University. He works in an Operations Research group led by Prof. Toshihide Ibaraki. Much of Prof. Fukushima's work appears in the Western literature. I am particularly interested in the work on nondifferentiable optimization (Ref 5-7) and our technical discussions focused on that work. During my visit I was also able to visit a group led by Prof. Nogi that has designed a distributed memory multiprocessor, ADENA.

Aside from the work listed above, Fukushima is active in several other areas in optimization. I will list some of these areas and provide a recent reference to the literature if one is available. These areas of activity include finite dimensional variational inequalities (Ref 8) with applications to traffic and market equilibrium problems, nonlinear network flow problems (Ref 9), homotopy methods (Ref 10), and ongoing work on combinatorial optimization and parallel methods for optimization. This latter project is joint work with Profs. Nguyen and Strodiot at the University of Namur in Belgium. All of Fukushima's work is directed toward numerical results.

The method of sequential quadratic programming (SQP) for nonlinear constrained optimization problems is a generalization of Newton's method as applied to nonlinear equations and unconstrained problems. In the case of equality constraints, one may think of SQP as the application of Newton's method to the gradient of the Lagrangian. A single SQP iterate solves a quadratic program that has as its objective function a quadratic model of the objective function at the current point and as its constraints the linearized constraints at the current point. Like Newton's method, if the initial data are sufficiently good, the iterates will converge quadratically, to the solution if typical smoothness and nonsingularity assumptions hold.

When the initial iterate is far from the solution, however, line search or trust region methods are used to modify

the Newton step in the unconstrained case to force "sufficient decrease" in the objective function. In the case of constrained optimization, the constraints require that decisions on step modification be based on how well the constraints are satisfied as well as the size of the objective function. Hence, steps are modified when the unmodified SQP step does not give sufficient decrease in a "merit" or "penalty" function. Such functions are constructed to measure a combination of constraint satisfaction with the value of the objective function. So called "exact" penalty functions, which are in general not differentiable, have the desirable property that their local minima coincide with those of the constrained problem to be solved.

Nonsmooth exact penalty functions may suffer from the "Maratos effect" when the merit function will not allow a full SQP step to be taken near the solution and therefore only q-linear convergence takes place. Fukushima's work (Ref 5) addresses this problem in the line search case by changing the linearized constraints for the QP subproblem in an appropriate way. He has extended this work to trust region methods (Ref 6) and to the case of nonsmooth objective function and equality constraints (Ref 7). He has a Fortran program that implements the algorithm as reported in Reference 5.

Fukushima's work in this area is related to work in the West by Fletcher, Powell, and Ynan, among others, in the United Kingdom and by Tolle, S.J. Wright, and others in the United States.

The Institute of Statistical Mathematics

My host for this visit was Prof. Kunio Tanabe, director of the Department of Prediction and Control. I have been aware of Prof. Tanabe's leading role in Japanese optimization for several years and discussed

optimization in Japan and his own work in development of a geometric framework for globally convergent optimization algorithms. While at the institute I also spoke with Prof. G. Kitagawa, who works in large scale time series analysis, and Dr. T. Tsuchiya, who works in automatic differentiation. I was impressed with the breadth of activities at the institute and the commitment the scientists there have to solving problems that are relevant to Japanese Government and industry. The institute has a 40-year history of success in solving problems for Japanese industry. According to Prof. Tanabe, this tradition encourages Japanese business to share data with the institute and rely on it for help.

Prof. Tanabe added to my list of people working on nonlinear optimization in Japan. Prof. T. Nakayama of Konan University works on nondifferentiable optimization and multiobjective programming. In the area of unconstrained smooth optimization, Tanaka mentioned N. Yamaki and M. Yabe of Tokyo Science University, noting that Yamaki is now in private industry; N. Sagara of Aichi University; K. Shimizu at Keio University, K. Tone at Saitama University, and M. Yamashita in private industry. As in the United States, the nonlinear optimization community is healthy and active but not as large as many other specialties in pure or applied mathematics.

Prof. Tanabe's research interests include numerical optimization and applications to such area as medicine, seismology, and statistics. Our conversations were about a geometric approach that he takes to the construction of globally convergent methods for nonlinear optimization that is motivated by interior point methods in linear programming. Key references are References 11-13.

In Reference 11, Tanabe introduces what he now calls a "center flattening transformation" in the context of singular systems as a way to numerically

follow the trajectories of the continuous Newton dynamical system to the solution. He has expanded these methods in subsequent work by describing a "center curve" subject of the feasible set of a constrained nonlinear programming problem. His "centered Newton" method (Ref 13) makes a connection between these geometric ideas and interior point methods in linear programming. The centered Newton method differs from conventional line search or trust region methods for optimization in that if a full Newton step fails to give sufficient decrease, then a new search direction biased toward the center curve is used. Tanabe believes that this approach will allow for longer steps to be taken than in more conventional implementations.

ACKNOWLEDGMENT

Thanks to A. Frommer (Univ. of Karlsruhe), E. Allgower (Colorado State Univ.), and J. Jerome (Northwestern Univ.) for many helpful comments.

