Clark's Theorem on linear programs holds for convex programs.
Given a linear minimization program, then there is an associated linear maximization program termed the dual. F. E. Clark proved the following theorem. "If the set of feasible points of one program is bounded, then the set of feasible points of the other program is unbounded." A convex program is the minimization of a convex function subject to the constraint that a number of other convex functions be nonpositive. As is well known, a dual maximization problem can be defined in terms of the Lagrange function. The dual objection function is the infimum of the Lagrange function. The feasible Lagrange multipliers are those satisfying: (i) the multipliers are nonnegative and (ii) the dual objective function is not negative infinity. It is found that Clark's Theorem applies unchanged to dual convex programs. Moreover, the programs have equal values.
Duffin
R J
RJ
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213.