We propose a proximal point algorithm to solve LAROS problem, that is the problem of finding a “large approximately rank-one submatrix”. This LAROS problem is used to sequentially extract features in data. We also develop a new stopping criterion for the proximal point algorithm, which is based on the duality conditions of \eps-optimal solutions of … Read more
In this work we propose solving huge-scale instances of the truss topology design problem with coordinate descent methods. We develop four efficient codes: serial and parallel implementations of randomized and greedy rules for the selection of the variable (potential bar) to be updated in the next iteration. Both serial methods enjoy an O(n/k) iteration complexity … Read more
We derive $C^2-$characterizations for convex, strictly convex, as well as uniformly convex functions on full dimensional convex sets. In the cases of convex and uniformly convex functions this weakens the well-known openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the … Read more
We propose a Newton method for solving smooth unconstrained vector optimization problems under partial orders induced by general closed convex pointed cones. The method extends the one proposed by Fliege, Grana Drummond and Svaiter for multicriteria, which in turn is an extension of the classical Newton method for scalar optimization. The steplength is chosen by … Read more
Iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets. When the objective consists of a loss function plus a nonsmooth regularization term, the solution often lies on a low-dimensional manifold of parameter space along which the regularizer is smooth. … Read more
The problem of finding a minimum l^1-norm solution to an underdetermined linear system is an important problem in compressed sensing, where it is also known as basis pursuit. We propose a heuristic optimality check as a general tool for l^1-minimization, which often allows for early termination by “guessing” a primal-dual optimal pair based on an … Read more
Recently, Nemirovski’s analysis indicates that the extragradient method has the $O(1/t)$ convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, we have developed a class of Fej\’er monotone projection and contraction methods. Until now, only convergence results are available to these projection and contraction methods, though … Read more
The rank function $\rank(\cdot)$ is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of $\rank(\cdot)$, and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the … Read more
The stable principal component pursuit (SPCP) problem is a non-smooth convex optimization problem, the solution of which has been shown both in theory and in practice to enable one to recover the low rank and sparse components of a matrix whose elements have been corrupted by Gaussian noise. In this paper, we first show how … Read more
This work introduces the use of the treed Gaussian process (TGP) as a surrogate model within the mesh adaptive direct search (MADS) framework for constrained blackbox optimization. It extends the surrogate management framework (SMF) to nonsmooth optimization under general constraints. MADS uses TGP in two ways: one, as a surrogate for blackbox evaluations; and two, … Read more