@MISC{Iwamoto_cross-dualon, author = {Seiichi Iwamoto}, title = {Cross-Dual on The Golden Optimum Solutions}, year = {} }

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Abstract

This paper owes its origin to two simple minimization problems. One is a shortest distance problem on the plane. The other is a ratio minimization problem over the unit interval. We associates each of two minimization problems with a counterpart “maximiza-tion problem. Thus we consider two couples of (minimization and maximization) problems. Further we associate the two couples with a third and common cross-dual couple. Finally we have tuto pairs between three couples. As a total we have six optimization problems, each of which is to optimize one two-variable quadratic objective function under another quadratic constraint. An optimum solution – optimum point and optimum value – is called Golden if both the slope and the op-timum value constitute the Golden ratio. We show two interesting features. One is the Golden optimum solution. All six problems have the Golden optimum slutions. The other is a cross-duality. The first pair has a cross 2-sum property. The second has a cross inverse property. We illustrate a generative one-variable curve. Finally we show that the curve generates a couple of two-variable optimization problems. 1