Piecewise line-search techniques for constrained minimization by quasi-Newton algorithms

Defining a consistent technique for maintaining the positive definiteness of the matrices generated by quasi-Newton methods for equality constrained minimization remains a difficult open problem. In this paper, we review and discuss the results obtained with a new technique based on an extension of the Wolfe line-search used in unconstrained optimization. The idea is to follow a piecewise linear path approximating a smooth curve, along which some reduced curvature conditions can be satisfied. This technique can be used for the quasi-Newton versions of the reduced Hessian method and for the SQP algorithm. A new argument based on geometrical considerations is also presented. It shows that in reduced quasi-Newton methods the appropriateness of the vectors used to update the matrices may depend on the type of bases representing the tangent space to the constraint manifold. In particular, it is argued that for orthonormal bases, it is better using the projection of the change in the gradient of the Lagrangian rather than the change in the reduced gradient. Finally, a strong q-superlinear convergence theorem for the reduced quasiNewton algorithm is discussed. It shows that if the sequence of iterates converges to a strong solution, it converges one step q-superlinearly. Conditions to achieve this speed of convergence are mild; in particular, no assumptions are made on the generated matrices. This result is, to our knowledge, the first extension to constrained optimization of a similar result proved by Powell in 1976 for unconstrained problems.