Summary: We consider a quadratic minimization (primal) problem with both fixed endpoints and its associated maximization (dual) problem from a view point of complementarity. We show that a new complementary identity produces the pair with an equality condition. The condition is a linear system of \((2n+1)\)-equation on \((2n+1)\)-variable. The system yields a couple of solutions; one is a minimum solution to \(n\)-variable primal and the other is a maximum one to \((n+1)\)-variable dual. Both the solutions turn out to be complementary. The optimal solution is characterized by the backward Fibonacci sequence. The duality is enhanced through conjugate function. The optimal solution is also given by dynamic programming.
For part I, see [S. Iwamoto et al., Bull. Kyushu Inst. Technol., Pure Appl. Math. 67, 1–28 (2020; Zbl 1469.90102)].