Derivation of duality in mathematical programming and optimization theory

Abstract

Consider the primal problem to minimize a function over a set of feasible solutions in Euclidean space. This paper derives a general dual problem directly from this primal problem. It shows that the Lagrangian, conjugate and Johri's formulations of duality in nonlinear programming, the linear programming duality, and the minimum norm duality principle in optimization theory are all special cases of this general duality formulation. This explains the duality — why two seemingly different problems provide the same answer — by showing that there really is no duality but that the dual problems are just different forms of the primal problem. All these results are derived in a simple, intuitive, unified manner.