REFERENCES

1. H. Kawasaki, "The upper and lower second order directional derivatives of a sup-type function," Math. Prog. 41, 327-29 (1988).

2. H. Kawasaki, “An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems," Math. Prog. 41, 73-96 (1988).

IN TOKYO

GMD also initiated the foundation of the International Computer Science Institute (ICST), headed by Jerome A. Feldman, at the University of California at Berkeley. This is a basic research institute for massively parallel systems, distributed systems, and the foundations of informatics.

The liaison office in Tokyo has three to four professionals. At the moment there are four; one is to rotate back to GMD in February 1991. Three have Ph.D. degrees, in physics, economics, and mathematics. (The director, Dr. Wattenberg, was away from the office when I visited.) A German-speaking Japanese administrative staff perform "research" and other tasks, and one person is also responsible for gathering and distributing government and other difficult-to-get documents relating to information technology. GMD also has a Washington office, larger than the one in Tokyo.

The director has been here since the office's inception, about 15 years. The remaining professional staff come for periods of 3 to 5 years. Except for the director, all of them expect to go back to an administrative arm of GMD.

During the first 6 months of a new person's tour, he/she spends about 2 days each week for 2 months each at differ

ent Japanese public organizations affiliated with the Ministry of International Trade and Industry (MITI). These organizations include MITI

headquarters; the Japan Information Processing Development Center (JIPDEC), which administers some of the MITI projects including the Institute for New Generation Computer Technology (ICOT), at least in the initial phase; and the Database Promotion Center (DPC). (This is permitted because the liaison office has the official sanction of both the German and Japanese Governments.) This gives new staff an opportunity to meet many key people and get national perspective. It is a tremendous orientation opportunity that would have been very valuable to me (still would be, even after a year). This, plus the continuity provided by the director's long tenure, assures that the staff do not step on each other's toes. The remaining 3 days each week are spent sharpening language skills and also doing "real" work. The main focus of the office is claimed to be research in computing, but they are also watching new releases in supercomputing, semiconductor technology, etc. that have commercial value for their laboratories back home. Although there is a science councillor in the German Embassy, some computing related liaison functions are handled in this office.

The office subscribes to a translation service that provides timely information about information technology news. One of the German professionals is responsible for transmitting a weekly summary, in German, and also providing an up-to-date list of meetings that have been announced. I think that the news service would be valuable because the compiler is a specialist who can rapidly check questionable or uninteresting items and get additional details when necessary. I was told that our office here tried a news service in the past but it was very expensive and not used much. Perhaps it was too general, or the scientists were not interested. I will consider

subscribing to a similar service again to see how useful it would be to me.

A major activity of the office is planning a workshop, formally titled "JapanGerman Forum on Information Technology," held every 18 months, alternating between Germany and Japan, with about 100 participants. Three topics are covered: microelectronics, computers, and new media. The workshop is a closed shop organized at the level of the German and Japanese Governments, and each country is responsible for supporting an appropriate number of their researchers. A Proceedings is produced, but it is only circulated to workshop participants, in government circles, and to certain industrial and academic research centers. The small GMD office staff would find it difficult to distribute copies of the individual papers to outside requestors, individual papers to outside requestors, but at the end of this article is a list of titles from the previous workshop to illustrate the material discussed. What is impressive about this list is that the level of papers allows most of them to be ofdirect interest to not only researchers but to those one step removed. As those of us who attend conferences know, carefully written survey papers are very rare.

The office also organizes most scientific visits for GMD staff and also of others from the German research community. The professional staff attend many meetings, participate in laboratory visits, write reports, etc., very much like I do. To the best of my knowledge, their reports are only distributed internally.

The office is housed in the German cultural center. GMD pays no rent for most of its space, but one overflow office is required and is expensive. The office has an internal computer network so that staff can communicate, print, etc. The office manager backs up files each week. They also have available electronic mail in three separate

forms: (1) a commercial local line to a service that allows them to contact GMD directly, (2) a commercial mail service, and (3) an x-400 mail service at a local commercial computer that is also connected to GMD. Their news translation services are sent to them electronically by (2) above. Current plans are to become a node on the rapidly growing Japanese equivalent of Internet. To maintain this requires a high level of sophistication. Luckily, each of the professionals is interested in computing and so they are willing and eager to take on this responsibility. (An equivalent activity in my office could be done but might be difficult because of the wide range of computer interests and expertise within the scientific staff.)

The GMD offices are attractive although not extravagant, looking more or less like those in the U.S. Embassy. Because they are in the cultural center, and open to everyone, they have a very benign feeling. This is helped by being next door to the equivalent of the German National Science Foundation, which is responsible for more general scientific exchanges. Overall, the offices give a very positive impression, in a country where these things count for more than they do in the United States.

The acting director informed me that a very rough estimate for their annual budget is DM1.5 million. For comparison, I am told that the equivalent budget for our office is about $1.0 million, although I am sure that an accurate comparison would require much more care.

This office impressed and surprised me, particularly the level of commitment with which a government laboratory is going after scientific information, research, and commercial opportunities. There are aspects of its operation that we should study carefully. This would be a worthwhile and easily arranged place for Western scientists to visit.