We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends logarithmically in the dimension. This technique is based on a recu...

We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends logarithmically in the dimension. This technique is based on a recu...

We report on the implementation and computational testing of several versions of a set covering algorithm, based on the family of cutting planes from conditional bounds discussed in the companion paper [2]. The algorithm uses a set of heuristics to find prime covers, another set of heuristics to find feasible solutions to the dual linear program which are needed to generate cuts, and subgradien...

We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the...

Dual decomposition has been successfully employed in a variety of distributed convex optimization problems solved by a network of computing and communicating nodes. Often, when the cost function is separable but the constraints are coupled, the dual decomposition scheme involves local parallel subgradient calculations and a global subgradient update performed by a master node. In this paper, we...

Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely many C functions, and the subgradients of the indicator of a set defined...

The most of nonsmooth optimization methods may be divided in two main groups: subgradient methods and bundle methods. Usually, when developing new algorithms and testing them, the comparison is made between similar kinds of methods. In this report we test and compare both different bundle methods and different subgradient methods as well as some methods which may be considered as hybrids of the...

We investigate collaborative optimization in a multi-agent setting, when the agents execute in a distributed manner using local information, while the communication topology used to exchange messages and information is modeled by a graph-valued random process, independent of other time instances. Specifically, we study the performance of the consensus-based multi-agent subgradient method, for t...

In this paper, we consider the problem of minimizing the sum of nondifferentiable, convex functions over a closed convex set in a real Hilbert space, which is simple in the sense that the projection onto it can be easily calculated. We present a parallel subgradient method for solving it and the two convergence analyses of the method. One analysis shows that the parallel method with a small con...