Optimization and Lagrange Multipliers: Non-C1 Constraints and "Minimal" Constraint Qualifications

Abstract

When do Lagrange multipliers exist at constrained maxima? In this paper we establish: a) Existence of multipliers, replacing C1 smoothness of equality constraint functions by differentiability (for Jacobian constraint qualifications) or, for both equalities and inequalities, by the existence of partial derivatives (for path-type constraint qualifications). This unifies the treatment of equality and inequality constraints. b) A notion of "minimal" Jacobian constraint qualifications. We give new Jacobian qualifications and prove they are minimal over certain classes of constraint functions. c) A path-type constraint qualification, weaker than previous constraint qualifications, that is necessary and sufficient for existence of multipliers. (It only assumes existence of partial derivatives.) A survey of earlier results, beginning with Lagrange's own multipliers for equality constraints is contained in the last section. Among others, it notes contributions and formulations by Weierstrass; Bolza; Bliss; Caratheodory; Karush; Kuhn and Tucker; Arrow, Hurwicz, and Uzawa; Mangasarian and Fromovitz; and Gould and Tolle.