A geometric representation for pairs of dual quadratic or linear programs

Abstract

The relation between a linear program and its dual has been elegantly described in terms of a Lagrangian function, in which each set of variables appears as a set of Lagrange multipliers for the other [l]. Lagrangian functions have also been defined for quadratic programs [2], and in analogy with the linear case, the quadratic Lagrangian has been used as a stepping-stone to arrive at quadratic duals [3,4]. Pairs of dual quadratic programs so defined, however, are strangely unsymmetrical. The present paper is the result of an effort to understand this lack of symmetry. A special quadratic program brought in by an engineer turned out to have a perfectly symmetrical geometric representation and a perfectly symmetrical associated problem which differed slightly from the dual based on the Lagrangian. This suggested a geometric definition of duality. From the geometric point of view the midpoint between primal and dual appears to be not the Lagrangian but a set of relations called here a decomposition and equivalent to the conditions of Kuhn and Tucker [2]. In cases more general than the symmetrical one just mentioned, the geometric definition of duality leads to asymmetric duals which agree in essence with those previously defined. These can, however, be reduced to a simple, symmetrical canonical form1 by changing coordinates. For the general quadratic program the coordinate change is computationally nontrivial, but for the special class of quadratic programs with which the paper is mainly concerned the coordinate change is merely a translation and is computationally easy. A computation procedure for the special class of quadratic programs considered here is presented at the end of the paper. This procedure is referred to as the projection method.