Fibonacci optimization and its related field --duality-- (II) (Study on Nonlinear Analysis and Convex Analysis)

Abstract

We consider a 2𝑛-variable parametric minimization problem, where a parameter λ＞0. Then it holds that the parametric minimization problem derives two problems, which are λ-parametric minimization problem (primal) and λ-parametric maximization problem (dual). Both the optimal solutions are expressed in terms of Gibonacci sequence, which is a parametric generalization of the Fibonacci one. Either solution is characterized by the backward Gibonacci sequence and its complementary --Hibonacci sequence-- . In particular, when a parameter λ＝1, we show that Gibonacci sequence and Hibonacci sequence are represented by Fibonacci number. Moreover, for λ＝4, both the sequences are represented by Sibonacci number.