Optimization appears not only in nature but throughout social activities and daily life. In this sense, optimization is a field of research that serves as a bridge between mathematics and the real world. Within optimization, the branch dealing with continuous quantities is called continuous optimization, and my research and teaching focus mainly on it. For example, the method of Lagrange multipliers, taught in first-year calculus, is one of the techniques discussed in continuous optimization. In what follows, I briefly introduce my research, education, and industry–academia collaboration, all centered on optimization.

　My research has centered on convex optimization, in particular semidefinite programming (SDP), a class of optimization problems with matrix variables (Fig. 1). Before joining IMI, I worked on proposing and implementing algorithms that solve difficult optimization problems—polynomial optimization among them—by reducing them to SDPs; the results include methods that exploit the sparsity of a problem to make computation more efficient, together with publicly released software. My focus has since shifted toward theory: I now study SDPs arising in control theory. Such problems are often ill-conditioned, that is, numerically difficult to handle, and a turning point came when I found that the cause can be described clearly in the language of systems theory (Fig. 2). Including the elucidation of the intriguing mathematical structure of the dual problems, I have continued to publish at the boundary between control and optimization. This style of pursuing both “deepening optimization theory itself” and “using optimization effectively in other fields” fits well with IMI’s mission of collaboration across disciplines and with industry, and I intend to keep opening up new research themes through collaboration with researchers in other fields and with companies.

　In supervising students, my policy is to avoid a shrinking reproduction of my own field: I encourage students to take on themes slightly outside my expertise, such as applications of integer programming and the development of algorithms for them. I learn a great deal from my students, and I am fortunate to be able to publish joint papers with them. I believe this style of supervision fosters people with a broad grounding in optimization who can thrive in both academia and industry. As for courses, although the undergraduate mathematics department offers no course on optimization, the graduate school does; there I teach the fundamentals of optimization theory over a wide range of topics, aiming to send out into society people who can put optimization to use.

　As for industry–academia collaboration, I have used IMI’s joint research framework to carry out funded joint research with partners in the automotive industry, manufacturing, government, and elsewhere. In doing so, I position students as co-researchers rather than inexpensive labor, and I make a point of working hands-on together with them. I have come to appreciate that such collaboration is valuable not only for solving companies’ problems but also as a venue for personnel exchange and human resource development (Fig. 3). My ideal of industry–academia collaboration is one in which industry, faculty, and students write papers together, leaving the knowledge they have gained to posterity, while the students earn degrees along the way and go on to thrive in academia and industry. I hope to build long-term partnerships with those who share this ideal. I will continue to actively seek opportunities for joint research and personnel exchange, and I warmly invite companies and organizations facing optimization-related challenges to contact me.