A Computationally Compact Sufficient Condition for Constrained Minima

This paper derives a second-order sufficient condition in differential form for a local minimum of the nonconvex nonlinear programming problem under the requirement of twice continuous differentiability. The condition is complementary to the differential form of the Kuhn-Tucker necessary conditions and involves the positive definiteness of a matrix of constrained second derivatives of rank and order equal to the number of variables having vanishing first constrained derivatives. By using linear information, the condition avoids consideration of the entire second-order portion of the Taylor expansion of the Lagrangian. An example illustrating the test is included.