Stampacchia variational inequality with weak convex mappings

ABSTRACT

In this paper, the concept of weak convex set-valued mapping is introduced and various conditions for a set-valued mapping to be weak convex are given. Then, existence theorems for the Stampacchia variational inequality problem are established, when the involved mapping is weak convex.     

A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations

ABSTRACT

In this paper, we develop a three-term conjugate gradient method involving spectral quotient, which always satisfies the famous Dai-Liao conjugacy condition and quasi-Newton secant equation, independently of any line search. This new three-term conjugate gradient method can be regarded as a variant of the memoryless Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method with regard to spectral quotient. By combining this method with the projection technique proposed by Solodov and Svaiter in 1998, we establish a derivative-free three-term projection algorithm for dealing with large-scale nonlinear monotone system of equations. We prove the global convergence of the algorithm and obtain the R-linear convergence rate under some mild conditions. Numerical results show that our projection algorithm is effective and robust, and is more competitive with the TTDFP algorithm proposed Liu and Li [A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 2016;53:427–450].     

On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACT

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP.     

A cyclic iterative method for solving a class of variational inequalities in Hilbert spaces

ABSTRACT

The purpose of this paper is to introduce a new iterative method for solving a variational inequality over the set of common fixed points of a finite family of sequences of nearly non-expansive mappings in a real Hilbert space. And, using this result, we give some applications to the problem of finding a common fixed point of non-expansive mappings or non-expansive semigroups and the problem of finding a common null point of monotone operators.     

An improved three-term conjugate gradient algorithm for solving unconstrained optimization problems

In this article, we present an improved three-term conjugate gradient algorithm for large-scale unconstrained optimization. The search directions in the developed algorithm are proved to satisfy an approximate secant equation as well as the Dai-Liao’s conjugacy condition. With the standard Wolfe line search and the restart strategy, global convergence of the algorithm is established under mild conditions. By implementing the algorithm to solve 75 benchmark test problems with dimensions from 1000 to 10,000, the obtained numerical results indicate that the algorithm outperforms the state-of-the-art algorithms available in the literature. It costs less CPU time and smaller number of iterations in solving the large-scale unconstrained optimization.     

A new equivalent single-level problem for bilevel problems

In this article, we give necessary optimality conditions for a bilevel optimization problem (P). An intermediate single-level problem (Q), which is equivalent to the bilevel optimization problem (P), has been introduced.     

Addendum to: A hybrid conjugate gradient method based on a quadratic relaxation of Dai–Yuan hybrid conjugate gradient parameter

Here, necessary corrections on computing the hybridization parameter of the quadratic hybrid conjugate gradient method of Babaie-Kafaki [S. Babaie-Kafaki, A hybrid conjugate gradient method based on a quadratic relaxation of Dai-Yuan hybrid conjugate gradient parameter, Optimization, DOI: 10.1080/02331934.2011.611512, 2011] are stated in brief. Throughout, we use the same notations and equation numbers as in Babaie-Kafaki (2011).     

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.