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The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function. [ABSTRACT FROM PUBLISHER]

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method. [ABSTRACT FROM AUTHOR]

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operatorFover a closed and convex setC. We assume that the setCcan be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions onF. We provide several examples for the case whereCis the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the split common fixed point problem for new demimetric mappings in two Banach spaces. Using the hybrid method, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in two Banach spaces. Furthermore, using the shrinking projection method, we obtain another strong convergence theorem for finding a solution of the problem in two Banach spaces. Using these results, we obtain well-known and new strong convergence theorems in Hilbert spaces and Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of cyclic antimonotonicity (known for scalar bifunctions) to the vector case, in order to obtain a vectorial equilibrium version of Ekeland's variational principle. We characterize the cyclic antimonotonicity in terms of a suitable approximation from below of the vector bifunction, which allows us to avoid the demanding triangle inequality property, usually required in the literature, when dealing with Ekeland's principle for bifunctions. Furthermore, a result for weak vector equilibria in the absence of convexity assumptions is given, without passing through the existence of approximate solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The split feasibility problem (SFP) is to find so that , where C and Q are non-empty closed convex subsets in Hilbert spaces and , respectively, and A is a linear bounded operator from to . Byrne proposed an iterative method called the CQ algorithm that involves the orthogonal projections onto C and Q. However, the projections onto C and Q might be hard to be implemented in general. In this paper, we propose a ball-relaxed projection method for the SFP. Instead of half spaces, we replace C and Q in the proposed algorithm by two properly chosen closed balls and . Since the projection onto the closed ball has closed form, the proposed algorithm is thus easy to be implemented. Under some mild conditions, we establish the weak convergence of the proposed algorithm to a solution of the SFP. As an application, we obtain new algorithms for solving the split equality problem. Preliminary numerical experiments show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new iterative method for finding the projection onto the intersection of two closed convex sets in the framework of Banach spaces is presented. It is a viscosity approximation method which produces a strongly convergent sequence. [ABSTRACT FROM AUTHOR]

In this paper, a new type of stepsize, approximate optimal stepsize, for gradient method is introduced to interpret the Barzilai–Borwein (BB) method, and an efficient gradient method with an approximate optimal stepsize for the strictly convex quadratic minimization problem is presented. Based on a multi-step quasi-Newton condition, we construct a new quadratic approximation model to generate an approximate optimal stepsize. We then use the two well-known BB stepsizes to truncate it for improving numerical effects and treat the resulted approximate optimal stepsize as the new stepsize for gradient method. We establish the global convergence andR-linear convergence of the proposed method. Numerical results show that the proposed method outperforms some well-known gradient methods. [ABSTRACT FROM PUBLISHER]

In this paper, a common zero of a finite family of monotone operators is approximated in Hadamard spaces. Some applications in convex minimization and fixed point theory are also presented. [ABSTRACT FROM AUTHOR]

This paper aims to solve the joint chance constrained programs (JCCP) by a DC (difference of two convex functions) function approach, which was established by Hong et al. [Oper. Res. 2011;59:617–630]. They used a DC function to approximate the chance constraint functionand constructed a sequential convex approximation method to solve the approximation problem. A disadvantage of this method is perhaps that the DC function they used is nonsmooth. In this article, we first propose a class of smoothing functions to approximate the maximum functionand the indicator function. Then, we construct the conservative smooth DC approximation function toand obtain the smooth DC approximation problems to JCCPs. We show that the solutions of a sequence of smooth approximation problems converge to some Karush–Kuhn–Tucker point of JCCPs under a certain asymptotic regime. [ABSTRACT FROM PUBLISHER]

This paper presents a method for approximating the nondominated set of a multiobjective linear fractional programming (MOLFP) problem. The resulting approximation set has acceptable quality measures, and is an-approximation to the nondominated set, whereis the tolerance vector and can be controlled by the decision-maker. The distance between every two resulting solutions is greater than or equal to a positive valueand hence there is no redundant information. Also, for a given upper bound the method controls the cardinality of the resulting approximation set. [ABSTRACT FROM PUBLISHER]

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions [ABSTRACT FROM PUBLISHER]

