Stochastic approximation techniques play an important role in solving many problems encountered in machine learning or adaptive signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct computation is too intensive, and they have thus to be estimated online from the observed signals. For batch optimization of … Read more
This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity … Read more
In this article we present a version of the proximal alternating direction method for a convex problem with linear constraints and a separable objective function, in which the standard quadratic regularizing term is replaced with an interior proximal metric for those variables that are required to satisfy some additional convex constraints. Moreover, the proposed method … Read more
In this paper, we present a conditional gradient type (CGT) method for solving a class of composite optimization problems where the objective function consists of a (weakly) smooth term and a (strongly) convex regularization term. While including a strongly convex term in the subproblems of the classical conditional gradient (CG) method improves its rate of … Read more
We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves … Read more
The Barzilai-Borwein method, an effective gradient descent method with clever choice of the step-length, is adapted from nonlinear optimization to Riemannian manifold optimization. More generally, global convergence of a nonmonotone line-search strategy for Riemannian optimization algorithms is proved under some standard assumptions. By a set of numerical tests, the Riemannian Barzilai-Borwein method with nonmonotone line-search … Read more
We study large scale extended trust region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS and that there are efficient algorithms for solving … Read more
In traditional nonlinear programming, the technique of converting a problem with inequality constraints into a problem containing only equality constraints, by the addition of squared slack variables, is well-known. Unfortunately, it is considered to be an avoided technique in the optimization community, since the advantages usually do not compensate for the disadvantages, like the increase … Read more
In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of “no-gap” … Read more
We develop algorithms for inner approximating the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation … Read more