The Lack of Positive Definiteness in the Hessian in Constrained Optimization

The use of the DFP or the BFGS secant updates requires the Hessian at the solution to be positive-definite. The second order sufficiency conditions insure the positive definiteness only in a sub-space of R(exp n). Conditions are given so the author can safely update with either update. The author proposes a new class of algorithms that generate a sequence converging 2-step q-superlinearly. He also proposes two specific algorithms. The first one converges q-superlinearly if the Hessian is positive-definite in R(exp n), and it converges 2-step q-superlinearly if the Hessian is positive-definite only in a subspace. The second one generates a sequence converging 1-step q-superlinearly. While the former costs one extra gradient evaluation, the latter costs one extra gradient evaluation and one extra function evaluation on the constraints.