Lagrangian relaxation is commonly used to generate bounds for mixed-integer linear programming problems. However, when the number of dualized constraints is very large (exponential in the dimension of the primal problem), explicit dualization is no longer possible. In order to reduce the dual dimension, different heuristics were proposed. They involve a separation procedure to dynamically … Read more
Given $\cA := \{a^1,\ldots,a^m\} \subset \R^n$, we consider the problem of reducing the input set for the computation of the minimum enclosing ball of $\cA$. In this note, given an approximate solution to the minimum enclosing ball problem, we propose a simple procedure to identify and eliminate points in $\cA$ that are guaranteed to lie … Read more
We extend a recently proposed alternating minimization algorithm to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1-norm. We derive the algorithm by … Read more
We modify the first order algorithm for convex programming proposed by Nesterov. The resulting algorithm keeps the optimal complexity obtained by Nesterov with no need of a known Lipschitz constant for the gradient, and performs better in practically all examples in a set of test problems. Citation Technical Report, Federal University of Santa Catarina, 2008. … Read more
The elegant results for strong duality and strict complementarity for linear programming, \LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a … 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
We study the common fixed points problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We propose a definition of sparseness of a family of operators and investigate a string-averaging algorithmic scheme that favorably handles the common fixed points problem when … Read more
Compressive (or compressed) sensing (CS) is an emerging methodology in computational signal processing that has recently attracted intensive research activities. At present, the basic CS theory includes recoverability and stability: the former quantifies the central fact that a sparse signal of length n can be exactly recovered from much less than n measurements via L1-minimization … Read more
After a brief introduction to Jordan algebras, we present a primal-dual interior-point algorithm for second-order conic optimization that uses full Nesterov-Todd-steps; no line searches are required. The number of iterations of the algorithm is $O(\sqrt{N}\log ({N}/{\varepsilon})$, where $N$ stands for the number of second-order cones in the problem formulation and $\varepsilon$ is the desired accuracy. … Read more
This paper considers a special but broad class of convex programing (CP) problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. We study two first-order penalty methods for solving the above class of problems, namely: the quadratic penalty method and … Read more