The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints

Optimality criteria form the foundations of mathematical programming both theoretically and computationally. In general, these criteria can be classified as either necessary or sufficient. Of course, one would like to have the same criterion be both necessary and sufficient. However, this occurs only under somewhat ideal conditions which are rarely satisfied in practice. In the absence of convexity, one is never assured, in general, of the sufficiency of any such optimality criterion. We are then left with only the necessary optimality criterion to face the vast number of mathematical programming problems which are not convex. The best-known necessary optimality criterion for a mathematical programming problem is the Kuhn-Tucker criterion [l]. However, the Fritz-John criterion [2], which predates the Kuhn-Tucker criterion by about three years, is in a sense more general. In order for the Kuhn-Tucker criterion to hold, one must impose a constraint-qualification on the constraints of the problem. On the other hand, no such qualification need be imposed on the constraints in order that the Fritz John criterion hold. Moreover, the Fritz John criterion itself can be used to derive a form of the constraint qualification for the Kuhn-Tucker criterion. Originally, Fritz John derived his conditions for the case of inequality constraints alone. If equality constraints are present and they are merely replaced by two inequality constraints, then the Fritz John original conditions become useless because every feasible point satisfies them. The new generalization of Fritz John’s conditions derived in this work treats equalities as equalities and does not convert them to inequalities. This makes it possible to handle equalities and inequalities together. Another contribution of the present work is a constraint qualification for equalities and inequalities together. Previous constraint qualifications treated equalities and inequalities separately, but not together. Since many