In this article, motivated by Rockafellar’s proximal point algorithm in Hilbert spaces, we discuss various weak and strong convergence theorems for resolvents of accretive operators and maximal monotone operators which are connected with the proximal point algorithm. We first deal with proximal point algorithms in Hilbert spaces. Then, we consider weak and strong convergence theorems for resolvents of accretive operators in Banach spaces which generalize the results in Hilbert spaces. Further, we deal with weak and strong convergence theorems for three types of resolvents of maximal monotone operators in Banach spaces which are related to proximal point algorithms. Finally, in Section 7, we apply some results obtained in Banach spaces to the problem of finding minimizers of convex functions in Banach spaces.

Mann's fixed point algorithm can be written as a line search method that generates a sequence $\{x_n\}$ through the recursive manner $x_{n+1}=x_n-\alpha_nv_n$, where $\alpha_n$ is the stepsize and where $v_n$ is the search direction given by $v_n=x_n-Tx_n$, with $T$ being a nonexpansive mapping. This line search method has widely been used in optimization, variational inequalities and fixed point problems. In this paper, we address the problem of selection of the sequence of parameters, $\{\alpha_n\}$, so as to have optimal convergence of this algorithm.

When the dual of a normed space $X$ is endowed with the weak$^*$ topology, the biconjugates of the proper convex lower semicontinuous functions defined on $X$ coincide with the functions themselves. This is not the case when $X^*$ is endowed with the strong topology. Working in the latter framework, we give formulae for the biconjugates of some functions that appear often in convex optimization, which hold provided the validity of some suitable regularity conditions. We also treat some special cases, rediscovering and improving recent results in the literature. Finally, we give a regularity condition that guarantees that the biconjugate of the supremum of a possibly infinite family of proper convex lower semicontinuous functions defined on a separated locally convex space coincides with the supremum of their biconjugates.

We introduce two pairs of nondifferentiable multiobjective second order symmetric dual problems with cone constraints over arbitrary closed convex cones, which is different from the one proposed by Mishra and Lai [12]. Under suitable second order pseudo-invexity assumptions we establish weak, strong and converse duality theorems as well as self-duality relations. Our symmetric duality results include an extension of the symmetric duality results for the first order case obtained by Kim and Kim [7] to the second order case. Several known results are abtained as special cases.

In this paper we extend some results given in [L. Rodríguez-Marín, M. Sama, $\tau^{w}$-contingent epiderivatives in reflexive spaces, Nonlinear Analysis, {68} (2008), 3780-3788]. Following the same approach, we associate a set-valued optimization problem with a family of simpler problems by using a decoupling of the ordering cone into half-spaces. In this context we give necessary and sufficient optimality conditions in terms of epiderivatives with respect to half-spaces. Moreover we obtain computation formulas for these conditions in term of derivatives of scalar set-valued maps.

This is to establish a new fixed point theorem for compact closed multimaps in the better admissible class defined on subsets of generalized convex uniform spaces and having Klee approximable ranges. Our new theorem unifies a large number of previous results. Some of them are listed in the chronological order.

In this paper, we prove Ekeland’s type of variational principle for a vector equilibrium problem, and present a Caristi-Kirk type fixed point theorem and an existence result for vector equilibrium solution.

Let $\{ \tau_n \}$ be a sequence of numbers recursively defined by $$ f(\tau_n) + f(\tau_n+\tau_{n-1}) + \dots + f( \tau_n+\tau_{n-1} + \dots +\tau_1 ) =1 , $$ where $f$ is a continuous and strictly decreasing function on $(0,\infty) $ with $f(0^+) \ge 1, $ and $ f(\infty)=0 .$ Assume the convexity of $\log f$ or $\log |f'|$. It can be shown that $ \{ \tau_n \} $ is increasing. Thus $ \lim \tau_n $ exists in $ (0, \infty]$.

