Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When ${\cal D}f$ satisfies the $L$-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton's method. As applications, we also obtain Kantorovich's type theorem, Smale's type theorem under the $\gamma$-condition and an extension of the theory of Smale's approximate zeros.

The image space approach is applied to the study of vector quasiequilibrium problems. Exploiting separation arguments in the image space, Lagrangian-type optimality conditions and gap functions for vector quasiequilibrium problems are derived.

In this paper we first study the relationship between local and global Minty-Browder monotone operators and then we show that these operators have generally convex preimages. Our results allow to show that positive semidefinitedness on the complement of a discrete set of the differential operator implies the Minty-Browder monotonicity of the operator itself. We also show that complex functions of one complex variable are Minty-Browder monotone under suitable conditions. Finally, we obtain some injectivity/univalency theorems that generalize some well-known results.

This research/survey paper firstly gives an overview of generalized convexity in calculus of variations and nonlinear elasticity, centered at the notions of quasiconvexity, polyconvexity, and rank-one-convexity. Then ${\cal A}$-convexity based on Young measures and relaxation are discussed. In this context a general version of the Jensen’s inequality for ${\cal A}$-convex functions is given that extends the classical Jensen’s inequality for convex functions.

Secondly new results for the unilateral contact problem in nonlinear elasticity are presented. In particular existence results are derived for the pure contact-traction problem under an appropriate recession condition for quasiconvex as well as for nonquasiconvex energy densities, using in the latter case the Young measure approach.

We consider well-posedness under perturbations of vector quasiequilibrium and bilevel-equilibrium problems. This kind of well-posedness relates Hadamard and Tikhonov well-posedness notions to sensitivity analysis and we apply largely techniques of the latter to establish sufficient conditions for wellposedness under perturbations. We also propose several new semicontinuity and quasiconvexity notions to weaken the imposed assumptions. Our results are new or include as special cases recent existing results. Many examples are provided for the illustration purpose.

We consider the monotone variational inequality of finding $x^*\in C$ such that $\langle (I-T)x^*,x-x^*\rangle \ge 0$ for $x\in C$, where $C$ is a closed convex subset of a real Hilbert space and $T$ is a nonexpansive self-mapping of $C$. Techniques of nonexpansive mappings are applied to regularize this variational inequality. The regularized solutions and an iteration process are shown to converge in norm to a solution of this variational inequality.

Many papers on both scalar and multiobjective optimization problems use the assumption that the objective and constraint functions are invex with respect to the same function $η$. In this note we characterize the finite families of functions for which this condition holds.

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.

The complementarity problem (CP) is one of the basic topics in nonlinear analysis. Since the constraint set of CP is a convex cone or a cone segment, weak order monotonicity properties can be utilized for its analysis instead of the usual norm monotonicity ones. Such nonlinear CPs with order monotonicity properties have a great number of applications, especially in economics and mathematical physics. Most solution methods were developed for the single-valued case, but this assumption seems too restrictive in many applications. In the paper, we consider extended concepts of multi-valued Z-mappings and examine a class of generalized mixed complementarity problems (MCPs) with box constraints, whose cost mapping is a general composition of multi-valued mappings possessing Z type properties. We develop a Gauss-Seidel algorithm for these MCPs. Some examples of computational experiments are also given.

In this paper, we formulate a general dual problem for a class of nondifferentiable multiobjective programs involving the support function of a compact convex set and linear functions. Fritz John and Kuhn-Tucker optimality conditions are presented. In addition, we establish weak and strong duality theorems for weakly efficient solutions under suitable generalized $(F,\alpha, \rho, d)$ convexity assumptions. Some special cases of our duality results are given.

The purpose of this paper is to introduce two hybrid proximal point algorithms to solve the constrained minimization problem for a convex functional in a uniformly convex and uniformly smooth Banach space. Using those iterative schemes, we establish the strong convergence theorems for relatively nonexpansive mappings which generalize the recent results in the literature.

In this paper, we study the connectedness and path-connectedness of the solution sets for symmetric vector equilibrium problems in locally convex Hausdorff topological vector spaces under some suitable assumptions. The results presented in this paper generalize some known results in [10, 14, 24, 32, 33].

The aim of this paper is to propose quasiconvexity concepts for discrete single variable functions and state some related optimality conditions. Four classes of discrete quasiconvex single variable functions are introduced, compared and characterized. Two different algorithm procedures for determining a minimum are provided.