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About 10,000 papers for “Optimization and Variational Analysis”

VARIATIONAL ANALYSIS IN OPTIMIZATION AND CONTROL

Boris S. Mordukhovich · 2008

Variational analysis has been recognized as a rapidly growing and fruitful area in mathematics concerning mainly the study of optimization and equilibrium problems, while also applying perturbation ideas and variational principles to a broad class of problems and situations that may be not of a variational nature. It can be viewed as a modern outgrowth of the classical calculus of variations, optimal control theory, and mathematical programming with the focus on perturbation/approximation techniques, sensitivity issues, and applications; see (1; 2; 3) One of the most characteristic features of modern variational analysis is the intrinsic presence of nonsmoothness, i.e., the necessity to deal with nondifferentiable functions, sets with nonsmooth boundaries, and set-valued mappings. Nonsmoothness naturally enters not only through initial data of optimization-related problems (particularly those with inequality and geometric constraints) but largely via variational principles and other optimization, approximation, and perturbation techniques applied to problems with even smooth data. In fact, many fundamental objects frequently appearing in the framework of variational analysis (e.g., the distance function, value functions in optimization and control problems, maximum and minimum functions, solution maps to perturbed constraint and variational systems, etc.) are inevitably of nonsmooth and/or set-valued structures requiring the development of new forms of analysis that involve generalized differentiation. It is important to emphasize that even the simplest and historically earliest problems of optimal control are intrinsically nonsmooth, in contrast to the classical calculus of variations. This is mainly due to pointwise constraints on control functions that often take only discrete values as in

Cited by 0

Some topics in variational analysis and optimization

Le Hoang Anh Nguyen · theses.fr (ABES) · 2014

Dans cette thèse, j’étudie d’abord la théorie des [gamma]-limites. En dehors de quelques propriétés fondamentales des [gamma]-limites, les expressions de [gamma]-limites séquentielles généralisant des résultats de Greco sont présentées. En outre, ces limites nous donnent aussi une idée d’une classification unifiée de la tangence et la différentiation généralisée. Ensuite, je développe une approche des théories de la différentiation généralisée. Cela permet de traiter plusieurs dérivées généralisées des multi-applications définies directement dans l’espace primal, tels que des ensembles variationnels,des ensembles radiaux, des dérivées radiales, des dérivées de Studniarski. Finalement, j’étudie les règles de calcul de ces dérivées et les applications liées aux conditions d’optimalité et à l’analyse de sensibilité.

Cited by 1Open access

Fixed point theory, variational analysis, and optimization

Al-Mezel, Saleh Abdullah R, Al-Solamy, Falleh Rajallah M, Ansari, Qamrul Hasan · 2015

<P>"There is a real need for this book. It is useful for people who work in areas of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics."<BR>-Nan-Jing Huang, Sichuan University, Chengdu, People's Republic of China</P>

Cited by 0

Fixed Point Theory, Variational Analysis, and Optimization

Saleh Abdullah Al-Mezel, Falleh Rajallah M. Al-Solamy, Qamrul Hasan Ansari · 2014

Fixed Point Theory, Variational Analysis, and Optimization not only covers three vital branches of nonlinear analysis-fixed point theory, variational inequalities, and vector optimization-but also explains the connections between them, enabling the study of a general form of variational inequality problems related to the optimality conditions invol

Cited by 32

Variational Gram Functions: Convex Analysis and Optimization

Amin Jalali, Maryam Fazel, Lin Xiao · SIAM Journal on Optimization · 2017

We introduce a new class of convex penalty functions, called variational Gram functions (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as regularizers in convex optimization problems arising from hierarchical classification, multitask learning, and estimating vectors with disjoint supports, among other applications. We study convexity of VGFs, and give characterizations for their convex conjugates, subdifferentials, proximal operators, and related quantities. We discuss efficient optimization algorithms for regularized loss minimization problems where the loss admits a common, yet simple, variational representation and the regularizer is a VGF. These algorithms enjoy a simple kernel trick, an efficient line search, as well as computational advantages over first order methods based on the subdifferential or proximal maps. We also establish a general representer theorem for such learning problems. Last, numerical experiments on a hierarchical classification problem are presented to demonstrate the effectiveness of VGFs and the associated optimization algorithms.

Cited by 11

Local Maximal Monotonicity in Variational Analysis and Optimization

Pham Duy Khanh, Vu Vinh Huy Khoa, Boris S. Mordukhovich et al. · Mathematics of Operations Research · 2025

The paper is devoted to a systematic study and characterizations of notions of local maximal monotonicity and their strong counterparts for set-valued operators that appear in variational analysis, optimization, and their applications. We obtain novel resolvent characterizations of these notions together with efficient conditions for their preservation under summation in broad infinite-dimensional settings. Further characterizations of these notions are derived by using generalized differentiation of variational analysis in the framework of Hilbert spaces. Funding: This research was supported by the U.S. National Science Foundation [Grants DMS-1808978 and DMS-2204519], the Australian Research Council Discovery Project [Grant DP-190100555], and Project 111 of China [Grant D21024].

Cited by 1

Variational Analysis and Optimization Analyse variationnelle et optimisation

Org Jiming Peng (macmaster, Et Jane Ye (victoria · 2008

On quasiconvex mathematical programming problems with equilibrium constraints A mathematical programming problem with equilibrium constraints (MPEC) is a constrained optimization problem with equality and inequality constraints and, additionally, some equilibrium constraints: (MPEC) min f(z) g(z) ≤ 0,

Cited by 0

Local Maximal Monotonicity in Variational Analysis and Optimization

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

The paper is devoted to a systematic study and characterizations of notions of local maximal monotonicity and their strong counterparts for set-valued operators that appear in variational analysis, optimization, and their applications. We obtain novel resolvent characterizations of these notions together with efficient conditions for their preservation under summation in broad infinite-dimensional settings. Further characterizations of these notions are derived by using generalized differentiation of variational analysis in the framework of Hilbert spaces.

Cited by 0Open access

Partial Second-Order Subdifferentials in Variational Analysis and Optimization

Boris S. Mordukhovich, Nguyen Mau Nam, Nguyen Thi Yen Nhi · Numerical Functional Analysis and Optimization · 2014

This article presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. The main results concern developing extended calculus rules for these second-order constructions in both finite-dimensional and infinite-dimensional frameworks. We also provide new applications of partial second-order subdifferentials to Lipschitzian stability of stationary point mappings in parametric constrained optimization and discuss some other applications.

Cited by 21

Some nonconvex geometric results in variational analysis and optimization.

CUHK Digital Repository (Chinese University of Hong Kong) · 2007

In this thesis, we consider the following two important subjects in the modern variational analysis for the corresponding nonconvex/nonmonotone and nonsmooth cases: geometric results and the variational inequality problem. By using the variational technique, we first present several nonsmooth (nonconvex) geometric results (including an approximate projection result, an extended extremal principle, nonconvex separation theorems, a nonconvex generalization of the Bishop-Phelps theorem and a separable point result) which extend some fundamental theorems in linear functional analysis, convex analysis and optimization theory. Then, by transforming the variational inequality problem into equivalent optimization problems, we establish some error bound result for the nonsmooth and nonmonotone variational inequality problem.

Cited by 0