Recovery Of Primal Solution In Dual Subgradient Schemes
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Recovery of Primal Solution in Dual Subgradient Schemes
In this thesis, we study primal solutions for general optimization problems. In particular, we employ the subgradient method to solve the Lagrangian dual of a convex constrained problem, and use a primal-averaging scheme to obtain near-optimal and near-feasible primal solutions. We numerically evaluate the performance of the scheme in the framework of Network Utility Maximization (NUM), which has recently drawn great research interest. Specifically for the NUM problems, which can have concave or nonconcave utility functions and linear constraints, we apply the dual-based decentralized subgradient method with averaging to estimate the rate allocation for individual users in a distributed manner, due to its decomposability structure. Unlike the existing literature on primal recovery schemes, we use a constant step-size rule in view of its simplicity and practical significance. Under the Slater condition, we develop a way to effectively reduce the amount of feasibility violation at the approximate primal solutions, namely, by increasing the value initial dual iterate; moreover, we extend the established convergence results in the convex case to the more general and realistic situation where the objective function is convex. In particular, we explore the asymptotical convergence properties of the averaging sequence, the tradeoffs involved in the selection of parameter values, the estimation of duality gap for particular functions, and the bounds for the amount of constraint violation and value of primal cost per iteration. Numerical experiments performed on NUM problems with both concave and nonconcave utility functions show that, the averaging scheme is more robust in providing near-optimal and near-feasible primal solutions, and it has consistently better performance than other schemes in most of the test instances.
Numerical Nonsmooth Optimization
Solving nonsmooth optimization (NSO) problems is critical in many practical applications and real-world modeling systems. The aim of this book is to survey various numerical methods for solving NSO problems and to provide an overview of the latest developments in the field. Experts from around the world share their perspectives on specific aspects of numerical NSO. The book is divided into four parts, the first of which considers general methods including subgradient, bundle and gradient sampling methods. In turn, the second focuses on methods that exploit the problem’s special structure, e.g. algorithms for nonsmooth DC programming, VU decomposition techniques, and algorithms for minimax and piecewise differentiable problems. The third part considers methods for special problems like multiobjective and mixed integer NSO, and problems involving inexact data, while the last part highlights the latest advancements in derivative-free NSO. Given its scope, the book is ideal for students attending courses on numerical nonsmooth optimization, for lecturers who teach optimization courses, and for practitioners who apply nonsmooth optimization methods in engineering, artificial intelligence, machine learning, and business. Furthermore, it can serve as a reference text for experts dealing with nonsmooth optimization.
Computational Science And Its Applications - Iccsa 2005
Author: Osvaldo Gervasi
language: en
Publisher: Springer Science & Business Media
Release Date: 2005-04-27
The four-volume set LNCS 3480-3483 constitutes the refereed proceedings of the International Conference on Computational Science and Its Applications, ICCSA 2005, held in Singapore in May 2005. The four volumes present a total of 540 papers selected from around 2700 submissions. The papers span the whole range of computational science, comprising advanced applications in virtually all sciences making use of computational techniques as well as foundations, techniques, and methodologies from computer science and mathematics, such as high performance computing and communication, networking, optimization, information systems and technologies, scientific visualization, graphics, image processing, data analysis, simulation and modelling, software systems, algorithms, security, multimedia etc.