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

ISBN-10: 0444863907

ISBN-13: 9780444863904

ISBN-10: 3642008127

ISBN-13: 9783642008122

ISBN-10: 3642008135

ISBN-13: 9783642008139

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**Extra resources for Algorithms for Constrained Minimization of Smooth Nonlinear Functions (Mathematical programming study)**

**Example text**

26) with p k = Hkgk. 28) D. e. after a finite number of iterations since {xk}~ x* satisfying the first-order optimality conditions). O; let x k§ = x k + 2-~d k + 2-2~e k. 6. Conclusions In this paper we have presented a superlinearly and globally convergent algorithm for the equality constrained nonlinear programming problem. This method can be viewed as a particularly efficient approximation of the quasi-Newton method along geodesics presented in [1] where the feasibility of the successive iterates is not enforced.

8). D. g. [17]) valid only when M is non-singular. It includes as a special case a formula used by Powell [18], Tapia [8] valid only when M is non-singular. 8). 3. 12) obviously the reduced gradient depends upon the change of coordinate defined by Sx. 13) where the subscripts and superscripts k indicate that the respective matrices and vectors are evaluated at the current iterate x k. 13) indicates that the quasi-Newton direction d k is a combination of a direction in the tangent space Tk to the submanifold Ck = c-I(c k) at x k and a direction pointing t o w a r d the constraint m a n i f o l d C = c-I(0) (since - A~ c k can be viewed as the first step of a Newton's method starting from x k to solve the system of equations c ( x ) = 0).

D. agrangian functions", Mathematical Programming 14 (1978) 224-248. D. A. , Numerical analysis (Springer-Verlag, Heidelberg 1978) pp. 144-157. [20] A. E. V. Levy, "Use of the augmented penalty function in mathematical programming problems, Part. 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.

### Algorithms for Constrained Minimization of Smooth Nonlinear Functions (Mathematical programming study) by A.G. Buckley, J-.L. Goffin

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