Search

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper deals with the generalized Clarke epiderivative of the extremum (or efficient point) multifunction in parametric vector optimization problems. The formulas for computing and/or estimating the generalized Clarke epiderivative of this extremum multifunction are given in terms of the Clarke tangent cone to the graph of a multifunction or the constraint mapping and/or the Fréchet derivative of the objective function. An application to semi-infinite programming is given. [ABSTRACT FROM AUTHOR]

Abstract: By constructing some suitable examples, Jeyakumar and Yen (2004)  have shown that the Robinson metric regularity (Rmr) and the Lipschitz-like property (Llp) of implicit multifunctions are not equivalent. This paper clarifies relationships between the two properties of implicit multifunctions. It turns out that the (reasonable) sufficient conditions for having (Rmr) (Llp) are quite different from those for the validity of the reverse implication. The implicit function theorem due to Yen and Yao (2009) serves as a tool for our analysis of (Rmr) and (Llp). [Copyright &y& Elsevier]

In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the strong convergence of an inexact proximalpoint algorithm. It is known that the proximal-point algorithm converges weakly to a solution of a maximal monotone operator, but fails to converge strongly. Solodov and Svaiter (Math. Program. 87:189–202, 2000) introduced a new proximal-type algorithm to generate a strongly convergent sequence and established a convergence result in Hilbert space. Subsequently, Kamimura and Takahashi (SIAM J. Optim. 13:938–945, 2003) extended the Solodov and Svaiter result to the setting of uniformly convex and uniformly smooth Banach space. On the other hand, Rockafellar (SIAM J. Control Optim. 14:877–898, 1976) gave an inexact proximal-point algorithm which is more practical than the exact one. Our purpose is to extend the Kamimura and Takahashi result to a new inexact proximal-type algorithm. Moreover, this result is applied to the problem of finding the minimizer of a convex function on a uniformly convex and uniformly smooth Banach space. [ABSTRACT FROM AUTHOR]

The article presents a discussion that looks into the wide class of inexact proximal-type methods for handling the generalized variational inequality problem with maximal monotone operator in a reflexive Banach space. An extension of Kamimura and Takahashi's algorithm is also provided with the aim of developing a generic inexact proximal point algorithm. A convergence analysis which unifies a broad class of proximal-type methods associated with finding Hilbert spaces or Banach spaces is also included.

Let Z and X be Hausdorff real topological vector spaces and let Lb(X, Z) be the space of continuous linear mappings from X into Z equipped with the topology of bounded convergence. In this paper, we define the (S)+ condition for operators from a nonempty subset of X into Lb(X, Z) and derive some existence results for vector variational inequalities with operators of the class (S)+. Some applications to vector complementarity problems are given. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new composite iterative scheme to approximate a zero of an -accretive operator defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of . The results presented in this paper substantially improve and extend the results due to Ceng et al. [L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. Math. (in press)], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643]. Our work provides a new approach for the construction of a zero of -accretive operators. [Copyright &y& Elsevier]