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Additional info for Algorithms for Constrained Minimization of Smooth Nonlinear Functions (Mathematical programming study)
We observe D. Gabay/ Reduced quasi-Newton methods 24 that V2~,(A) is negative definite (since Ax~) has full rank m); hence ~r is locally concave for any r -> r* and we can define the dual problem Max XEB(A*,~ 2) qZr(h). 19) 3. 1. 1) where Ak denotes theJacobian matrix of the map c at x k. 4) and its convergence analysis is well known. To avoid computations of second order derivatives, Mk is defined as an approximation to the Hessian L(x k, A k) based upon accumulated first order information. 4) has been approximated.
Using the dual form of QP~(x) (see (18)) the following result can be established : Proposition 3. (x). Then x E Dc if and only if x E D. z Hence satisfying $(x) = 0. Q. Mayne and E.
J. Mort, "Quasi-Newton methods, motivation and theory", SIAM Review 19 (1977) 46--89. E. Byrd, "Local convergence of the diagonalized method of multipliers", Journal off Optimization Theory and Applications 26 (1978) 485-500. Mathematical Programming Study 16 (1982) 45-61 North-Holland Publishing Company A SUPERLINEARLY CONVERGENT ALGORITHM FOR CONSTRAINED OPTIMIZATION PROBLEMS* D. Q. MAYNE Department of Electrical Engineering, Imperial College of Science and Technology, London SW7 2BZ, Great Britain E.
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