24 research outputs found
Tight Error Bounds for Nonnegative Orthogonality Constraints and Exact Penalties
For the intersection of the Stiefel manifold and the set of nonnegative
matrices in , we present global and local error bounds
with easily computable residual functions and explicit coefficients. Moreover,
we show that the error bounds cannot be improved except for the coefficients,
which explains why two square-root terms are necessary in the bounds when for the nonnegativity and orthogonality, respectively. The error bounds
are applied to penalty methods for minimizing a Lipschitz continuous function
with nonnegative orthogonality constraints. Under only the Lipschitz continuity
of the objective function, we prove the exactness of penalty problems that
penalize the nonnegativity constraint, or the orthogonality constraint, or both
constraints. Our results cover both global and local minimizers
Stationarity and regularity concepts for set systems
Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle) are presented for the case of an Asplund space
Transversality Properties: Primal Sufficient Conditions
The paper studies 'good arrangements' (transversality properties) of
collections of sets in a normed vector space near a given point in their
intersection. We target primal (metric and slope) characterizations of
transversality properties in the nonlinear setting. The Holder case is given a
special attention. Our main objective is not formally extending our earlier
results from the Holder to a more general nonlinear setting, but rather to
develop a general framework for quantitative analysis of transversality
properties. The nonlinearity is just a simple setting, which allows us to unify
the existing results on the topic. Unlike the well-studied subtransversality
property, not many characterizations of the other two important properties:
semitransversality and transversality have been known even in the linear case.
Quantitative relations between nonlinear transversality properties and the
corresponding regularity properties of set-valued mappings as well as nonlinear
extensions of the new transversality properties of a set-valued mapping to a
set in the range space due to Ioffe are also discussed.Comment: 33 page
National Natural Science Foundation of China (11101248, 71101140), Shandong Province Natural Science Foundation (ZR2010AQ026), and Young Teacher
Abstract. In this paper, we deal with the semi-infinite complementarity problems (SICP), in which several important issues are covered, such as solvability, semismoothness of residual functions, and error bounds. In particular, we characterize the solution set by investigating the relationship between SICP and the classical complementarity problem. 1 Furthermore, we show that the SICP can be equivalently reformulated as a typical semiinfinite min-max programming problem by employing NCP functions. Finally, we study the concept of error bounds and introduce its two variants, ε-error bounds and weak error bounds, where the concept of weak error bounds is highly desirable in that the solution set is not restricted to be nonempty. Key words. semi-infinite complementarity problem, semidifferentiable and semismooth, error bounds, weak error bounds
Error bounds for vector-valued funtions on metric spaces
In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new primal space derivative-like objects – slopes – are introduced and a classification scheme of error bound criteria is presented