Mathematical Analysis

Download Acta Numerica 1995 (Volume 4) by Arieh Iserles PDF

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By Arieh Iserles

Acta Numerica has proven itself because the leading discussion board for the presentation of definitive studies of numerical research issues. Highlights of this year's factor comprise articles on sequential quadratic programming, mesh adaption, unfastened boundary difficulties, and particle tools in continuum computations. The invited papers will enable researchers and graduate scholars alike to speedy seize the present traits and advancements during this box.

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Since by A4 a reduced Hessian at the solution is positive definite, it follows that if (xk,uk) is close enough to (x*,u*) then the reduced Hessian at (xk, uk) is positive definite. 27) it can be seen that the constraint equation of (ECQP) becomes Vh(xk)tYkpY = -h(xk), which by virtue of A2 can be solved to obtain pY = -[Vh(xk)tYk]-1h(xk). 28) The minimization problem (ECQP) is now an unconstrained problem in n — m variables given by minimize \pzxZkxBkZk pz + (Vf{xkf + pYxBk)Zk pz. Pz The matrix in this unconstrained problem is an approximation to the reduced Hessian of the Lagrangian at xk.

See also Bonnans et al. (1992)) The fact that the augmented Lagrangian merit function does not suffer from this problem has been shown by many authors. 6. SQP Trust Region Methods Trust region algorithms have become a part of the arsenal for solving unconstrained optimization problems, so it is natural to attempt to extend the ideas to solving constrained optimization problems and, in particular, to SQP methods. In this section an outline of the basic ideas of this approach will be provided. As the subject is still in a state of flux, no attempt will be made to give a comprehensive account of the algorithms that have been proposed.

Schnabel (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey. M. M. El-Alem (1991), 'A global convergence theory for the Celis-Dennis-Tapia trust region algorithm for constrained optimization', SIAM Journal on Numerical Analysis 28, 266-290. M. M. El-Alem (1992), 'A robust trust region algorithm with nonmonotonic penalty parameter scheme for constrained optimization', Department of Mathematical Sciences 92-30, Rice University.

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