Vol. 15, No. 3, 1965

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Symmetric dual nonlinear programs

George Bernard Dantzig, E. Eisenberg and Richard Warren Cottle

Vol. 15 (1965), No. 3, 809–812

Consider a function K(x,y) continuously differentiable in x Rn and y Rm. We form two problems:

PRIMAL: Find (x,y) 0 and Min F such that

F = K (x,y)− yTDyK (x,y), DyK (x,y) ≦ 0

DUAL: Find (x,y) 0 and Max G such that

G = K (x,y)− xTDxK (x,y), DxK (x,y) ≧ 0

where DyK(x,y) and DxK(x,y) denote the vectors of partial derivatives DyiK(x,y) and DxjK(x,y) for i = 1,,m and j = 1,,n. Our main result is the existence of a common extremal solution (x0,y0) to both the primal and dual systems when (i) an extremal solution (x0,y0) to the primal exists, (ii) K is convex in x for each y, concave in y for each x and (iii) K, twice differentiable, has the property at (x0,y0) that its matrix of second partials with respect to y is negative definite.

Mathematical Subject Classification
Primary: 90.58
Received: 23 January 1964
Published: 1 September 1965
George Bernard Dantzig
E. Eisenberg
Richard Warren Cottle