A Complementarity Partition Theorem for Multifold Conic Systems

Consider a homogeneous multifold convex conic system $$Ax = 0, \; x\in K_1\times...\times K_r$$ and its alternative system A\transp y \in K_1^*\times...\times K_r^*, where $K_1,..., K_r$ are regular closed convex cones. We show that there is canonical partition of the index set ${1,...,r}$ determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming.Comment: 12 pages, 2 figure

Refining the partition for multifold conic optimization problems

In this paper, we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in Bonnans and Ramirez [Perturbation analysis of second-order cone programming problems, Math Program. 2005;104(2-30):205-227] for conic optimization problems, and in Pena and Roshchina [A complementarity partition theorem for multifold conic systems, Math Program. 2013;142(1-2):579-589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular, that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions.This research was partially supported by ANID (Chile) under REDES project number 180032 and by the Australian Research Council grant DE150100240. The second author was supported by FONDECYT (Fondo de Fomento al Desarrollo Cientifico y Tecnologico) regular projects 1160204 and 1201982, and Basal Program CMM-AFB 170001 (Comision Nacional de Investigacion Cientifica y Tecnologica), all from ANID (Chile)

Bad semidefinite programs: they all look the same

Conic linear programs, among them semidefinite programs, often behave pathologically: the optimal values of the primal and dual programs may differ, and may not be attained. We present a novel analysis of these pathological behaviors. We call a conic linear system $Ax <= b$ {\em badly behaved} if the value of $\sup { | A x <= b }$ is finite but the dual program has no solution with the same value for {\em some} $c.$ We describe simple and intuitive geometric characterizations of badly behaved conic linear systems. Our main motivation is the striking similarity of badly behaved semidefinite systems in the literature; we characterize such systems by certain {\em excluded matrices}, which are easy to spot in all published examples. We show how to transform semidefinite systems into a canonical form, which allows us to easily verify whether they are badly behaved. We prove several other structural results about badly behaved semidefinite systems; for example, we show that they are in $NP \cap co-NP$ in the real number model of computing. As a byproduct, we prove that all linear maps that act on symmetric matrices can be brought into a canonical form; this canonical form allows us to easily check whether the image of the semidefinite cone under the given linear map is closed.Comment: For some reason, the intended changes between versions 4 and 5 did not take effect, so versions 4 and 5 are the same. So version 6 is the final version. The only difference between version 4 and version 6 is that 2 typos were fixed: in the last displayed formula on page 6, "7" was replaced by "1"; and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by "A_3 - A_2 - A_1

Factors Influencing Customer Satisfaction towards E-shopping in Malaysia

Online shopping or e-shopping has changed the world of business and quite a few people have decided to work with these features. What their primary concerns precisely and the responses from the globalisation are the competency of incorporation while doing their businesses. E-shopping has also increased substantially in Malaysia in recent years. The rapid increase in the e-commerce industry in Malaysia has created the demand to emphasize on how to increase customer satisfaction while operating in the e-retailing environment. It is very important that customers are satisfied with the website, or else, they would not return. Therefore, a crucial fact to look into is that companies must ensure that their customers are satisfied with their purchases that are really essential from the ecommerce’s point of view. With is in mind, this study aimed at investigating customer satisfaction towards e-shopping in Malaysia. A total of 400 questionnaires were distributed among students randomly selected from various public and private universities located within Klang valley area. Total 369 questionnaires were returned, out of which 341 questionnaires were found usable for further analysis. Finally, SEM was employed to test the hypotheses. This study found that customer satisfaction towards e-shopping in Malaysia is to a great extent influenced by ease of use, trust, design of the website, online security and e-service quality. Finally, recommendations and future study direction is provided. Keywords: E-shopping, Customer satisfaction, Trust, Online security, E-service quality, Malaysia