On the cone eigenvalue complementarity problem for higher-order tensors

In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we given an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing an upper bound of cone eigenvalues of tensors. Moreover, some new results concerning the bounds of number of eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least, an implementable projection algorithm for solving TGEiCP is also developed for the problem under consideration. As an illustration of our theoretical results, preliminary computational results are reported.Comment: 26 pages, 2 figures, 3 table

Hong Kong cinema under "one country, two systems" : production, reception and policy

Since the People’s Republic of China (PRC) resumed sovereignty over Hong Kong in 1997, the unprecedented ―one country, two systems (OCTS) policy has been put into practice. While this policy is usually considered from political, economic and legal perspectives, this study proposes a cultural studies approach to the understanding of this political formula through the examination of post-1997 Hong Kong cinema, particularly its production and reception in relation to the policies of both the central and local governments. Crossing and combining the disciplines of cultural studies and film studies, this dissertation has two primary aims: to understand this ―OCTS era as a peculiar cultural-historical conjuncture through the lens of Hong Kong cinema; and to explore the impact and influence of the OCTS policy on Hong Kong cinema as a social, economic and cultural institution. Embedding a textual analysis within contextual inquiry, this study will unravel the interplay between the reflection of the OCTS in the Hong Kong cinematic imaginary and its impact on the industrial operation, commercial performance, and critical response to post-1997 Hong Kong cinema. This thesis will address the production and reception of post-1997 Hong Kong cinema, and its significance for the analytical understanding of the OCTS policy through a number of perspectives. First, in its newly-claimed PRC market, Hong Kong cinema tends to be censored or self-censored. The resulting ―one movie, two versions phenomenon illustrates how Hong Kong and the PRC collaborate economically on the basis of ―one country while, at the same time, they diverge politically under the ―two systems. Second, the prominent presence of Mainland actresses in the thriving film co-productions is an indication of the changing dynamics in the Hong Kong-PRC relationship as a result of China’s economic takeoff. However, in an effort to retain a xv distinct local identity, some Hong Kong filmmakers are deliberately ignoring the lucrative PRC market in order to keep Hong Kong cinema unchanged. These ―not for the PRC‖ films are instrumental in monitoring the fulfillment of the ―no change in Hong Kong for fifty years promise made by the OCTS arrangement. Furthermore, either through the cinematic portrayals of Macao and Taiwan, or through the industrial linkages to Singapore and Malaysia, Hong Kong cinema has demonstrated a variety of ―Chineseness-es outside the PRC. The unshakable connections between Hong Kong cinema and Chinese diasporas have posed a serious challenge to the notion of equating ―China to the PRC as defined in the OCTS policy. Finally, the economic integration of Greater China has brought about the emergence of a pan-Chinese cinema mainly based on the Hong Kong martial arts genre. Taking advantage of their common history, cultural heritage and anti-imperialist Chinese nationalism, these pan-Chinese martial arts films have made a significant contribution to an imaginary ―unified cultural China, although the ultimate goal of the OCTS policy— the grand political reunification under the rule of the PRC — is still a dream yet to be fulfilled. By addressing the complexity of the PRC’s ―split reunification‖ with Hong Kong under the OCTS, this study challenges the simple dichotomy of ―PRC socialism vs. Hong Kong capitalism, probing overlapping concepts of ―China— the PRC, the Greater China or the imagined cultural China. Finally, it makes a broader contribution to research on ―national‖ cinemas in the context of dynamic geo-political and socio-cultural change within regions and across the globe

A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programming with Application to Imaging Data

In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized DC (Difference-of-Convex functions) programming, which is minimizing the sum of two separable DC parts and one two-block-variable coupling function. To circumvent the nonconvexity and nonseparability of the problem under consideration, we accordingly introduce a Unified Bregman Alternating Minimization Algorithm (UBAMA) by maximally exploiting the favorable DC structure of the objective. Specifically, we first follow the spirit of alternating minimization to update each block variable in a sequential order, which can efficiently tackle the nonseparablitity caused by the coupling function. Then, we employ the Fenchel-Young inequality to approximate the second DC components (i.e., concave parts) so that each subproblem reduces to a convex optimization problem, thereby alleviating the computational burden of the nonconvex DC parts. Moreover, each subproblem absorbs a Bregman proximal regularization term, which is usually beneficial for inducing closed-form solutions of subproblems for many cases via choosing appropriate Bregman kernel functions. It is remarkable that our algorithm not only provides an algorithmic framework to understand the iterative schemes of some novel existing algorithms, but also enjoys implementable schemes with easier subproblems than some state-of-the-art first-order algorithms developed for generic nonconvex and nonsmooth optimization problems. Theoretically, we prove that the sequence generated by our algorithm globally converges to a critical point under the Kurdyka-{\L}ojasiewicz (K{\L}) condition. Besides, we estimate the local convergence rates of our algorithm when we further know the prior information of the K{\L} exponent.Comment: 44 pages, 7figures, 5 tables. Any comments are welcom

von Neumann type trace inequality for dual quaternion matrices

As a powerful tool to represent rigid body motion in 3D spaces, dual quaternions have been successfully applied to robotics, 3D motion modelling and control, and computer graphics. Due to the important applications in multi-agent formation control, this paper addresses the concept of spectral norm of dual quaternion matrices. We introduce a von Neumann type trace inequality and a Hoffman-Wielandt type inequality for general dual quaternion matrices, where the latter characterizes a simultaneous perturbation bound on all singular values of a dual quaternion matrix. In particular, we also present two variants of the above two inequalities expressed by eigenvalues of dual quaternion Hermitian matrices. Our results are helpful for the further study of dual quaternion matrix theory, algorithmic design, and applications

An efficient symmetric primal-dual algorithmic framework for saddle point problems

In this paper, we propose a new primal-dual algorithmic framework for a class of convex-concave saddle point problems frequently arising from image processing and machine learning. Our algorithmic framework updates the primal variable between the twice calculations of the dual variable, thereby appearing a symmetric iterative scheme, which is accordingly called the {\bf s}ymmetric {\bf p}r{\bf i}mal-{\bf d}ual {\bf a}lgorithm (SPIDA). It is noteworthy that the subproblems of our SPIDA are equipped with Bregman proximal regularization terms, which make SPIDA versatile in the sense that it enjoys an algorithmic framework covering some existing algorithms such as the classical augmented Lagrangian method (ALM), linearized ALM, and Jacobian splitting algorithms for linearly constrained optimization problems. Besides, our algorithmic framework allows us to derive some customized versions so that SPIDA works as efficiently as possible for structured optimization problems. Theoretically, under some mild conditions, we prove the global convergence of SPIDA and estimate the linear convergence rate under a generalized error bound condition defined by Bregman distance. Finally, a series of numerical experiments on the matrix game, basis pursuit, robust principal component analysis, and image restoration demonstrate that our SPIDA works well on synthetic and real-world datasets.Comment: 32 pages; 5 figure; 7 table