Analytic Functions Optimizing Competing Constraints

J. William Helton, Andrei E. Vityaev
1997 SIAM Journal on Mathematical Analysis  
Optimization of sup-norm-type performance functions over the space of H ∞ functions is an area of extensive research. In electrical engineering, it is central to the subject of H ∞ 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 ∞ -type optimum is frequency independent (flat). In this paper, we study simultaneous (Pareto) optimization of several competing performances Γ 1 , . . . , Γ
more » ... . We find under strong assumptions on the performance functions that if we are optimizing over N functions (f 1 , . . . , f N ) in H ∞ and have l performance measures with l ≤ N , then at a nondegenerate Pareto optimum (f * 1 , . . . , f * N ), 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 ∞ 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 [
doi:10.1137/s0036141095293086 fatcat:nuq4o7painhxtkmq7cfgl3ogdm