Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2019 | Book

SOC Functions Read chapter Read first chapter

SOC Functions and Their Applications

Author: Prof. Jein-Shan Chen

Publisher: Springer Singapore

Book Series : Springer Optimization and Its Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This book covers all of the concepts required to tackle second-order cone programs (SOCPs), in order to provide the reader a complete picture of SOC functions and their applications. SOCPs have attracted considerable attention, due to their wide range of applications in engineering, data science, and finance. To deal with this special group of optimization problems involving second-order cones (SOCs), we most often need to employ the following crucial concepts: (i) spectral decomposition associated with SOCs, (ii) analysis of SOC functions, and (iii) SOC-convexity and -monotonicity.

Moreover, we can roughly classify the related algorithms into two categories. One category includes traditional algorithms that do not use complementarity functions. Here, SOC-convexity and SOC-monotonicity play a key role. In contrast, complementarity functions are employed for the other category. In this context, complementarity functions are closely related to SOC functions; consequently, the analysis of SOC functions can help with these algorithms.

Table of Contents

Frontmatter
Chapter 1. SOC Functions
Abstract
During the past two decades, there have been active research for second-order cone programs (SOCPs) and second-order cone complementarity problems (SOCCPs). Various methods had been proposed which include the interior-point methods [1, 102, 109, 123, 146], the smoothing Newton methods [51, 63, 71], the semismooth Newton methods [86, 120], and the merit function methods [43, 48]. All of these methods are proposed by using some SOC complementarity function or merit function to reformulate the KKT optimality conditions as a nonsmooth (or smoothing) system of equations or an unconstrained minimization problem. In fact, such SOC complementarity functions or merit functions are closely connected to so-called SOC functions. In other words, studying SOC functions is crucial to dealing with SOCP and SOCCP, which is the main target of this chapter.
Jein-Shan Chen
Chapter 2. SOC-Convexity and SOC-Monotonity
Abstract
In this chapter, we introduce the SOC-convexity and SOC-monotonicity which are natural extensions of traditional convexity and monotonicity. These kinds of SOC-convex and SOC-monotone functions are also parallel to matrix-convex and matrix-monotone functions, see [21, 74]. We start with studying the SOC-convexity and SOC-monotonicity for some simple functions, e.g., \(f(t)=t^2, t^3, 1/t, t^{1/2}, |t|\), and \([t]_{+}\). Then, we explore characterizations of SOC-convex and SOC-monotone functions.
Jein-Shan Chen
Chapter 3. Algorithmic Applications
Abstract
In this chapter, we will see details about how the characterizations established in Chap. 2 be applied in real algorithms. In particular, the SOC-convexity are often involved in the solution methods of convex SOCPs; for example, the proximal-like methods. We present three types of proximal-like algorithms, and refer the readers to [115, 116, 118] for their numerical performance.
Jein-Shan Chen
Chapter 4. SOC Means and SOC Inequalities
Abstract
In this chapter, we present some other types of applications of the aforementioned SOC-functions, SOC-convexity, and SOC-monotonicity. These include so-called SOC means, SOC weighted means, and a few SOC trace versions of Young, Hölder, Minkowski inequalities, and Powers–Størmer’s inequality. We believe that these results will be helpful in convergence analysis of optimizations involved with SOC. Many materials of this chapter are extracted from [36, 77, 78], the readers can look into them for more details.
Jein-Shan Chen
Chapter 5. Possible Extensions
Abstract
It is known that the concept of convexity plays a central role in many applications including mathematical economics, engineering, management science, and optimization theory. Moreover, much attention has been paid to its generalization, to the associated generalization of the results previously developed for the classical convexity, and to the discovery of necessary and/or sufficient conditions for a function to have generalized convexities.
Jein-Shan Chen
Backmatter
Metadata
Title
SOC Functions and Their Applications
Author
Prof. Jein-Shan Chen
Copyright Year
2019
Publisher
Springer Singapore
Electronic ISBN
978-981-13-4077-2
Print ISBN
978-981-13-4076-5
DOI
https://doi.org/10.1007/978-981-13-4077-2