Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new resu...

We analyze proximal methods based on entropy-like distances for the minimization of convex functions subject to nonnegativity constraints. We prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator. We also consider linearly constrained convex problems and establish a qua...

We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a ...

We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost over a finite horizon. Hard constraints are introduced first, and then reformulated in terms of probabilistic constraints. It is shown that, for a suitable parametrization of the control policy, a wide class of the resulting op...

Min–max problems have broad applications in machine learning, including learning with non-decomposable loss and robustness to data distribution. Convex–concave min–max problem is an active topic of research efficient algorithms sound theoretical foundations developed. However, it remains a challenge design provably for non-convex or without smoothness. In this paper, we study family problems, w...

This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to have an appropriate w...

We present in this paper the alternating minimization method on CAT(0) spaces for solving unconstraints convex optimization problems. Under the assumption that the function being minimize is convex, we prove that the sequence generated by our algorithm converges to a minimize point. The results presented in this paper are the first ones of this type that have been given for theses spaces and al...

The cyclic block coordinate descent-type (CBCD-type) methods, which performs iterative updates for a few coordinates (a block) simultaneously throughout the procedure, have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications include many popular statistical machine learning methods such as elastic-net regression, ridge penalized log...

In this paper we derive efficiency estimates of the regularized Newton’s method as applied to constrained convex minimization problems and to variational inequalities. We study a one-step Newton’s method and its multistep accelerated version, which converges on smooth convex problems as O( 1 k3 ), where k is the iteration counter. We derive also the efficiency estimate of a second-order scheme ...