The difference equation above is motivated by a heat conduction problem studied in Myshkis (1997) and Chen, Chow and Hsieh (2006).

In this paper, we propose new hybrid proximal-type algorithms for a maximal monotone operator in a Banach spaces and establish some strong convergence results. An application to the problem of finding a minimizer of a convex function is given.

We revisit optimization theory involving set functions which are defined on a family of measurable subsets in a measure space. In this paper, we focus on a minimax fractional programming problem with subdifferentiable set functions. Using nonparametric necessary optimality conditions, we introduce generalized $(\mathcal{F},\rho, \theta)$-convexity to establish several sufficient optimality conditions for a minimax programming problem, and construct a new dual model to unify the Wolfe type dual and the Mond-Weir type dual as special cases of this dual programming problem. Finally we establish a weak, strong, and strict converse duality theorem.

Management by objectives (MBO) is an effective way for enterprise management. By setting the targets of the productivity the companies try their best, including adjustment of resource allocation and competence, to reach the targets. Within the same framework of productivity and of resources the targets may not be attainable. However, by stretching a little bit, human capacity, resources, the production coefficients, and other relevant parameters may be adjusted so as to make the target feasible. In this article, we formulate the program into linear programming model and study how to optimally adjust the relevant coefficients so that the target solution could be attainable. In case the target is unattainable, we may either utilize the bisection method or the fuzzy linear programming techniques to revise the target as to make it a reachable one.

In this paper, to decide the best point of many efficient points in vector optimization, we consider an evaluate method of efficient points for solutions in vector optimization problem. We introduce an evaluate function of efficient points, and show properties of the evaluate function.

This paper concerns sensitivity analysis for general parametric constrained problems of multiobjective optimization in infinite-dimensional spaces by using advanced tools of modern variational analysis and generalized differentiation. We pay the main attention to computing and estimating coderivatives of frontier and efficient solution maps in parametric multiobjective problems with respect to generalized order optimality that include a vast majority of conventional multiobjective problems in the presence of geometric, operator, functional, and equilibrium constraints. The obtained results are new in both finite-dimensional and infinite-dimensional spaces.

This short paper surveys recent several applications of an order preserving operator inequality, especially, logarithmic trace inequalities are presented.

A representation theorem for all norms equivalent to the original norm in a complex Hilbert space is established by using ellipsoids.

Nowadays one of the most concerned topics in the field of complex networks is to find how the synchronizability depends on various parameters of networks. Complex networks with coupling delays have gained increasing attention in various fields of science and engineering in the past decade. One interesting problem is to investigate the effect of delay on dynamical behaviors and to determine the range of delay, in which, the synchronization stability can be achieved. In this paper, based on the qualitative theory of linear time-delay systems, the synchronization stability in complex dynamical networks with coupling delays is considered and some stability criteria of synchronization state are obtained. It is shown that by virtue of these obtained criteria, the range of delay on synchronization stability of complex networks with coupling delays can be analytically estimated. Finally, a couple of examples are illustrated which agree well with our theoretical results.

In this paper, we introduce a class of new iterative procedures with errors approximating the common fixed point for three generalized asymptotically quasi-nonexpansive mappings and prove several strong convergence theorems for the iterative procedures with errors in uniformly convex Banach spaces. Our results extend and improve the recent corresponding theorems obtained by Chang-Kim-Jin [6], Ghosh-Debnath [7], Kim-Kim-Kim [14], Li-Kim-Huang [17], Liu ([21, 22]), Nammanee-Noor-Suantai [24], Suantai [27], Tan-Xu [28], Xu-Noor [30], Zeng-Wong-Yao [32], Zeng-Yao [33], Zhou-Cho-Grabiec [34] and Zhou-Guo-Hwang-Cho [35].

We study the controllability problem for a system governed by a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. Notice that we are not assuming that the generalized semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows to obtain the general controllability principle in the terms of the topological degree theory for condensing multivalued operators. Two realizations of this principle are considered.