Suliman Al-Homidan

Quantum statistics of three modes coupled oscillators

From a quantum optics point of view the problem of three modes time-dependent coupled oscillators is considered. The connection related to the directional coupler is given, and the solution in the Heisenberg picture is obtained. The Glauber second-order correlation function has been used to discuss the bunching and antibunching. The phenomena of squeezing as well as the quasiprobability distribution functions (Wigner function and Q-function) are examined.

Nonsmooth variational-like inequalities and nonsmooth vector optimization

In this paper, we consider nonsmooth vector variational-like inequalities and nonsmooth vector optimization problems. By using the scalarization method, we define nonsmooth variational-like inequalities by means of Clarke generalized directional derivative and study their relations with the vector optimizations and the scalarized optimization problems. Some existence results for solutions of our nonsmooth variational-like inequalities are presented under densely pseudomonotonicity or pseudomonotonicity assumption.

Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings

The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a relatively nonexpansive mapping , and the set of zeros of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in . These new results represent the improvement, generalization, and development of the previously known ones in the literature.

Algorithms of Common Solutions for Generalized Mixed Equilibria, Variational Inclusions, and Constrained Convex Minimization

We introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Frechet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.

SQP algorithms for solving Toeplitz matrix approximation problem

Given an n × n matrix F, we find the nearest symmetric positive semi-definite Toeplitz matrix T to F. The problem is formulated as a non-linear minimization problem with positive semi-definite Toeplitz matrix as constraints. Then a computational framework is given. An algorithm with rapid convergence is obtained by l1 Sequential Quadratic Programming (SQP) method. Copyright

Nonsmooth Invexities, Invariant Monotonicities and Nonsmooth Vector Variational-Like Inequalities with Applications to Vector Optimization

It is well known that the convexity of functions plays a vital role in mathematical economics, engineering, management, optimization theory, etc. This concept in linear spaces relies on the possibility of connecting any two points of the space by the line segment between them. Since convexity is often not enjoyed the real problems, several classes of functions have been defined and studied for the purpose of weakening the limitations of convexity. In 1981, Hanson [33] realized that the convexity requirement, utilized to prove sufficient optimality conditions for a differentiable mathematical programming problem, can be further weakened by substituting the linear term y − x appearing in the definition of differentiable convex, pseudoconvex and quasiconvex functions with an arbitrary vector-valued function. In view of this idea, Hanson [33] (see also Craven [12]) introduced the concept of invexity by replacing the linear term y − x in the definition of convex function by a vector-valued function η(y, x).

Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized (F,α ,ρ,θ)-d-V-univex functions

In this paper, we introduce a new class of generalized (F,@a,@r,@q)-d-V-univex functions for a nonsmooth multiobjective programming problem. Sufficient optimality conditions under generalized (F,@a,@r,@q)-d-V-univex functions are established for a feasible solution to be an efficient solution. Appropriate duality theorems for a Mond-Weir-type dual are also presented under the aforesaid assumptions.

An iterative method for common solutions of equilibrium problems and hierarchical fixed point problems

AbstractThe purpose of this paper is to investigate the problem of finding an approximate point of the common set of solutions of an equilibrium problem and a hierarchical fixed point problem in the setting of real Hilbert spaces. We establish the strong convergence of the proposed method under some mild conditions. Several special cases are also discussed. Numerical examples are presented to illustrate the proposed method and convergence result. The results presented in this paper extend and improve some well-known results in the literature. MSC:49J30, 47H09, 47J20, 49J40.

Hybrid methods for approximating Hankel matrix

Hybrid methods for minimizing least distance functions with Hankel positive semi-definite matrix constraints are considered. Our approach is based on (i) a projection algorithm which converges globally but slowly; and (ii) the Newton method which is faster. Hybrid methods that attempt to combine the best features of both methods are then considered. Comparative numerical results are reported.

Collectively fixed point and maximal element theorems in topological semilattice spaces

In this article, we establish a collectively fixed point theorem and a maximal element theorem for a family of multivalued maps in the setting of topological semilattice spaces. As an application of our maximal element theorem, we prove the existence of solutions of generalized abstract economies with two constraint correspondences. We consider the system of (vector) quasi-equilibrium problems (in short, (S(V)QEP)) and system of generalized vector quasi-equilibrium problems (in short, (SGVQEP)). We first derive the existence result for a solution of (SQEP) and then by using this result, we prove the existence of a solution of system of a generalized implicit quasi-equilibrium problems. By using existence result for a solution of (SQEP) and weighted sum method, we derive an existence result for solutions of (SVQEP). By using our maximal element theorem, we also establish some existence results for the solutions of (SGVQEP). Some applications of our results to constrained Nash equilibrium problem for vector-valued functions with infinite number of players and to semi-infinite problems are also given.