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By applying some techniques of set-valued and variational analysis, we study solution stability of nonhomogeneous split equality problems and nonhomogeneous split feasibility problems, where the constraint sets need not be convex. Necessary and sufficient conditions for the Lipschitz-likeness of the solution maps of the problems are given and illustrated by concrete examples. The obtained results complement those given in [Huong VT, Xu HK, Yen ND. Stability analysis of split equality and split feasibility problems. arXiv:2410.16856.], where classical split equality problems and split feasibility problems have been considered.

In this paper, for the first time in the literature, we study the stability of solutions of two classes of feasibility (i.e., split equality and split feasibility) problems by set-valued and variational analysis techniques. Our idea is to equivalently reformulate the feasibility problems as parametric generalized equations to which set-valued and variational analysis techniques apply. Sufficient conditions, as well as necessary conditions, for the Lipschitz-likeness of the involved solution maps are proved by exploiting special structures of the problems and by using an advanced result of B.S. Mordukhovich [J. Global Optim. 28, 347--362 (2004)]. These conditions stand on a solid interaction among all the input data by means of their dual counterparts, which are transposes of matrices and regular/limiting normal cones to sets. Several examples are presented to illustrate how the obtained results work in practice and also show that the existence of nonzero solution assumption made in the necessity conditions cannot be lifted.

In this paper, we first introduce and study the notion of random Chebyshev centers. Further, based on the recently developed theory of stable sets, we introduce the notion of random complete normal structure so that we can prove the two deeper theorems: one of which states that random complete normal structure is equivalent to random normal structure for an $L^0$-convexly compact set in a complete random normed module; the other of which states that if $G$ is an $L^0$-convexly compact subset with random normal structure of a complete random normed module, then every commutative family of nonexpansive mappings from $G$ to $G$ has a common fixed point. We also consider the fixed point problems for isometric mappings in complete random normed modules. Finally, as applications of the fixed point theorems established in random normed modules, when the measurable selection theorems fail to work, we can still prove that a family of strong random nonexpansive operators from $(\Omega,\mathcal{F},P)\times C$ to $C$ has a common random fixed point, where $(\Omega,\mathcal{F},P)$ is a probability space and $C$ is a weakly compact convex subset with normal structure of a Banach space.

We extend the concept of well-posedness to the split equilibrium problem and establish Furi-Vignoli-type characterizations for the well-posedness. We prove that the well-posedness of the split equilibrium problem is equivalent to the existence and uniqueness of its solution under certain assumptions on the bifunctions involved. We also characterize the generalized well-posedness of the split equilibrium problem via the Kuratowski measure of noncompactness. We illustrate our theoretical results by several examples.

Motivated by the randomized version of the classical Bolzano--Weierstrass theorem, in this paper we first introduce the notion of a random sequentially compact set in a random normed module and develop the related theory systematically. From these developments, we prove the corresponding Schauder fixed point theorem: let $E$ be a random normed module and $G$ a random sequentially compact $L^0$--convex set of $E$, then every $\sigma$--stable continuous mapping from $G$ to $G$ has a fixed point, which unifies all the previous random generalizations of the Schauder fixed point theorem. As one of the applications of the theorem, we prove the existence of Nash equilibrium points in the context of conditional information. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a random sequentially compact set is generally noncompact., Comment: 48 pages

Parallel and cyclic projection algorithms are proposed for minimizing the sum of a finite family of convex functions over the intersection of a finite family of closed convex subsets of a Hilbert space. These algorithms are of predictor-corrector type, with each main iteration consisting of an inner cycle of subgradient descent process followed by a projection step. We prove the convergence of these methods to an optimal solution of the composite minimization problem under investigation upon assuming boundedness of the gradients at the iterates of the local functions and the stepsizes being chosen appropriately, in the finite-dimensional setting. We also discuss generalizations and limitations of the proposed algorithms and our techniques.

The Frank-Wolfe algorithm, a very first optimization method and also known as the conditional gradient method, was introduced by Frank and Wolfe in 1956. Due to its simple linear subproblems, the Frank-Wolfe algorithm has recently been received much attention for solving large-scale structured optimization problems arising from many applied areas such as signal processing and machine learning. In this paper we will discuss in detail the convergence analysis of the Frank-Wolfe algorithm in Banach spaces. Two ways of the selections of the stepsizes are discussed: the line minimization search method and the open loop rule. In both cases, we prove the convergence of the Frank-Wolfe algorithm in the case where the objective function $f$ has uniformly continuous (on bounded sets) Fr\'echet derivative $f'$. We introduce the notion of the curvature constant of order $\sigma\in (1,2]$ and obtain the rate $O(\frac{1}{k^{\sigma-1}})$ of convergence of the Frank-Wolfe algorithm. In particular, this rate reduces to $O(\frac{1}{k^{\nu}})$ if $f'$ is $\nu$-H\"older continuous for $\nu\in (0,1]$, and to $O(\frac{1}{k})$ if $f'$ is Lipschitz continuous. A generalized Frank-Wolfe algorithm is also introduced to address the problem of minimizing a composite objective function. Convergence of iterates of both Frank-Wolfe and generalized Frank-Wolfe algorithms are investigated., Comment: 27 pages

The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image & signal processing and machine learning. In convex settings, numerous convergence results have been established for ADMM as well as its varieties. However, due to the absence of convexity, the convergence analysis of nonconvex ADMM is generally very difficult. In this paper we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and often leads to an improvement of the performance of the algorithm. Under certain assumptions, we prove that the iterative sequence generated by BADMM converges to a stationary point of the associated augmented Lagrangian function. The obtained results underline the feasibility of ADMM in applications under nonconvex settings., Comment: 17 pages, 2 figures