and $A\in {\mathbb R}^{m\mbox{\tiny\texttimes} n}, b\in {\mathbb R}^m$. We present an FPTAS for this problem by reformulating the QP ($\Pi$) as a parametrized LP and “rounding” the optimal solution. … Read more
Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving a linear program with linear complementarity constraints, i.e., an LPCC. Alternatively, this task can be … Read more
This paper considers constraint propagation methods for continuous constraint satisfaction problems consisting of linear and quadratic constraints. All methods can be applied after suitable preprocessing to arbitrary algebraic constraints. The basic new techniques consist in eliminating bilinear entries from a quadratic constraint, and solving the resulting separable quadratic constraints by means of a sequence of … Read more
We consider a regression problem where uncertainty affects to the dependent variable of the elements of the database. A model based on the standard epsilon-Support Vector Regression approach is given, where two hyperplanes need to be constructed to predict the interval-valued dependent variable. By using the Hausdorff distance to measure the error between predicted and … Read more
We study linear optimization problems over the cone of copositive matrices. These problems appear in nonconvex quadratic and binary optimization, for instance the maximum clique problem and other combinatorial problems can be reformulated as such a problem. We present new polyhedral inner and outer approximations of the copositive cone which we show to be exact … Read more