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Second order sufficient conditions for optimizing with equality constraints. (English) Zbl 0767.49019

The two basic approaches for investigating the extreme values of a function subject to equality constraints are the Jacobean approach, involving the constrained gradient, and the Lagrangean approach, which considers the unconstrained extrema of the Lagrangean function. The equivalence of the first order necessary conditions obtained using the two methods is well known. In this paper, second order sufficient conditions are obtained using both approaches. The Jacobean approach involves the Constrained Hessian Matrix of the given function while the Lagrangean approach involves the so called Bordered Hessian Matrix of the Lagrangean function. The equivalence of the two sets of conditions is proved by showing that both conditions are equivalent to a third (apparently new) second order sufficient condition involving the Reduced Hessian Matrix of the Lagrangean function. It is also proved that \(H_ C(f)=H_ R(L)\). This new sufficiency condition has computational advantages over the two. All three conditions are of the most general type, that is, they are inconclusive only if higher order tests are necessary.

MSC:

49K99 Optimality conditions
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