Variational problems have proved of value in many image processing and analysis applications. However increase of sensor resolution as occurred in medical imaging and experimental fluid dynamics demand for adapted solving strategies to handle the huge amount of data. In this paper we address the decomposition of the general class of quadratic variational problems, which includes several importa...

The applicant develops new tools of variational analysis; in particular, two new versions of the Ekeland variational principle and subdifferential variational principle for set-valued mappings, as well as new constructions of extremal set systems. They play a significant role to studying an important class of constrained optimization problems called Multiobjective Optimization Problems with Equ...

We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known threestep forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for so...

The main purpose of this paper is to analyze the variational iteration method for solving the two-point boundary value problems. We show that Wang [Shu-Qjang Wang, A variational approach to nonlinear two-point boundary value problems, Comput. Math. Appl. 58 (2009) 2452-2455] demonstrated some incorrect solutions for two-point boundary value problems studied by means of the variational approach....

We introduce a discrete-time fractional calculus of variations on the time scale hZ, h > 0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of ...

In this paper we survey the relationships between scalar and vector variational inequalities (of differential type) and the underlying optimization problem. We show that the variational inequalities of Stampacchia type can be viewed as necessary optimality conditions, while the variational inequalities of Minty type can be considered as sufficient optimality conditions. Concerning this last sta...

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational ...

In this paper‎, ‎we study the existence of solutions for a‎ ‎ fractional boundary value problem‎. ‎By using critical point theory‎ ‎ and variational methods‎, ‎we give some new criteria to guarantee‎ ‎ that‎ ‎ the problems have at least one solution and infinitely many solutions.

Fractional action-like variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multi-dimensional fractional action-like problems of the calculus of variations. 2000 Mathematics Subject Classification: 49K10, 49S05.

For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-vari...