We investigate projected scaled gradient (PSG) methods for convex minimization problems. These methods perform a descent step along a diagonally scaled gradient direction followed by a feasibility regaining step via orthogonal projection onto the constraint set. This constitutes a generalized algorithmic structure that encompasses as special cases the gradient projection method, the projected Newton method, … Read more
We focus on a class of nonconvex cooperative optimization problems that involve multiple participants. We study the duality framework and provide geometric and analytic character- izations of the duality gap. The dual problem is related to a market setting in which each participant pursuits self interests at a given price of common goods. The duality … Read more
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
METRIC Approximation is a popular model for supply chain management. We prove that it has a unimodal objective function when the demands of the n retailers are normally distributed. That allows us to solve it with a convergent sequence. This optimization method leads us to a closed-form equation of computational complexity O(n). Its solutions are … Read more
We present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Flexible Coordinate Descent (FCD). At each iteration of … 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
We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in … Read more
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
In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization … 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