Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets. Our result improves and extends the corresponding results announced by many others.     

A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others.     

Strong convergence for a minimization problem on solutions of systems of equilibrium problems and common fixed points of an infinite family and semigroup of nonexpansive mappings

In this paper, we introduce a suitable Mann type algorithm for finding a common element of the set of solutions of systems of equilibrium problems and the set of common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme to the unique solution of the minimization problem on the solution of systems of equilibrium problems and the common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings. Our results extend and improve the recent result of Colao and Marino [V. Colao and G. Marino, Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings, Nonlinear Anal. 73 (2010) 3513–3524] and many others.     

Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature.     

Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

The purpose of the paper is to introduce modified Halpern and Ishikawa iteration for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in Banach spaces. We also consider two strong convergence theorems for relatively nonexpansive mappings with some proper restriction.     

Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces

In this paper, we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k_i-strict pseudocontractions in a Banach space. We obtain some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others.     

On the existence of polynomial-time approximation schemes for the reoptimization of discrete optimization problems

It is shown that no polynomial-time approximation scheme exists for the reoptimization of the set covering problem in inserting an element into or eliminating it from any set. A similar result is obtained for the minimum graph coloring problem in inserting a vertex with at most two incidence edges and for the minimal bin packing problem in eliminating any element.     

A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem

The purpose of this paper is to introduce and study an iterative algorithm for solving a general split equality problem. The problem consists of finding a common element of the set of common zero points for a finite family of maximal monotone operators, the set of common fixed points for a finite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in the setting of infinite-dimensional Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence theorem for approximating a solution of the split equality problem. As special cases, we shall utilize our results to study the split equality equilibrium problems and the split equality optimization problems. Our result complements and extends some related results in literature.     

A triangular finite difference scheme for large deflection problems

A lumped triangular element formulation is developed based on a finite difference approach for the large deflection analysis of plates and shallow shells. The presented formulation is independent of the boundary condition (unlike the finite difference formulation) and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using an incremental Newton-Raphson iterative procedure. A study of the large deflection behaviour of thin plates is made for various edge conditions and aspect ratios, and the results obtained are compared with those using a finite element scheme. Representative nondimensional solutions for deflections and stresses are presented in the form of graphs.     

Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces

We prove the existence of a solution of the mixed equilibrium problem (MEP) by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solutions set of a mixed equilibrium problem and the fixed points set of a countable family of nonexpansive mappings in the frameworks of Banach spaces. By using a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solutions set of MEP and common fixed points set of nonexpansive mappings. Moreover, some applications concerning the equilibrium and the convex minimization problems are obtained.     

Computer evaluation of high order numerical schemes to solve advective transport problems

This study evaluates the performance of three representative high-order finite difference schemes to solve two sets of simple one-dimensional benchmark problems in terms of their ability to resolve spurious oscillation, numerical spreading, and peak clipping. Three models, namely QUICKEST, ULTIMATE, and ENO were constructed to represent the classical high-order schemes without a flux limiter, TVD with a flux limiter, and TVB schemes, respectively. Three sets of results generated by QUICKEST, ULTIMATE, and ENO were compared with the analytical solutions. The first set indicated that none of these high-order schemes could yield satisfactory simulations when the grid size and time-step size specified by the benchmark problems were used. The second set showed that all three numerical schemes generated excellent computations when the grid size was reduced to one-tenth and the time-step size was reduced to one-fifth of those specified by the benchmark problems. The third set demonstrated that the results obtained by these schemes deteriorated even with the reduced grid size and time-step size when 100 folds of simulation times was conducted. The ENO and ULTIMATE schemes successfully eliminated spurious oscillations for all cases as expected. The QUICKEST scheme alleviated the problem of spurious oscillations only when the reduced grid and time-step sizes were used. In terms of numerical spreading and peak clipping, none of the three schemes produced satisfactory results unless the reduced grid and time-step were used. Peak clipping poses a more severe problem for these high order schemes than numerical spreading. A common set of benchmark problems is needed for the evaluation and testing of any numerical scheme.     

Numerical verification of solutions for elasto-plastic torsion problems

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for the elasto-plastic torsion problems governed by the variational inequality. Based upon the finite element approximations and the explicit a priori error estimates for a simple problem, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. This paper is an extension of the previous paper [1] in which we mainly dealt with the obstacle problems, but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem. A numerical example is illustrated.     

