This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property ( A ) . Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward–backward splitting method version of our algorithm and minimization problem via Douglas–Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.

ABSTRACT In this paper, we first introduce a Bregman prox-convexity notion as the generalization of the convexity for Bregman prox-convexity in Banach spaces. We verify the single-valudeness and Bregman firm nonexpansiveness of the Bregman proximity operator of the functions and provide examples of the notion. Using this notion, we show that the Bregman proximal point scheme remains convergent under appropriate conditions on the functions to be minimized in the setting of Banach spaces.

We investigate the existence of generalized variational solutions to abstract hemivariational inequality driven by a coercive and locally Lipschitz operator with no further monotonicity conditions imposed on it. A finite dimensional Ritz type approximation scheme is derived. Applications to competitive systems driven by the generalized ( p , q ) - Laplacian with convection are given.

We establish a general method for simultaneously perturbing a convergent sequence of functions in such a way that the sequence of strong minima of the perturbed functions tends to the strong minimum of their limit.

ABSTRACT In this work, we revisit a version of the Tseng extragradient method for pseudomonotone variational inequalities and give sufficient conditions for its convergence. We first study a modified inertial Tseng method for variational inequalities to obtain weak convergence results under pseudomonotonicity and Lipschitz continuity assumptions. The advantage of the proposed method is that the inertial factor is chosen in [ 0 , 1 ] . Second, linear convergence is obtained for the proposed method under strong pseudomonotonicity and Lipschitz continuity assumptions. The main results obtained in this paper extend and improve some related works in the literature.

The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L 1 -norm control cost with a weight γ > 0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L 2 -norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L 1 -norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L 2 -sense.

ABSTRACT A classical solution approach to semi-infinite programming, which is easy to implement, is based on discretizing the semi-infinite index set. The Blankenship and Falk algorithm adaptively chooses a small discretization. On every iteration, a solution based on the current discretization is calculated. In a second step, the most violated constraint is determined and added to the discretization. In a previous work, the authors showed that the algorithm has a slow convergence and introduced a new method that exhibits a quadratic rate [Seidel and Küfer, An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence. Optimization. 2022;71(8):2211–2239]. In this paper, we further investigate the introduced method. The method implements new constraints that can cut off parts of the feasible set. We will study the effect on local minima. We assume that in each iteration local solutions to the discretized problems are computed. We will give an example showing that in general a limit point is not necessarily a local solution of the original semi-infinite problem, but only of an approximate problem. We then study second-order conditions and show that they coincide for both problems. We use this to develop conditions under which local solutions converge to a local solution in the limit. Finally, we present quadratic convergence results for the case of local solutions.

In this paper, we give primal and dual Kuhn-Tucker necessary conditions for the existence of a strict local minimum of order two for vector optimization problems with equality and inequality constraints under some new regularity conditions. First, we improve the existing primal necessary conditions for such minima. Then, we apply an alternative theorem to derive dual Kuhn-Tucker necessary conditions of second and higher-order. To compare our results to the ones in the literature, we provide some examples.