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Considering a closed multifunction $\Psi$ between two Banach spaces, it is known that metric regularity and strong metric subregularity of $\Psi$ are, respectively, stable with respect to “small Li...

For an inequality system defined by an infinite family of proper lower semicontinuous convex functions in normed linear space, we consider the Farkas--Minkowski (FM for short) type qualification an...

This paper shows, in the metric space setting, that the p-order metric regularity is preserved under perturbation by locally Holder continuous mappings of order 1/p, extending a classical result which says that 1-order metric regularity is preserved under perturbation by locally Lipschitz mappings (see Theorem 3.3 in (Dontchev et al., Trans. Am. Math. Soc. 355:493–517, 2003) and Theorem 1 in (Dontchev, Optimization 64:41–48, 2015)) by allowing p ∈ (0, 1]. The argument is based on a new result of fixed point theorem type.

When the objective function undergoes both a tilt perturbation and a general parameter perturbation, this paper considers the notions of a fully stable Holder minimizer, a uniform Holder growth con...

Using the error bound results established in the present paper for approximate solutions, we study the stability issues when perturbed by possibly nonaffine smooth maps $E$ for the abstract inequal...

For a finite/infinite family of closed convex sets with nonempty intersection in Hilbert space, we consider the (bounded) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem. Several sufficient conditions are provided to ensure the bounded linear regularity in terms of the interior-point conditions and some finite codimension assumptions. A unified projection method, called Algorithm B-EMOPP, for solving the convex feasibility problem is proposed, and by using the bounded linear regularity, the linear convergence results for this method are established under a new control strategy introduced here.

Using the convex process theory and the majorizing technique, we study the convergence issues of the iterative sequences generated by the extended Newton method for solving the conic inequality system F ≥ C 0 defined by a cone C and a Frechet differentiable function F satisfying the (extended) weak γ-condition. Convergence criterion of the extended Newton method is presented. As an application, we establish, when F is analytic, an extended α-theory similar to Smale's α-theory for nonlinear analytic equations.

Using the Borwein–Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Holder metric subregularity and Holder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Holder minimizers are established.

In this paper, we introduce and study the notions of Holder weak sharp minimizers, stable Holder weak sharp minimizers and Holder tilt-stable weak minimizers for a proper lower semicontinuous function f on a Banach space. In terms of the Holder metric subregularity/regularity of ∂ f , we consider optimality conditions for Holder weak sharp minimizers and stable Holder weak sharp minimizers. We prove that x is a stable Holder weak sharp minimizer (resp. a Holder tilt-stable weak minimizer) of f if and only if it is a stable Holder sharp minimizer (resp. a Holder tilt-stable minimizer) of f .