On Secant Updates for Use in General Constrained Optimization

Abstract

Abstract.In this paper we present two new classes of successive quadratic programming (SQP) secant methods for the equality-constrained optimization problem. One class of methods uses the SQP augmented Lagrangian formulation, while the other class uses the SQP Lagrangian formulation.We demonstrate, under the standard assumptions, that in both cases the BFGS and DFP versions of the algorithm are locally q-superlinearly convergent. To our knowledge this is the first time that either local or (j-superlinear convergence has been established for an SQP Lagrangian secant method which uses either the BFGS or DFP updating philosophy and assumes no more than the standard assumptions. Since the standard assumptions do not require positive definiteness of the Hessian of the Lagrangian at the solution, it is no surprise that our BFGS and DFP updates possess the hereditary positive definiteness property only on a proper subspace.

A convergent secant method for constrained optimization

“…If t*k]lPkgkll < Ak, then ~ = --t*kPkgk is a feasible solution of problem (Sk), and hence t* ~k < 0k (~) = Ck (k) 1 IIPkgkll ~…”

“…For the cla~s of constrained nonlinear optimization problem min f(x) (1) s.t. c(x) ~= (Cl(X),...,Cm(X)) =0…”

“…where f,c~ : ~n ___+ S}~I (i = 1,...,m) are twice continuously differentiable functions and m < n, secant method [1]' [2] is one of the commonly used methods. In [3] Fontecilla proposed a new family of secant methods which can be summarized as follows.…”

“…and call it the e-stationary point set of problem(1). Obviously, if e = 0 then f2~ is just the set of Karush-Kuhn-Tucker points.…”

Recursive quadratic programming methods based on the augmented lagrangian

“…Powell (1985) has noted that his (1977) strategy may in fact produce badly conditioned matrices and the same may be true of (28)-(31). Other authors, including Pantoja (1984), Tapia (1984) and BartholomewBiggs (1985), have suggested alternative updating schemes which appear promising and which may, in due course, supersede both the techniques mentioned above.…”