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A new inclusion problem is introduced using generalized Cayley operator and we call it Cayley inclusion problem. We also study its corresponding resolvent equation problem. By using a generalized resolvent operator and generalized Yosida approximation operator, first we establish a fixed point formulation for Cayley inclusion problem. An algorithm is defined to find the solution of Cayley inclusion problem. An existence and convergence result is proved. Secondly, we have shown the equivalence of Cayley inclusion problem with a resolvent equation. We define an iterative algorithm with some of its equivalent forms for solving resolvent equation problem. A numerical example is constructed and a convergence graph is shown by using MATLAB program. [ABSTRACT FROM AUTHOR]

This paper presents two new theorems on Geoffrion's properly efficient solutions and seven examples illustrating their applications to linear fractional vector optimization problems with unbounded constraint sets. Provided that all the components of the objective function are properly fractional, the first theorem gives sufficient conditions for the efficient solution set to coincide with the Geoffrion properly efficient solution set. Admitting that the objective function can have some affine components, in the second theorem we give sufficient conditions for an efficient solution to be a Geoffrion's properly efficient solution. The recession cone of the constraint set, the derivatives of the scalar objective functions, but no tangent cone to the constraint set at the efficient point, are used in the second theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method. Under suitable conditions, some strong convergence theorems are proved. As applications, we utilize our results to study the convex minimization problem, variational inequality problem. [ABSTRACT FROM AUTHOR]

We introduce two new iterative algorithms with line-search process for solving a variational inequality problem with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms in a real Hilbert space. [ABSTRACT FROM AUTHOR]

This article proposes two general inertial algorithms with line-search process for finding a solution of a variational inequality problem with fixed-point constraints via a subgradient-extragradient approach. Common solution theorems are obtained in Hilbert spaces. [ABSTRACT FROM AUTHOR]

E.G. Belousov and V.G. Andronov [Solvability and Stability of Problems of Polynomial Programming (in Russian), Publishing House of the Moscow University, Moscow, p. 4, 1993] have observed that the notions of structural convexity and ravine of a polynomial function, which were introduced by themselves, are useful for studying stability and solvability of convex polynomial mathematical programming problems, as well as for the investigation of the distribution of integer points in convex sets. The results given in Chapters 3-5 of the book justify their observations. This paper presents some facts about structural convexity and ravines of quadratic functions. Among other things, we obtain a verifiable criterion for structural convexity of a quadratic function and show that such a function cannot have ravines along linear subspaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study two modified hybrid proximal-point algorithms for finding a common element of the solution set EP of a generalized equilibrium problem and the set for two maximal monotone operators and defined on a Banach space . Strong and weak convergence theorems for these two modified hybrid proximal-point algorithms are established.

First-order methods such as proximal gradient, which use Forward–Backward Splitting techniques have proved to be very effective in solving nonsmooth convex minimization problem, which is useful in solving various practical problems in different fields such as machine learning and image processing. In this paper, we propose few new forward–backward splitting algorithms, which consume less number of iterations to converge to an optimum. In addition, we derive convergence rates for the proposed formulations and show that the speed of convergence of these algorithms is significantly better than the traditional forward–backward algorithm. To demonstrate the practical applicability, we apply them to two real-world problems of machine learning and image processing. The first issue deals with the regression on high-dimensional datasets, whereas the second one is the image deblurring problem. Numerical experiments have been conducted on several publicly available real datasets to verify the obtained theoretical results. Results demonstrate the superiority of our algorithms in terms of accuracy, the number of iterations required to converge and the rate of convergence against the classical first-order methods. [ABSTRACT FROM AUTHOR]

A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented. [ABSTRACT FROM AUTHOR]

In the present paper, differential inclusions with a causal multioperator are considered. We apply the method of nonsmooth integral guiding potentials to the study of the problem on the asymptotic behaviour of solutions for such inclusions. We investigate both the case of a closed and convex-valued right-hand part and the case of a nonconvex-valued and lower semicontinuous right-hand part. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified inertial subgradient extragradient method for solving a variational inequality problem with Lipschitz pseudomonotone mapping and a common fixed-point problem of a family of nonexpansive mappings. Under mild conditions, we obtain strong convergence theorems in a real Hilbert space. An application is also provided. [ABSTRACT FROM AUTHOR]