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Convex Analysisの二,三の進展について

This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.

Proximal Algorithms

Power Aware Routing in Mobile Ad Hoc Networks

Nonlinear Programming: A Unified Approach

Given a maximal monotone operator T in a Banach space, a family of enlargements E(T) of T has been introduced by Svaiter. He also defined a sum and a positive scalar multiplication of enlargements. The first aim of this work is to further study the properties of these operations. Burachik and Svaiter studied a family of convex functions H(T) which is in a one to one correspondence with E(T). The second

Operating enlargements of monotone operators: new connections with convex functions

We consider a primal optimization problem in a reflexive Banach space and a duality scheme via generalized augmented Lagrangians. For solving the dual problem (in a Hilbert space), we introduce and analyze a new parameterized Inexact Modified Subgradient (IMSg) algorithm. The IMSg generates a primal-dual sequence, and we focus on two simple new choices of the stepsize. We prove that every weak accumulation point of the primal sequence is a primal solution and the dual sequence converges weakly to a dual solution, as long as the dual optimal set is nonempty. Moreover, we establish primal convergence even when the dual optimal set is empty. Our second choice of the stepsize gives rise to a variant of IMSg which has finite termination.

An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y∈T(x)) satisfying x∈A and y∈A ⊥ An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn The resulting generalized equation can be solved by using the proximal point algorithm We consider instead the use of hybrid proximal methods Hybrid methods use enlargements of operators, close in spirit to the concept of ϵ-subdifferentials We characterize the enlargement of the partial inverse operator in terms of the enlargement of T itself We present a new algorithm of resolution that combines Spingarn and hybrid methods, we prove for this method global convergence only assuming existence of solutions and maximal monotonicity of T We also show that, under standard assumptions, the method has a linear rate of convergence For the important problem of finding a zero of a sum of maximal monotone operators T 1, …, T m , we present a highly paralleliza

An inexact method of partial inverses and a parallel bundle method

Variational Analysis and Generalized Differentiation in Optimization and Control

We study some theoretical properties of augmented Lagrangian duality schemes for nonconvex and nonsmooth optimization. These schemes may not have primal convergence, in the sense that limit points of the primal sequence may not solve the primal problem. First, we prove that primal convergence holds if and only if the dual function is differentiable at the dual limit. Second, we relate the latter differentiability property with the existence of exact penalty parameters. Finally, we give a condition on the Lagrangian which ensures that every point supports an exact penalty representation.

Regina S. Burachik

Papers

Operating enlargements of monotone operators: new connections with convex functions

An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians

An inexact method of partial inverses and a parallel bundle method

Variational Analysis and Generalized Differentiation in Optimization and Control

On primal convergence for augmented Lagrangian duality