Given ${\cal A} := \{a^1,\ldots,a^m\} \subset \R^n$ with corresponding positive weights ${\cal W} := \{\omega_1,\ldots,\omega_m\}$, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point $c_{\cal A} \in \R^n$ that minimizes the maximum weighted Euclidean distance from $c_{\cal A}$ to each point in ${\cal … 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
The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously … Read more
In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [17], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and … Read more
Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of Computational Convex Analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is … Read more
Mean-variance (MV) analysis is often sensitive to model mis-specification or uncertainty, meaning that the MV efficient portfolios constructed with an estimate of the model parameters (i.e., the expected return vector and covariance of asset returns) can give very poor performance for another set of parameters that is similar and statistically hard to distinguish from the … Read more
This paper concerns a fractional function of the form $x^Ta/\sqrt{x^TBx}$, where $B$ is positive definite. We consider the game of choosing $x$ from a convex set, to maximize the function, and choosing $(a,B)$ from a convex set, to minimize it. We prove the existence of a saddle point and describe an efficient method, based on … Read more
Radiation therapy is an important modality in treating various cancers. Various treatment planning and delivery technologies have emerged to support Intensity Modulated Radiation Therapy (IMRT), creating significant opportunities to advance this type of treatment. We investigate the possibility of including the dose prescription, specified by the DVH, within the convex optimization framework for inverse IMRT … Read more
In this work, a regression problem is studied where the elements of the database are sets with certain geometrical properties. In particular, our model can be applied to handle data affected by some kind of noise or uncertainty and interval-valued data, and databases with missing values as well. The proposed formulation is based on the … Read more
In this paper, we analyze the numerical behaviour of a separable Augmented Lagrangian algorithm with multiple scaling parameters, different parameters associated with each dualized coupling constraint as well as with each subproblem. We show that an optimal superlinear rate of convergence can be theoretically attained in the twice differentiable case and propose an adaptive scaling … Read more