In this paper, we consider the quasi-convex feasibility problem (QFP), which is to find a common point of a family of sublevel sets of quasi-convex functions.By employing the Bregman projection mapping, we propose a unified framework of Bregman quasi-subgradient methods for solving the QFP.This paper is contributed to establish the convergence theory, including the global convergence, iteration complexity, and convergence rates, of the Bregman quasi-subgradient methods with several general control schemes, including the α-most violated constraints control and the s-intermittent control.Moreover, we introduce a notion of the Hölder-type bounded error bound property relative to the Bregman distance for the QFP, and use it to establish the linear (or sublinear) convergence rates for Bregman quasi-subgradient methods to a feasible solution of the QFP.

In this study, we investigate an uncertain multiobjective optimization problem through a setvalued optimization problem, and introduce a Newton method to find robust weakly efficient points of the considered uncertain optimization problem.We assume that the problem under consideration has uncertainty only in the objective function, and the involved uncertainty set is of finite cardinality.Also, for each uncertain scenario, the components of the objective function of the problem are assumed to be twice continuously differentiable and locally strong convex.Utilizing the concept of a partition set from set optimization, we formulate a class of vector optimization problems to solve the formulated set optimization problem pertaining to the considered uncertain multiobjective optimization.We derive a Newton method to solve this class of vector optimization problems that facilitates generating a sequence of points whose any limit point is a weakly robust efficient solution of the considered problem.The proposed method is found to have a local superlinear convergence rate under standard hypotheses with a regularity condition.Additionally, assuming Lipschitz continuity of the Hessian of the objective function for all scenarios, we show local quadratic convergence of the method.Finally, we provide numerical examples to discuss and illustrate the performance of the proposed method.

The aim of this paper is to study the path connectedness of the minimal and weak minimal solution sets for set optimization problems equipped with partial set order relations defined on the family of bounded sets by means of Minkowski difference.We propose some new notions of generalized convexity for set-valued maps and a lower level map.Further, by investigating several properties of this level map, we study path connectedness of the solution sets for set optimization problems with respect to partial order relations.Finally, we give an application of the derived results to Nash equilibrium games with vector-valued maps under uncertainty.

In a Hilbert space H , we investigate the long time behavior of the trajectories of the following second-order differential system (t)and the time scale parameter is a positive function that tends to infinity as time approaches infinity.We based on Tikhonov regularization techniques and an appropriate Lyapunov energy function simultaneously prove fast convergence rates of the objective function to the global minimum, strong convergence of trajectories to the minimizer of the minimum norm, as well as convergence rates for the velocity and the gradient.Numerical examples are also given to illustrate these results.Finally, we extend our results to the non-smooth case.

In this paper, we consider a planar clustering problem with location constraints for cluster centers.A simple adaptation of the k-means algorithm solving the presented problem to local optimality is outlined.We further show that our problem can be stated as a MIQCP (mixed-integer-quadratically-constrained-programming) problem, and some results to solve the formulated problem to global optimality with a popular solver are presented.We also present a specialized enumeration algorithm which can be used to find the global optima of the problem and our numerical experiments indicate that this approach is a better choice in comparison to solving the formulated MIQCP problem with a general solver.

In the paper, we introduce several accelerated iterative algorithms for solving the multiple-set split common fixed-point problem of quasi-nonexpansive operators.Based on the primal-dual method, we construct several iterative algorithms in a way that combines inertial technology and self-adaptive stepsizes such that the implementation of the algorithms does not need any prior information about bounded linear operator norm.Under suitable assumptions, we obtain the weak convergence of the proposed algorithms in real Hilbert spaces.As applications, we obtain relative iterative algorithms to solve the multiple-set split feasibility problem.Finally, the performance of the proposed algorithms is illustrated by numerical experiments.

In this paper, we state that generalized convexity, invex sets, and the Fréchet differentiability assumption associated with curvilinear integral type functionals represent some necessary and sufficient mathematical tools for establishing various connections between the solutions of some robust (weak) vector commanded variational inequalities and (weak, proper) robust efficient solutions of the corresponding multi-objective variational control problem.In addition, the physical motivation of the problem under investigation is formulated in the illustrative application given at the end of this paper.

This paper discusses the k-sparse complex signal recovery from quadratic measurements via the p -minimization model, where 0 < p ≤ 1. We establish the p restricted isometry property over simultaneously low-rank and sparse matrices, which is a weaker restricted isometry property to guarantee the successful recovery in the p case. The main result is to demonstrate that p -minimization can recover complex k-sparse signals from m k + pk log(n/k) complex Gaussian quadratic measurements with high probability. The resulting sufficient condition is met by fewer measurements for smaller p and reaches m k when p turns to zero. Furthermore, an iteratively-reweighted algorithm is proposed. Numerical experiments also demonstrate that p minimization with 0 < p < 1 performs better than 1 minimization.