Positivity

We study the facial structures of the cone of all decomposable positive linear maps from the matrix algebra M m into M n . Especially, we completely determine the faces of the cone which arise from the dual of positive linear maps.

In this work we study the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We establish some characterizations and basic properties of these constructions, and then develop calculus including sum rules and chain rules involving smooth functions. As an application, we also explore the upper estimates of the directional Mordukhovich subdifferentials and singular subdifferentials of marginal functions.

Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for ℓ n p balls is given.

In this paper, Levitin–Polyak (in short LP) well-posedness in the set and scalar sense are defined for a set optimization problem and a relationship between them is found. Necessary and sufficiency criteria for the LP well-posedness in the set sense are established. Some characterizations in terms of Hausdorff upper semicontinuity and closedness of approximate solution maps for the LP well-posedness have been obtained. Further, a sequence of solution sets of scalar problems is shown to converge in the Painlevé–Kuratowski sense to the minimal solution sets of the set optimization problem. Finally, the perturbations of the ordering cone and the feasible set of the set optimization problem are considered and the convergence of its weak minimal and minimal solution sets in terms of Painlevé–Kuratowski convergence is discussed.

The main result of the paper extends the classical result of E. Odell on Schreier unconditionality to arrays in Banach spaces. An application is given on the “multiple of the inclusion plus compact" problem which is further applied to a hereditarily indecomposable Banach space constructed by N. Dew.

This study is devoted to the semidefinite optimization problems with inequality constraints. We use the notion of convexificators to establish optimality conditions for nonsmooth semidefinite optimization problems. Moreover, we introduce appropriate constraint qualifications to present the Karush–Kuhn–Tucker multipliers.

The analysis in the Image Space allows one to extend the applications of maximum and variational principles for constrained optimization. Such principles are embedded in a separation scheme, in the Image Space, which can be seen as a common root from which they are derived. In particular, Ekeland and Auchmuty Variational Principles are analysed.

This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi: 10.1006/jmaa.2000.6813 ). We introduce and characterize two seemingly new natural concepts of $$\varepsilon $$ -minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function.

The paper contains several characterizations of Banach lattices $$E$$ with the dual positive Schur property (i.e., $$0 \le f_n \xrightarrow {\sigma (E^*,E)} 0$$ implies $$\Vert f_n\Vert \rightarrow 0$$ ) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford–Pettis property.

We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from Lp into Lq. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices.