Center for Economic Research, Department of Economics, University of Minnesota
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.
Hurwicz, L. and Richter, M.K., (1995), "Optimization and Lagrange Multipliers: Non-C1 Constraints and "Minimal" Constraint Qualifications", Discussion Paper No. 280, Center for Economic Research, Department of Economics, University of Minnesota.
Hurwicz, Leonid; Richter, Marcel K..
Optimization and Lagrange Multipliers: Non-C1 Constraints and "Minimal" Constraint Qualifications.
Center for Economic Research, Department of Economics, University of Minnesota.
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