We propose a unifying framework for polyhedral approximation in convex optimization. It subsumes classical methods, such as cutting plane and simplicial decomposition, but also includes new methods, and new versions/extensions of old methods, such as a simplicial decomposition method for nondifferentiable optimization, and a new piecewise linear approximation method for convex single commodity network flow … Read more
The approach of estimate sequence offers an interesting rereading of a number of accelerating schemes proposed by Nesterov. It seems to us that this framework is the most appropriate descriptive framework to develop an analysis of the sensitivity of the schemes to approximations. We develop in this work a simple, self-contained, and unified framework for … Read more
In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems has been receiving increasing attention in the literature as it allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with … Read more
We consider a recently proposed optimization formulation of multi-task learning based on trace norm regularized least squares. While this problem may be formulated as a semidefinite program (SDP), its size is beyond general SDP solvers. Previous solution approaches apply proximal gradient methods to solve the primal problem. We derive new primal and dual reformulations of … Read more
This paper considers a special class of convex programming (CP) problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold. We present first-order methods for this class of problems based on an inexact version of the classical augmented Lagrangian (AL) approach, where the subproblems are approximately solved by means … Read more
Recently the proximal bundle method for minimizing a convex function has been extended to an inexact oracle that delivers function and subgradient values of unknown accuracy. We adapt this method to a partially inexact oracle that becomes exact only when an objective target level for a descent step is met. In Lagrangian relaxation, such oracles … Read more
We introduce a general-purpose learning machine that we call the “Guaranteed Error Machine”, or GEM, and two learning algorithms, a “real GEM algorithm” and an “ideal GEM algorithm”. The real GEM algorithm is for use in real applications, while the ideal GEM algorithm is introduced as a theoretical tool; however, these two algorithms have identical … Read more
In this paper we propose two modifications to Nesterov’s algorithms for minimizing convex functions in relative scale. The first is based on a bisection technique and leads to improved theoretical iteration complexity, and the second is a heuristic for avoiding restarting behavior. The fastest of our algorithms produces a solution within relative error O(1/k) of … Read more
In this paper we study the link between a semi-infinite chance-constrained optimization problem and its randomized version, i.e. the problem obtained by sampling a finite number of its constraints. Extending previous results on the feasibility of randomized convex programs, we establish here the feasibility of the solution obtained after the elimination of a portion of … Read more
This paper considers an important class of convex programming problems whose objective function $\Psi$ is given by the summation of a smooth and non-smooth component. Further, it is assumed that the only information available for the numerical scheme to solve these problems is the subgradient of $\Psi$ contaminated by stochastic noise. Our contribution mainly consists … Read more