The Chinese University of Hong Kong

About: The Chinese University of Hong Kong is a education organization based out in Hong Kong, China. It is known for research contribution in the topics: Population & Computer science. The organization has 43411 authors who have published 93672 publications receiving 3066651 citations.

Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls

Abstract: This paper deals with an optimal stochastic linear-quadratic (LQ) control problem in infinite time horizon, where the diffusion term in dynamics depends on both the state and the control variables. In contrast to the deterministic case, we allow the control and state weighting matrices in the cost functional to be indefinite. This leads to an indefinite LQ problem, which may still be well posed due to the deep nature of uncertainty involved. The problem gives rise to a stochastic algebraic Riccati equation (SARE), which is, however, fundamentally different from the classical algebraic Riccati equation as a result of the indefinite nature of the LQ problem. To analyze the SARE, we introduce linear matrix inequalities (LMIs) whose feasibility is shown to be equivalent to the solvability of the SARE. Moreover, we develop a computational approach to the SARE via a semi-definite programming associated with the LMIs. Finally, numerical experiments are reported to illustrate the proposed approach.

On characterization of entropy function via information inequalities

Abstract: Given n discrete random variables /spl Omega/={X/sub 1/, /spl middot//spl middot//spl middot/, X/sub n/}, associated with any subset /spl alpha/ of (1, 2, /spl middot//spl middot//spl middot/, n), there is a joint entropy H(X/sub /spl alpha//) where X/sub /spl alpha//={X/sub i/:i/spl epsiv//spl alpha/}. This can be viewed as a function defined on 2/sup {1, 2, /spl middot//spl middot//spl middot/, n}/ taking values in (0, +/spl infin/). We call this function the entropy function of /spl Omega/. The nonnegativity of the joint entropies implies that this function is nonnegative; the nonnegativity of the conditional joint entropies implies that this function is nondecreasing; and the nonnegativity of the conditional mutual information implies that this function has the following property: for any two subsets /spl alpha/ and /spl beta/ of {1, 2, /spl middot//spl middot//spl middot/, n} H/sub /spl Omega//(/spl alpha/)+H/sub /spl Omega//(/spl beta/)/spl ges/H/sub /spl Omega//(/spl alpha//spl cup//spl beta/)+H/sub /spl Omega//(/spl alpha//spl cap//spl beta/). These properties are the so-called basic information inequalities of Shannon's information measures. Do these properties fully characterize the entropy function? To make this question more precise, we view an entropy function as a 2/sup n/-1-dimensional vector where the coordinates are indexed by the nonempty subsets of the ground set {1, 2, /spl middot//spl middot//spl middot/, n}. Let /spl Gamma//sub n/ be the cone in R/sup 2n-1/ consisting of all vectors which have these three properties when they are viewed as functions defined on 2/sup {1, 2, /spl middot//spl middot//spl middot/, n}/. Let /spl Gamma//sub n/* be the set of all 2/sup n/-1-dimensional vectors which correspond to the entropy functions of some sets of n discrete random variables. The question can be restated as: is it true that for any n, /spl Gamma/~/sub n/*=/spl Gamma//sub n/? Here /spl Gamma/~/sub n/* stands for the closure of the set /spl Gamma//sub n/*. The answer is "yes" when n=2 and 3 as proved in our previous work. Based on intuition, one may tend to believe that the answer should be "yes" for any n. The main discovery of this paper is a new information-theoretic inequality involving four discrete random variables which gives a negative answer to this fundamental problem in information theory: /spl Gamma/~*/sub n/ is strictly smaller than /spl Gamma//sub n/ whenever n>3. While this new inequality gives a nontrivial outer bound to the cone /spl Gamma/~/sub 4/*, an inner bound for /spl Gamma/~*/sub 4/ is also given. The inequality is also extended to any number of random variables.

