Advances in Nonlinear Programming: Proceedings of the 96 by M. J. D. Powell (auth.), Ya-xiang Yuan (eds.)

By M. J. D. Powell (auth.), Ya-xiang Yuan (eds.)

About 60 scientists and scholars attended the ninety six' overseas convention on Nonlinear Programming, which was once held September 2-5 at Institute of Compu­ tational arithmetic and Scientific/Engineering Computing (ICMSEC), Chi­ nese Academy of Sciences, Beijing, China. 25 members have been from outdoors China and 35 from China. The convention was once to have a good time the 60's birthday of Professor M.J.D. Powell (Fellow of Royal Society, college of Cambridge) for his many contributions to nonlinear optimization. On behalf of the chinese language Academy of Sciences, vp Professor Zhi­ hong Xu attended the hole rite of the convention to precise his hot welcome to all of the contributors. After the hole rite, Professor M.J.D. Powell gave the keynote lecture "The use of band matrices for moment by-product approximations in belief quarter methods". thirteen different invited lectures on contemporary advances of nonlinear programming got in the course of the 4 day assembly: "Primal-dual tools for nonconvex optimization" through M. H. Wright (SIAM President, Bell Labs), "Interior aspect trajectories in semidefinite programming" by means of D. Goldfarb (Columbia college, Editor-in-Chief for sequence A of Mathe­ matical Programming), "An method of spinoff unfastened optimization" through A.

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Primal-dual interior methods can be interpreted in several ways; see, for example, [8] and the recent book [28] by S. Wright on primal-dual methods A PRIMAL-DUAL INTERIOR METHOD FOR NLP 35 for linear programming. 1) j=l where Il is a positive scalar. Interior methods for constrained optimization, many based on this barrier function, have been the subject of intense research since their revival in 1984; see [17; 23; 28]. 2) where e = (1,1, ... , l)T, all vector and matrix functions are evaluated at x, and D(x) denotes the diagonal matrix diag(dj(x)).

S+1 that this expression is at most ~(2£-s-I)2/(s+I). (n-s-2) when n-s is even and £=p+l=~(n-s-l) when n-s is odd. 4) is no greater than the number n-s-2+ (n-2s-2)2 + (n-2s-4)2 n2-2n+2 - . 5) cannot be achieved. Thus we deduce the upper bound t(n-l)2/(s+l) on the total number of Givens rotations that occur. 3) and by setting £=p= t(n-s-2), we find that the number ofrotations is bounded below by t(n-l) /(s+l) -t(s+I). 2), we may wish to apply further Givens rotations that reduce the band-width to 2s+ 1.

8). 7) is a poor estimate of the distance to the boundary. If dt(x+a(J) LlX) ::; 0 for any i, a steplength a(1+ 1 ) that produces a strictly feasible point is found using a one-dimensional secantlike iteration. After a strictly feasible point is found, subsequent steplengths are modified using a simple Armijo-like rule; if further infeasibility is encountered, the steplength is modified accordingly. Following a successful line search, x and Z retain their final values from the line search. 10) A PRIMAL-DUAL INTERIOR METHOD FOR NLP 6 47 CHOOSING THE BARRIER PARAMETER An important issue in any primal-dual method is the choice of barrier parameter.

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