An ODE-based approach to nonlinearly constrained minimax problems

Abstract

We consider the following problem: Choosex 1, ...,x n to

wherem 1,m 2,m 3 are integers with 0≤m 1m 2m 3, thef i are given real numbers, and theg i are given smooth functions. Constraints of the formg i (x 1, ...,x n )=0 can also be handled without problem. Each iteration of our algorithm involves approximately solving a certain non-linear system of first-order ordinary differential equations to get a search direction for a line search and using a Newton-like approach to correct back into the feasible region when necessary. The algorithm and our Fortran implementation of it will be discussed along with some examples. Our experience to date has been that the program is more robust than any of the library routines we have tried, although it generally requires more computer time. We have found this program to be an extremely useful tool in diverse areas, including polymer rheology, computer vision, and computation of convexity-preserving rational splines.

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Kaufman, E.H., Leeming, D.J. & Taylor, G.D. An ODE-based approach to nonlinearly constrained minimax problems. Numer Algor 9, 25–37 (1995). https://doi.org/10.1007/BF02143925

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Keywords

  • Polymer
  • Differential Equation
  • Real Number
  • Computer Time
  • Ordinary Differential Equation