Newsletter No. 161

CUHK Newsletter No. 161 19th March 2000 3 EXPERTS GATHER TO COMBAT OSTEOPOROSIS H undreds of doctors from all over the Asia Pacific Region gathered in the Hong Kong Convention and Exhibition Centre on 4th March 2000 to attend the opening ceremony of the first certificate course offered by the Asian Pacific Os t e o p o r o s is F o u n d a t i o n. The foundation was established by a group of doctors led by the Department of Community and Family Medicine in response to the urgent need f or practitioners in all specialities to be equipped with cutting-edge knowledge for the diagnosis, treatment, prevention, rehabilitation, and patient education of the disease. Participants of the certificate course w i ll be introduced to the risk factors, current problems, and the latest methods of treatment and diagnosis of osteoporosis. Research conducted by the Faculty of Medicine revealed that bone mass loss in Chinese women occurs as early as 31- 32 years of age. During a lifetime, women lose 58 per cent of their bone mass while men lose 39 per cent. I f unattended, the disease w i l l weigh heavily on the provision of medical services. The World Health Organization has designated the upcoming decade as the 'Bone and Joint Decade'. Nat i onal Screening Programmes for Cervical Cancer Under Scrut iny A bout 80 pathologists and cytologists from Hong Kong and mainland China participated in a symposium entitled 'Cervical Cancer Screening in the New Millennium' organized by the Department of Anatomical and Cellular Pathology on 15th January 2000 at the Prince of Wales Hospital. Cervical cancer is the fourth commonest cancer among women in Hong Kong and the Pap smear is an effective means of detecting and preventing its development. The symposium focussed on the logistics of an effective national screening programme as well as new methods of screening. Prof. Kenneth Suen f r om the Faculty of Medicine of the University of British Columbia, who is in charge of one o f the oldest na t i onal screening programmes in the world, recounted the screening experience in Canada. Dr. Susan Fan, director of the Hong Kong Family Planning Association, described the situation in Hong Kong, while Prof. Chen Lezhen and Prof. Song Lei from the Chinese People's Liberation A rmy General Hospital reviewed cervical cancer screening in Be i j i ng. Prof. Alexander Chang f r om the CUHK Department of Anatomical and Cellular Pathology conducted a live demonstration of remote control telepathology with Beijing on problem cases of cervical smears. Prof. Zhou Xunyu studied mathematics at Fudan University, obtaining his M.Sc. and Ph.D. in 1984 and 1989 respectively. He then spent four years as a postdoctoral fellow at Kobe University in Japan and the University of Toronto in Canada. He joined The Chinese University in 1993 as a lecturer in the Department of Systems Engineering (now renamed Department of Systems Engineering and Engineering Management), becoming senior lecturer in 1998. Prof. Zhou is a senior member of IEEE and a member of SIAM, and serves as an associate editor to Operations Research and IEEE Transactions on Automatic Control. common belief that control cost has to be positive, or the problem would be trivial like the two quoted in the opening of this article. And certainly, a positive control cost has a clear physical meaning. Deterministic LQ problems with positive control cost can be elegantly solved using the ubiquitous Riccati equation. For instance, in missile manufacture, precision necessitates the investment of control, i.e., money, researchers, time, etc. And the more control input, the higher the precision and the control cost. Here the goal is measured by the square of the difference between the missile's current position and that of its target, while the cost is measured by the square of the control. Prof. Zhou explained, 'A situation with positive control cost is one where more control input w i l l bring greater returns, yet it isn't advisable to maximize the control level because of the positive cost involved. One needs to strike a balance.' Returning to the missile analogy, i f a high precision missile means having to invest half a country's budget, should it still be done i f the country was not hemmed in on all sides by enemies? The missile example, hence, is a deterministic model that has a positive control cost, so an overwhelming i nves tment in mi ss i le manufacturing w i l l be avoided in an optimal decision. I f the control cost is negative, meaning there is a reward to the control, increasing the control would result in more direct returns, e.g. the more savings in the bank the more the interest. In such a case, the optimal strategy is simply 'the larger the control the better' and the problem becomes trivial or meaningless. Due to this reason, LQ control problems with negative control cost were often neglected in the past. Prof. Zhou's observations, however, showed that a stochastic LQ problem with negative control cost may still be non-trivial i f the control is influencing the level of uncertainty of the model, in which case 'the-larger-the-control-the- better' strategy is no longer valid. For instance, it does not appear to be a best strategy to put all the $1,000,000 in a stock no matter how good its performance has been in the past. The reason is obvious: the potential gain due to a larger amount of money invested may not outweigh the potential loss due to greater risk. This kind of situation is prevalent in real-world systems. The crucial factor distinguishing a stochastic LQ problem with negative control from its deterministic counterpart is the fact that the decision-maker in the latter case has control over the level of uncertainty. In 1997 Prof. Zhou received an earmarked grant of HK$ 1,134,000 from the Research Grants Council to study the stochastic LQ problem with indefinite, i.e., possibly positive or negative, control cost and to obtain a complete solution to the problem. A related issue under study is: i f the control cost is negative, how negative can it be before the problem becomes meaningless? Other researchers on the project team include Prof. David Yao and Prof. Duan Li, two postdoctoral fellows and two doctoral candidates of the Department of Systems Engineering and Engineering Management, as well as a handful of researchers from Australia, the US, the Netherlands, and mainland China. Three Phases of the Project The project was divided into three phases. In the first phase, which was completed in late 1998, the stochastic LQ model in a finite time horizon, i.e., a relatively 'short' time span, was tackled and the foundation of the whole theory was laid. Using a new differential equation called the Stochastic Riccati Equation (SRE), the extra cost incurred by uncertainty was precisely calculated. An estimate of how negative the control cost could be and when the cost of uncertainty w i ll start to outweigh the benefit of larger control was given. An algorithm of computing the solution to th SRE was also presented. In the second phase, completed in late 1999, the stochastic LQ problem in an infinite time horizon, i.e., a relatively 'long' time span, was studied with an emphasis on solving it numerically. In this case, the associated Riccati equation is an algebraic, rather than differential, equation. Algebraic equations, unlike d i f f e r en t i al equations, contain no derivatives of the unknown. This enabled the researchers to apply the techniques of linear matrix inequality ( LMI) and semidefinite programming (SDP), two very active research topics at present in the area of mathematical programming, to solve the Riccati equation. It was found that LQ problems can be solved computationally using powerful SDP solvers. In the final phase, which is expected to be completed by late 2000, the focus is on the application of the theory developed in the first phase to problems in finance, as well as its implications for them. Since the Nobel prize-winning Black-Scholes model for evaluating options on assets is exactly a linear diffusion model, the research team applied the stochastic LQ model as a f r amewo rk to study fundamental problems in finance such as portfolio selection, options pricing, and risk hedging. 'The inherent linear-quadratic structure of the Black-Scholes model makes it fall nicely into the application domain of the stochastic LQ theory we developed,' Prof. Zhou said. The results derived would enable uncertainty or risk to be quantified and evaluated with great precision via either Riccati equations or semidefinite programming. They w i ll have important implications for financial risk management. Prof. Zhou pointed out that the project deals w i th a very exciting research area. Many fundamental and important problems remain unresolved and their resolution is expected in turn to give rise to new problems. The findings o f the p r o j ect have been h i g h ly commended in academic circles and its topic described as pleasantly 'surprising'. The project has also generated articles which have been published in major academic journals in the field including SIAM Journal on Control and Optimization and IEEE Transactions on Automatic Control. Piera Chen