Complementary Duals—Both Fixed Points (III)—

We consider a quadratic minimization problem with both fixed endpoints and its associated maximization problem from a viewpoint of complementarity. We show that a complementary identity with an elementary inequality generates a pair of dn _ 1T-variable minimization problem (primal) and dn t 1T-variable maximization one (dual). The identity produces an equality condition, which is a linear system of 2n-equation on 2n-variable. The condition splits into a pair of linear systems of dn _ 1T-equation on dn _ 1T-variable and of dn t 1T-equation on dn t 1T-variable. The former solves the primal, while the latter does the dual. Both the optimal solutions are characterized by the Fibonacci sequence. The solutions are also given through dynamic programming.