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In this paper, we establish through an openness condition the metric subregularity of a multimapping with normal ω (·) -regularity of either the graph or values. Various preservation results for prox-regular and subsmooth sets are also provided. [ABSTRACT FROM AUTHOR]

This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as the sum of a function twice differentiable in the extended sense and a subdifferentially continuous, prox-regular, twice epi-differentiable function. For such a function, which is not necessarily prox-regular, it is shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. In addition, other characterizations of the quadratic growth and the strong metric subregularity of the subdifferential are also given. Besides, properties of functions twice differentiable in the extended sense are examined. [ABSTRACT FROM AUTHOR]

In this paper, we provide in a general Hilbert space new characterizations of uniform prox-regularity involving outside but sufficiently close points of considered sets. We show that the complement of a prox-regular set is nothing but the union of closed balls with common radius. We derive from this that the prox-regularity of a given closed set is equivalent to the semiconvexity property of its distance function. Various estimates involving the metric projection mapping to a prox-regular set are also established. [ABSTRACT FROM AUTHOR]

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to introduce a viscosity type algorithm for solving proximal split feasibility problems and prove the strong convergence of the sequences generated by our iterative schemes in Hilbert spaces. First, we prove strong convergence result for a problem of finding a point which minimizes a convex function f such that its image under a bounded linear operator A minimizes another convex function g. Secondly, we prove another strong convergence result for the case where one of the two involved functions is prox-regular. In all our results in this work, our iterative schemes are proposed by way of selecting the step sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm $\|A\|$ is not an easy task. Finally, we give a numerical example to study the efficiency and implementation of our iterative schemes. Our results complement the recent results of Moudafi and Thakur (Optim. Lett. 8:2099-2110, 2014, doi:) and other recent important results in this direction. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to introduce iterative algorithms and prove their strong convergence for solving proximal split feasibility problems and fixed point problems for $$k$$ -strictly pseudocontractive mappings in Hilbert spaces. The sequence generated by our first iterative scheme converges strongly to an approximate common solution of convex minimization feasibility problem and fixed point problem. Furthermore, our second algorithm generates a strongly convergent sequence to an approximate common solution of non-convex minimization feasibility problem and fixed point problem. In all our results in this work, our iterative schemes are proposed with a way of selecting the step-sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm $$||A||$$ is very difficult, if not an impossible task. Our result complements many recent and important results in this direction. [ABSTRACT FROM AUTHOR]

Around a solution of an optimization problem, an 'identifiable' subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are 'minimal'. This new notion lays a broad and intuitive variational-analytic foundation for optimality conditions, sensitivity, and active set methods. [ABSTRACT FROM AUTHOR]

In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function $$f$$ such that its image under a bounded linear operator $$A$$ minimizes another convex function $$g$$ . Based on an idea introduced in Lopez (Inverse Probl 28:085004, ), we propose a split proximal algorithm with a way of selecting the step-sizes such that its implementation does not need any prior information about the operator norm. Because the calculation or at least an estimate of the operator norm $$\Vert A\Vert $$ is not an easy task. Secondly, we investigate the case where one of the two involved functions is prox-regular, the novelty of this approach is that the associated proximal mapping is not nonexpansive any longer. Such situation is encountered, for instance, in numerical solution to phase retrieval problem in crystallography, astronomy and inverse scattering Luke (SIAM Rev 44:169-224, ) and is therefore of great practical interest. [ABSTRACT FROM AUTHOR]