Modeling and solving rich quay crane scheduling problems

The quay crane scheduling problem is a core task of managing maritime container terminals. In this planning problem, discharge and load operations of containers of a ship are scheduled on a set of deployed quay cranes. In this paper, we provide a rich model for quay crane scheduling that covers important issues of practical relevance like crane-individual service rates, ready times and due dates for cranes, safety requirements, and precedence relations among container groups. Focus is put on the incorporation of so-called unidirectional schedules into the model, by which cranes move along the same direction, either from bow to stern or from stern to bow, when serving the vessel. For solving the problem, we employ a branch-and-bound scheme that is known to be the best available solution method for a class of less rich quay crane scheduling problems. This scheme is extended by revising and extending the contained lower bounds and branching criteria. Moreover, a novel Timed Petri Net approach is developed and incorporated into the scheme for determining the starting times of the discharge and load operations in a schedule. Numerical experiments are carried out on both, sets of benchmark instances taken from the literature and real instances from the port of Gioia Tauro, Italy. The experiments confirm that the new method provides high quality solutions within short runtimes. It delivers new best solutions for some of the benchmark problems from the literature. It also shows capable of coping with rich real world problem instances where it outperforms the planning approach applied by practitioners.     

Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

In the present work, error indicators for the potential and elastostatic problems are used in a combined fashion to implement an adaptive meshing scheme for the solution of two-dimensional steady-state thermoelastic problems using the Boundary Element Method. These error indicators exploit in their formulation the possibility of generating two different numerical solutions from just one analysis using Hermite elements. The first solution is the standard one obtained from an analysis using Hermite elements. The second is a “reduced” solution obtained representing the field variables inside an element using some of the degrees of freedom of the Hermite element together with Lagrangian shape functions. The basic idea behind the computation of the error indicator is to compare these two solutions, on an element by element basis, to obtain an estimate of the magnitude of the error in the numerical solution corresponding to the Hermite elements. In this sense, it is assumed that the bigger the difference between these two solutions, the bigger the error in the original solution with Hermite elements. Since the thermoelastic problem in its uncoupled fashion is considered, the former approach is applied to both problems, heat conduction and thermoelastic. Since both numerical solutions for each one of these problems are obtained from just one analysis, the computational cost of the proposed error indicators is very low.     

Discrete particle swarm optimization based on estimation of distribution for terminal assignment problems

Terminal assignment problem (TEAP) is to determine minimum cost links to form a network by connecting a given set of terminals to a given collection of concentrators. This paper presents a novel discrete particle swarm optimization (PSO) based on estimation of distribution (EDA), named DPSO-EDA, for TEAP. EDAs sample new solutions from a probability model which characterizes the distribution of promising solutions in the search space at each generation. The DPSO-EDA incorporates the global statistical information collected from personal best solutions of all particles into the PSO, and therefore each particle has comprehensive learning and search ability. In the DPSO-EDA, a modified constraint handling method based on Hopfield neural network (HNN) is also introduced to fit nicely into the framework of the PSO and thus utilize the merit of the PSO. The DPSO-EDA adopts the asynchronous updating scheme. Further, the DPSO-EDA is applied to a problem directly related to TEAP, the task assignment problem (TAAP), in order to show that the DPSO-EDA can be generalized to other related combinatorial optimization problems. Simulation results on several problem instances show that the DPSO-EDA is better than previous methods.     

On the stability of solutions to minimal and nonminimal design problems

A partial resolution to the question of stability of solutions to the minimal design problem is given in terms of transfer matrix factorizations employing the new notions of common system poles and common systems zeros as well as the fixed poles of all solutions and the fixed poles of minimal solutions. The results are employed to more directly and easily resolve questions involving the attainment of stable solutions to the model matching problem and stable minimal-order state observers.     

Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method

In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters.     

Inverse spectral problems of the theory of identification of linear dynamic systems

The problem of restoration of the parameters of the linear operator from a finite set of eigenvalues was considered. A new scheme of its solution was proposed, both the necessary and sufficient conditions for existence of solutions were presented, and a procedure of their numerical construction was substantiated. Consideration was given to the applied problems of identification and controllability of the linear dynamic systems.     

Local search for load balancing problems for servers with large dimension

We consider a new load balancing model that arises in the processing of user requests for files located on a given set of servers. The optimization criterion is the total excess of actual load over the limit load. In order to redistribute the load and minimize the criterion, files can be moved between the servers. We show that if there are no other constraints related to the stage of moving the files, then this problem is equivalent to a problem previously considered in literature. For this special case of this problem, we propose a stochastic local search scheme that combines a special procedure for fast querying of the neighborhoods and a procedure of non-aggravating modification of intermediate solutions. Results of numerical experiments show that the proposed approach is able to find high-quality solutions for instances of large dimension under tight time constraints.