The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm’s rate of convergence … Read more
This paper shows that $p$-order metric regularity is preserved under perturbation of H\”older continuous mapping of order $1/p$, which answers affirmatively a problem posed recently by Dontchev. CitationTechnical report, Department of Mathematics, Chinese University of Hong Kong, 07/2015
For reducing costs of market frictions, investors need to build a small-scale portfolio by solving a cardinality constrained portfolio selection problem which is NP-hard in general and not easy to be solved eciently for a large-scale problem. In this paper, we propose a penalty proximal alternat- ing linearized minimization method for the large-scale problems in … Read more
We address the minimization of the sum of a proper, convex and lower semicontinuous with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The … Read more
We propose an algorithm for solving nonsmooth, nonconvex, constrained optimization problems as well as a new set of visualization tools for comparing the performance of optimization algorithms. Our algorithm is a sequential quadratic optimization method that employs Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton Hessian approximations and an exact penalty function whose parameter is controlled using a steering strategy. … Read more
We propose ARock, an asynchronous parallel algorithmic framework for finding a fixed point to a nonexpansive operator. In the framework, a set of agents (machines, processors, or cores) update a sequence of randomly selected coordinates of the unknown variable in an asynchronous parallel fashion. As special cases of ARock, novel algorithms for linear systems, convex … Read more
This paper describes a bundle method for (approximately) minimizing complicated nonsmooth convex functions with additive structure, with the primary goal of computing bounds on the solution values of difficult optimization problems such as stochastic integer programs. The method combines features that have appeared in previously proposed bundle methods, but not in the particular configuration we … Read more
In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generates by the algorithm is convergent for the pseudomonotone case and weakly convergent for the quasimonotone ones. This approach unifies the … Read more
In this paper we analyze boosting algorithms in linear regression from a new perspective: that of modern first-order methods in convex optimization. We show that classic boosting algorithms in linear regression, namely the incremental forward stagewise algorithm (FS-epsilon) and least squares boosting (LS-Boost-epsilon), can be viewed as subgradient descent to minimize the loss function defined … Read more
This paper describes an algorithm for solving structured nonsmooth convex optimization problems using OSGA, a first-order method with the complexity $O(\eps^{-2})$ for Lipschitz continuous nonsmooth problems and $O(\eps^{-1/2})$ for smooth problems with Lipschitz continuous gradient. If the nonsmoothness of the problem is manifested in a structured way, we reformulate the problem in a form that … Read more