Algorithms for Constrained Minimization of Smooth Nonlinear by A.G. Buckley, J-.L. Goffin

By A.G. Buckley, J-.L. Goffin

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II", Journal of, Optimization Theory and Applications 8 (1971) 336-349. 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.

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. t. of solving a sequence of QP defined successively; since such problems can be efficiently solved in a finite number of iterations, this approach presents a clear advantage over primal t By a solution of the QP we mean the couple consisting of a minimizing element and of the corresponding vector of Lagrange multipliers associated with the constraints.

Clearly 0'(X, C) = 0(X, J~(X, C), C) -~-(89 c)T/'~p(X, C). Q. Mayne and E. Polak/ Superlinearly convergent algorithm 51 A convergent algorithm, employing/~(x, H ) as a search direction if conditions (14) are satisfied and `6(x, c) otherwise, can be constructed. However a superlinearly convergent algorithm requires, inter alia, that, near a solution, the search direction/~(x, H ) is always employed and that the step length is unity. The test conditions (14) are such that the former requirement is satisfied.

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