Gibonacci Optimization : duality (Mathematical Decision Making Under Uncertainty and Related Topics)

Abstract

We show that a parametric linear system of equations plays a fundamental part in establishing a mutual relation between minimization problem (primal) and maximization problem (dual). The system is of 2n-equation on 2n-variable, called zero-minimum condition. It yields a couple of second-order finite (n-) linear difference equation on n-variable, which constitute the respective optimal conditions. The respective equations have a mimimum solution for primal and a maximum one for 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--.