A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

Optimality Conditions for Group Sparse Constrained Optimization Problems

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

ON THE REGULARITY OF SETS IN RIEMANNIAN MANIFOLDS

This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.

Cryptically galled infructescence: a new sheoak gall type in Allocasuarina luehmannii and Casuarina pauper (Casuarinaceae)

Insect galls formed within the infructescences (cones) of Allocasuarina luehmannii and Casuarina pauper in southern New South Wales, Australia, are described. The galling was internal within the infested cones, which were small and irregularly developed, but could appear superficially normal except that they had a higher than normal proportion of samaras retained on bracteole dehiscence. Cross-sections revealed abnormal morphology and wasp larval chambers. All exit holes found were between bracteole pairs of either fertile or infertile florets. Emergent wasps were tentatively identified as Eurytoma sp. sensu lato (Hymenoptera: Eurytomidae). These cryptically galled infructescences represent a previously undescribed gall type in the Casuarinaceae and, for Eurytoma, potentially a rare instance of phytophagy in Australia. Infested cones were found in a season when normal cones in this drought affected area were not easily found. It was concluded that this phytophagy could negatively impact the regeneration potential of two already compromised sheoak species.

Topological Properties of E-Metric Spaces with Applications to Fixed Point Theory

The purpose of this paper is to present some topological properties in E-metric spaces such as the properties of e-sequences, the decision conditions of e-Cauchy sequences, the characteristics of non-normal cones, and so on. Moreover, the theorem of nested closed-balls in such spaces is displayed. In addition, some principal applications to fixed point theory are also given.

Optimal harvesting of a hierarchical age-structured population system

We investigate an optimal harvesting problem for age-structured species, in which elder individuals are more competitive than younger ones, and the population is modeled by a highly nonlinear integro-partial differential equation with a global feedback boundary condition. The existence of optimal strategies is established by means of compactness and maximizing sequences, and the maximum principle obtained via an adjoint system, tangent-normal cones and a new continuity result. In addition, some numerical experiments are presented to show the effects of the price function and younger’s weight on the optimal profits.