In this paper, we first discuss the Painlevé–Kuratowski set convergence of (weak) minimal point set for a convex set, when the set and the ordering cone are both perturbed. Next, we consider a convex vector optimization problem, and take into account perturbations with respect to the feasible set, the objective function and the ordering cone. For this problem, by assuming that the data of the approximate problems converge to the data of the original problem in the sense of Painlevé–Kuratowski convergence and continuous convergence, we establish the Painlevé–Kuratowski set convergence of (weak) minimal point and (weak) efficient point sets of the approximate problems to the corresponding ones of original problem. We also compare our main theorems with existing results related to the same topic. [ABSTRACT FROM PUBLISHER]

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to generalize and improve some topological properties of solutions set to the set-valued vector equilibrium problems by using the scalar characterization method. Moreover, the Lipschitz continuity of an approximate solution mapping for the parametric set-valued vector equilibrium problems is studied. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to obtain strong convergence result for approximation of solution to constrained convex minimization problem using a new iterative scheme in a real Hilbert space. Furthermore, we give numerical analysis of our iterative scheme. [ABSTRACT FROM AUTHOR]

In this paper, we first study in a Hilbertian framework the weak convergence of a general Gradient Projection Algorithm for minimizing a convex function of classover a convex constraint set. The way of selecting the stepsizes corresponds to the one used by López et al. for the particular case of the Split Feasibility Problem. This choice allows us to avoid the computation of operator norms. Afterwards, a relaxed version of the Gradient Projection Algorithm is considered where the feasible set is approximated by half-spaces making the projections explicit. Finally, to get the strong convergence, each step of the general Gradient Projection Method is combined with a viscosity step. This is done by adapting Halpern’s algorithm to our problem. The general scheme is then applied to the Split Equality Problem, and also to the Split Feasibility Problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce generalized Levitin–Polyak well-posedness of symmetric strong vector quasi-equilibrium problem. We give characterizations for generalized Levitin–Polyak well-posedness of the symmetric strong vector quasi-equilibrium problem and the symmetric weak vector quasi-equilibrium problem by closed graph of the approximating solution mapping. Our results improve the main result presented in Zhang. [ABSTRACT FROM AUTHOR]

Sufficient conditions for a weak local minimizer in the classical calculus of variations can be expressed without reference to conjugate points. The local quadratic convergence of Newton’s method follows from these sufficient conditions. Newton’s method is applied in the minimization form; that is, the step is generated by minimizing the local quadratic approximation. This allows the extension to a globally convergent line search based algorithm (which will be presented in a future paper). [ABSTRACT FROM PUBLISHER]

In this paper, we consider an optimal co with state and control constraints, where a prespecified sequence of active subsystems is given. Both the switching instants and the control function are to be chosen such that the cost functional is minimized. Firstly, we convert the problem into a conventional optimal control problem based on parameterization of the switching instants. Next, through discretizing the control space, applying a time-scaling transformation and using the smoothing technique, the optimal control problem is approximated by a non-linear optimization problem with linear constraints and bounds on the variables. Furthermore, an algorithm that finds a suboptimal solution to the original problem is proposed. One major advantage with the new algorithm implementation is the only need to solve a non-linear optimization problem with linear constraints and bounds on the variables. The gradients of these linear constraints and the objective function can be easily obtained. Therefore, the original optimization problem can be solved efficiently using any gradient-based method, such as sequential quadratic programming algorithm. Finally, a flight planning example illustrates the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic facility location problem with submodular penalties. By exploring the structural properties of submodular function, we present a primal-dual-approximation algorithm for the problem. [ABSTRACT FROM AUTHOR]

This paper aims at studying, in the image space, an approximation of a vector optimization problem obtained by substituting the involved functions with their-derivatives. It is shown that, under the hypothesis of-differentiability, the existence of a lower semistationary point is equivalent to the linear separation between the image of the approximated problem and a suitable convex subset of the image space. Applications to optimality conditions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of a linear fractional differential equation by using the method of regularization of Tikhonov. New concepts and results about controllability are established. Then, under the condition of the positivity of the controllability operator, we obtain that the linear system can be steered to an arbitrary small neighbourhood of the fractional integral of the state at final time. [ABSTRACT FROM AUTHOR]