In this paper we study general $l_p$ regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of first- and second-order stationary points, and hence also of local minimizers of the $l_p$ minimization problems. We extend some existing iterative reweighted $l_1$ (IRL1) and $l_2$ (IRL2) minimization methods to solve these problems and … Read more
This paper improves bounds for restricted isometry constant (RIC) in compressed sensing. Let \phi be a m*n real matrix and k be a positive integer with k
In this work we propose a trust-region algorithm for the problem of minimizing a function within a convex closed domain. We assume that the objective function is differentiable but no derivatives are available. The algorithm has a very simple structure and allows a great deal of freedom in the choice of the models. Under reasonable … Read more
We compare different approaches of optimization under uncertainty in the context of pricing strategies for conspicuous consumption products in recession periods of uncertain duration and strength. We consider robust worst-case ideas and how the concepts of Value at Risk (VaR) and Conditional Value at Risk (CVaR) can be incorporated efficiently. The approaches are generic in … Read more
We address the numerical problem of recovering large matrices of low rank when most of the entries are unknown. We exploit the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on a single Grassmann manifold. We then apply second-order Riemannian trust-region methods (RTRMC 2) and Riemannian conjugate gradient methods … Read more
As in most Data Mining procedures, how to tune the parameters of a Support Vector Machine (SVM) is a critical, though not sufficiently explored, issue. The default approach is a grid search in the parameter space, which becomes prohibitively time-consuming even when just a few parameters are to be tuned. For this reason, for models … Read more
This article explains, again, why radius of stability models, such as info-gap’s robustness model, are models of local robustness and why they are therefore unsuitable for the treatment of severe uncertainty. Citation Working Paper SM-12-2, Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, Australia. Article Download View Risk Analysis 101: fooled by … Read more
The Cauchy distribution has no moments (expected value, variance, etc.), because the defining integrals diverge. A way to “concentrate” the Cauchy distribution, in order to get finite moments, is suggested by an elementary problem in mechanics, giving the Cauchy distribution as a special case. The concentrated distribution has finite moments of all orders, while keeping … Read more
Dantzig-Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That … Read more
This paper formalizes and adapts the well known concept of Pareto efficiency in the context of the popular robust optimization (RO) methodology. We argue that the classical RO paradigm need not produce solutions that possess the associated property of Pareto optimality, and illustrate via examples how this could lead to inefficiencies and sub-optimal performance in … Read more