Stochastic-approximation gradient methods are attractive for large-scale convex optimization because they offer inexpensive iterations. They are especially popular in data-fitting and machine-learning applications where the data arrives in a continuous stream, or it is necessary to minimize large sums of functions. It is known that by appropriately decreasing the variance of the error at each … Read more
We consider models of two-stage stochastic programming with a quantile second stage criterion and optimization models with a chance constraint on the second stage objective function values. Such models allow to formalize requirements to reliability and safety of the system under consideration, and to optimize the system in extreme conditions. We suggest a method of … Read more
In this paper, we present new methods for black-box convex minimization. They do not need to know in advance the actual level of smoothness of the objective function. Their only essential input parameter is the required accuracy of the solution. At the same time, for each particular problem class they automatically ensure the best possible … Read more
We present an inexact bundle method for minimizing an unconstrained convex sup-function with an open domain. Under some mild assumptions, we reformulate a convex conic programming problem as such problem in terms of the support function. This method is a first-order method, hence it requires much less computational cost in each iteration than second-order approaches … Read more
We propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations. Most bundle algorithms use a cutting-plane model of the objective function to formulate a subproblem whose solution gives the next iterate. Proximal bundle methods employ the model in the objective function of the subproblem, while level … Read more
The main aim of this short note is to show that the sub\-differentiability and duality results established by Laghdir (2005), Laghdir and Benabbou (2007), and Alimohammady \emph{et al.}\ (2011), stated in Fréchet spaces, are consequences of the corresponding known results using Moreau–Rockafellar type conditions. Article Download View On the use of semi-closed sets and functions … Read more
Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann’s theorem on matrix norms is an early example. We discuss the example of “identifiability”, a common property of nonsmooth functions associated with the existence of a smooth manifold of approximate critical points. Identifiability (or its synonym, “partial smoothness”) is the … Read more
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 … Read more
We consider optimization problems with an objective function that is the sum of two convex terms: one is smooth and given by a black-box oracle, and the other is general but with a simple, known structure. We first present an accelerated proximal gradient (APG) method for problems where the smooth part of the objective function … Read more
Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. We extend Fourier-Motzkin elimination to semi-infinite linear programs which are linear programs with finitely many variables and infinitely many constraints. Applying projection leads to new characterizations of important properties for primal-dual pairs of semi-infinite programs such as zero duality gap, feasibility, boundedness, and solvability. … Read more