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About 10,000 papers for “Variational analysis”

Variational Analysis

Wim Stefanus Ackooij, Welington Luis Oliveira · RePEc: Research Papers in Economics

Abstract In this chapter, we will discuss tools from variational analysis that will be useful when analysing the convergence of algorithms. Our main focus will be on locally Lipschitzian functions, for which generalized directional derivatives and subdifferential can be defined. For the latter, our focus will mostly be on Clarke’s subdifferential, but other possibilities exist also. We will briefly mention these as well. The chapter also covers the topic of constraint qualification, in other words, conditions allowing us to express the geometry of a given set of constraints in optimality conditions, in terms of nominal underlying data.

Cited by 0

Variational Analysis

Shouchuan Hu, Νικόλαος Παπαγεωργίου · Birkhäuser advanced texts. Basler Lehrbücher · 2022

Cited by 0

Variational Numerical Analysis

Luke, D. Russell · GoeScholar The Publication Server of the Georg-August-Universität Göttingen (Georg-August-Universität Göttingen) · 2023

This monograph covers numerical analysis using the tools of variational analysis to handle nonsmooth and nonconvex problems in continuous optimization

Cited by 0

Co-Variational Analysis

Joachim Blatter, Markus Haverland · Palgrave Macmillan UK eBooks · 2012

We call the first research approach to small-N studies co-variational analysis (COV). This methodological approach presents empirical evidence of the existence of co-variation between an independent variable X and a dependent variable Y to infer causality. This approach has dominated the thinking about small-N research designs since the 1970s. We present it here as the first approach because it has been central to most of the methodological reflections on small-N research designs. Nevertheless, as we will see at the end of this chapter, because of its inherent limitations, it seems wise to complement this approach with other approaches, especially with elements from the causal-process tracing approach (CPT).

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Quasi-variational analysis

Masahiro Kubo · Sugaku Expositions · 2017

We introduce the concept of quasi-variational principles (QVPs) and present general methods of analysis using QVPs for problems that are not necessarily of Euler–Lagrange type. We also introduce a new class of nonlinear operators named “quasi-subdifferential operators” for the mathematical analysis of variational and quasi-variational inequalities, and derive their existence theorems from QVPs. Furthermore, we explain the notion of quasi-subdifferential evolution equations.

Cited by 2

Variational and Strong Variational Convexity in Infinite-Dimensional Variational Analysis

Pham Duy Khanh, Vu Vinh Huy Khoa, Boris S. Mordukhovich et al. · arXiv (Cornell University) · 2023

This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by Rockafellar, the importance of them has been already recognized and documented in finite-dimensional variational analysis and optimization. Here we address general infinite-dimensional settings and derive comprehensive characterizations of both variational and strong variational convexity notions by developing novel techniques, which are essentially different from finite-dimensional counterparts. As a consequence of the obtained characterizations, we establish new quantitative and qualitative relationships between strong variational convexity and tilt stability of local minimizers in appropriate frameworks of Banach spaces.

Cited by 0Open access

Partially Smooth Variational Analysis

Jonathan M. Borwein, Variational Principles, D. Minimality et al. · 1999

I will describe the basic ideas from Banach space theory that allow one to fruitfully apply notions from smooth analysis in spaces which do not admit smooth renorms (or "bumps") and so to perform Partially Smooth Variational Analysis. I shall also mention some open functional analytic questions relating to best approximation, viscosity and variational analysis.

Cited by 0

SUBGRADIENTS AND VARIATIONAL ANALYSIS

R. T. Rockafellar · 2010

Abstract. The study of problems of maximization or minimization subject to constraints has been a fertile field for the development of mathematical analysis from classical times. In recent decades, convexity has come forward as an important tool, and the geometry of convexity has been translated into notions of directional derivatives and subgradients of functions that may not be differentiable in the ordinary sense. Now there has emerged a form of analysis able to deal robustly, even in the absence of convexity, with the phenomena of nonsmoothness that arise in variational problems. The Need for Variational Analysis Differential calculus has been so successful in treating a variety of physical phenomena that mathematics has long relied on differentiable functions as the main tools of analysis. The domain of such a function, in a finite-dimensional setting, is typically an open subset of IR n or of a differentiable manifold such as might be defined by a system of equations in IR n and coordinatized locally by IR d for some d < n. Many of the systems and phenomena that have come under mathematical scrutiny in recent decades, however, are of interest especially at their frontiers of feasibility. They involve functions and mappings whose domains may be

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