Necessary and sufficient conditions are obtained for the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting. Properties of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the key to these results. A previous characterization of tilt stability comes out as a sp...

In terms of the normal cone and the coderivative, we provide some necessary and/or sufficient conditions of metric subregularity for (not necessarily closed) convex multifunctions in normed spaces. As applications, we present some error bound results for (not necessarily lower semicontinuous) convex functions on normed spaces. These results improve and extend some existing error bound results. ...

The purpose of this paper is to consider the set-valued optimization problem in Asplund spaces without convexity assumption. By a scalarization function introduced by Tammer and Weidner (J Optim Theory Appl 67:297–320, 1990), we obtain the Lagrangian condition for approximate solutions on set-valued optimization problems in terms of the Mordukhovich coderivative.

This paper sheds new light on regularity of multifunctions through various characterizations of directional Hölder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Hölder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directi...

The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend results to general Banach/Asplund spaces and other classes perturbations, sharpen coderivative tools used analysis robustness well-posedness mathematical problems related regularity properties mappings involved statements. ...

This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained constrained problems of nonsmooth optimization by using tools variational analysis generalized differentiation. Both are coderivative-based employ Hessians (coderivatives subgradient mappings) associated with objective functions, which either class $${{\mathcal {C}}}^{1,1}$$ , or represented in ...

We use the standard notations: differential forms ap, their exterior algebra A, exterior derivative d<p, Hodge's star operator *, coderivative 3f = (_ 1)nP + n 1*d*(P, Laplace-Beltrami operator A = d3 + Md, exterior product (pAg,, inner product (so,*) = fJmpA*4p, and the Dirichlet norm D(sp) = (dep, dp) + (a<p, 5op). For 0forms u the norm reduces to D(u) = (du, du), and Au has the representation

We give a systematic account of the exterior algebra of forms on q-Minkowski space, introducing the q-exterior derivative, q-Hodge star operator, q-coderivative, q-LaplaceBeltrami operator and the q-Lie-derivative. With these tools at hand, we then give a detailed exposition of the q-d’Alembert and q-Maxwell equation. For both equations we present a q-momentum-indexed family of plane wave solut...

Solution stability of a class of linear generalized equations in finite dimensional Euclidean spaces is investigated by means of generalized differentiation. Exact formulas for the Fréchet and the Mordukhovich coderivatives of the normal cone mappings of perturbed Euclidean balls are obtained. Necessary and sufficient conditions for the local Lipschitz-like property of the solution maps of such...

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunctio...