Extensions of Lagrange Multipliers in Nonlinear Programming

defined for x > 0, u > 0. A saddle point of (1.2) is defined to be a point (x*, u*), x* > Q, u* > 0 such that +(x, u*) < +(x*, u*) _ +(x*, u) for all x _ 0, u ? 0. One of the Kuhn-Tucker results is that a saddle point of(1.2) provides a solution to (1.1). Under cer-eain additional regularity assumptions the equivalence of the saddle point problem and the program (1.1) has been shown [1], [10]. In particular, if the feasible set in (1.1) has an interior point (a nonnegative x such that g(x) < b),3 and if the objective function is concave and the constraint functions are convex, then x* is a solution to (1.1) if and only if there is a nonnegative u* such that (x*, u*) is a saddle point of (1.2). This is the so-called Kuhn-Tucker equivalence theorem of nonlinear programming. In this paper Lagrange multipliers are generalized from the usual constants to (possibly nonlinear) multiplier functions. This leads to the statement of several general equivalence results under quite weak assumptions. From a somewhat different point of view, Everett [4] drops differentiability assumptions and discusses a procedure for solving nonlinear programs in terms of