We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model of the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle … Read more
The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing in particular several characterizations of … Read more
With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based ${\cal C}^1$-smooth univariate cubic $L_1$ splines. An $L_1$ spline minimizes the $L_1$ norm of the difference between the first-order derivative of the spline and the local divided difference of … Read more
The traffic assignment problem with elastic demands can be formulated as an optimization problem, whose objective is sum of a congestion function and a disutility function. We propose to use a variant of the Analytic Center Cutting Plane Method to solve this problem. We test the method on instances with different congestion functions (linear with … Read more
We use the proximal point method with the phi-divergence given by phi(t) = t – log t – 1 for the minimization of quasiconvex functions subject to nonnegativity constraints. We establish that the sequence generated by our algorithm is well-defined in the sense that it exists and it is not cyclical. Without any assumption of … Read more
Given a set of points $\cS = \{x^1,\ldots,x^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze an algorithm for the problem of computing a $(1 + \eps)$-approximation to the the minimum volume axis-aligned ellipsoid enclosing $\cS$. We establish that our algorithm is polynomial for fixed $\eps$. In addition, the algorithm returns a small … Read more
In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on the Euclidean space and the nonnegative orthant. For the unconstrained minimization problem, assumming that the function is bounded from below and lower semicontinuous we prove that iterations fxkg given by 0 2 b@(f(:)+(k=2)jj:xk1jj2)(xk) are … Read more
An algorithm is developed for minimizing nonsmooth convex functions. This algorithm extends Elzinga-Moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to Elzinga-Moore’s algorithm what a proximal bundle method is to Kelley’s algorithm. Instead of lower approximations … Read more
We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in … Read more
We present a target-following framework for semidefinite programming, which generalizes the target-following framework for linear programming. We use this framework to build weighted path-following interior-point algorithms of three distinct flavors: short-step, predictor-corrector, and large-update. These algorithms have worse-case iteration bounds that parallel their counterparts in linear programming. We further consider the problem of finding analytic … Read more