A Generalized Utility Model of Disappointment and Regret Effects on Post-Choice Valuation

Abstract: In this paper we show that performance information about “forgone” alternatives i.e., alternative that were considered but not chosen can have a significant impact on post-choice valuation. Our approach introduces a new and parsimonious way of looking at satisfaction that combines the literature on post-choice valuation with research regarding generalized expected utility theory. While the post-valuation literature focuses on the selected brand as the valuation's basis e.g., Anderson and Sullivan [Anderson, E. W., M. W. Sullivan. 1993. The antecedents and consequences of customer satisfaction for firms. Marketing Sci.12Spring 125--143.], Bolton and Drew [Bolton, R., J. Drew. 1991. A multistage model of customers' assessments of service quality and value. J. Consumer Res.17 375--384.], we draw on a stream of research in generalized expected utility theory that considers both chosen and forgone alternatives as the basis for valuation e.g., Bell [Bell, D. 1983. Risk premiums for decision regret. Management Sci.29 1156--1166 and Bell, D. 1985. Disappointment in decision making under uncertainty. Oper. Res.33 1--27.]; Loomes and Sugden [Loomes, G., R. Sugden. 1982. Regret theory: An alternative theory of rational choice under uncertainty. The Econom. J.92 805--824 and Loomes, G., R. Sugden. 1986. Disappointment and dynamic consistency in choice under uncertainty. Rev. Econom. Stud.53 271--282.]. The result is a combined model of post-choice valuation that explicitly incorporates both concepts. Specifically, we extend the existing paradigm of post-choice valuation to include buyers' regret concerning forgone alternatives, proposing a generalized utility theory-based treatment of post-choice product assessment that uses the intuitively appealing concepts of disappointment and regret as the basis. We propose a model for conceptualizing post-choice valuation that is consistent with the existing literature, discuss how this model extends the construct to consider the influence of forgone alternatives, and report results of an empirical test that contrasts our model to important recent work in the area e.g., Boulding et al. [Boulding, W., A. Kalra, R. Staelin, V. A. Zeithaml. 1993. A dynamic process model of service quality: From expectations to behavioral intentions. J. Marketing Res.30February 7--27.]. Our generalized model of post-choice valuation is based on the sum of three components that represent factors that may contribute to consumers' assessment of a chosen product or service. The first component is expected performance. The second component is disappointment, which captures the discrepancy between actual and expected performance much as the disconfirmation construct in traditional satisfaction research. The third component is regret, which captures the difference between the performance of the chosen product/service and the performance of a forgone product/service. This perspective is useful in that risk is captured by the disappointment and regret terms, providing an intuitively appealing decomposition of post-choice valuation and offering several advances over previous representations of disappointment and regret. We test our model via a choice experiment. Participants in the empirical study were asked to make choices between successive lottery pairs. They were then given outcome feedback on the forgone alternative as well as on the chosen alternative in each lottery pair. Immediately following outcome feedback for each choice, subjects were asked to evaluate their decision. Our results clearly suggest an effect of regret on post-choice valuation---information about the forgone alternative influenced subjects' valuation of the chosen alternative. We also find evidence that, as predicted, the effects of disappointment and regret on post-choice valuation are asymmetric. Specifically, the negative effect of disappointment on post-choice valuation was greater than the positive effect of elation. Similarly, the negative effect of regret was greater than the positive impact of rejoicing. Our research offers five contributions to the literature on post-choice valuation. First, our results illustrate the advantage of using generalized utility theory as the basis for conceptualizing and modeling post-choice valuation. We derive a model of post-choice valuation that formally captures the components of disappointment and regret and show that the outcome of both the chosen alternative through disappointment and a forgone alternative through regret can influence the chosen alternative's valuation. Second, we formally integrate the concepts of disappointment and regret, which have been examined separately for many years, into a single model based on a multiattribute preference structure. Third, we argue that the effects of disappointment and regret on post-choice valuation are asymmetric and present empirical evidence in this regard. Fourth, our results suggest that word-of-mouth regarding forgone alternatives may affect post-choice valuation, extending research that has not heretofore considered forgone outcomes' role in this process. Finally, our development provides a preconsumption measure of “potential disappointment and regret” in modeling choice. At the time of choice, consumers may visualize the feelings of disappointment and/or regret that will be derived at consumption, taking into account both the chosen brand and forgone brands. Our generalized utility theory-based approach to the post-choice valuation construct can be useful in examining the role of disappointment and regret as preconsumption constructs.

Classification of solutions of higher order conformally invariant equations

Abstract: Recently, there have been much analytic work on the conformally invariant operators as well as its associated differential equations. A well known second order conformally invariant operator comes from the Yamabe problem or, more generally, the problem of prescribed scalar curvature. Given a smooth positive function K defined on a compact Riemannian manifold (M, g0) of dimension n ≥ 2, we ask whether there exists a metric g conformal to g0 such that K is the scalar curvature of the new metric g. Let g = eg0 for n = 2 or g = u 4 n−2 g0 for n ≥ 3, then the problem is reduced to find solutions of the following nonlinear elliptic equations: