Robust optimization programs are hard to solve even when the constraints are convex. In previous contributions, it has been shown that approximately robust solutions (i.e. solutions feasible for all constraints but a small fraction of them) to convex programs can be obtained at low computational cost through constraints randomization. In this paper, we establish new … Read more
A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the … Read more
We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concordant barriers. This approach extends the reach of the modern theory of interior-point methods, and lays the foundation for new ways to treat … Read more
A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually and of a smooth term modeling the data formation model as well … Read more
The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a … 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
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
In http://www.optimizationonline.org/DB_HTML/2005/05/1136.html 05/25/05, we have demonstrated that the family of all affine non-anticipative output-based control laws in a discrete time linear dynamical system affected by uncertain disturbances is equivalent, as far as state-control trajectories are concerned, to the family of all affine non-anticipative “purified-output-based” control laws. The advantage of the latter representation of affine controls … Read more
The inclusion of transaction costs is an essential element of any realistic portfolio optimization. In this paper, we consider an extension of the standard portfolio problem in which convex transaction costs are incurred to rebalance an investment portfolio. In particular, we consider linear, piecewise linear, and quadratic transaction costs. The Markowitz framework of mean-variance efficiency … Read more