Analytic Functions Optimizing Competing Constraints

• Published in: SIAM journal on mathematical analysis 1997-05, Vol.28 (3), p.749-767
• Main Authors: Helton, J. William, Vityaev, Andrei E.
• Format: Article
• Language: English
• Subjects: Applied mathematics; Computers; Pareto optimum
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/S0036141095293086

Optimization of sup-norm-type performance functions over the space of $H^\infty$ functions is an area of extensive research. In electrical engineering, it is central to the subject of $H^\infty$ design, while in several complex variables, it is often required to produce analytic discs with valuable properties. It has been known for many years that an $H^\infty$-type optimum is frequency independent (flat). In this paper, we study simultaneous (Pareto) optimization of several competing performances $\Gamma_1, \dots , \Gamma_l$. We find under strong assumptions on the performance functions that if we are optimizing over $N$ functions $(f_1, \ldots ,f_N)$ in $H^\infty$ and have $l$ performance measures with $l \le N$, then at a nondegenerate Pareto optimum $(f^*_1, \ldots ,f^*_N)$, {\it every} performance is flat. Besides flatness, there are other gradient--alignment conditions which must hold at an optimum. The article presents these and thus gives the precise first-derivative test for a natural class of $H^\infty$ Pareto optima. Such optimality conditions are valuable for assessing how iterations in a computer run are progressing. Also, in the traditional case, optimality conditions have been the base of highly sucessful computer algorithms; see [J. W. Helton, O. Merino, and T. Walker, Indiana U. Math. J., 42 (1993), pp. 839--874].