Global convergence

J.H. Wilkinson, Global convergence of tridiagonal QR algorithm with origin shift. Algorithms Applic., 1 (1968)409 20. 

A feasible path optimization approach can be very expensive because an iterative calculation is required to solve the undetermined model. A more efficient way is to use an unfeasible path approach to solve the NLP problem however, many of these large-scale NLP methods are only efficient in solving problems with few degrees of freedom. A decoupled SQP method was proposed by Tjoa and Biegler (1991) that is based on a globally convergent SQP method. 

This guarantees the boundedness of all error signals. By invoking Barbalat s lemma [3], it can be recognized that V -> 0, which implies the global convergence to 0 of both x and e, while the parameters estimation errors 60 and 6C are only guaranteed to be uniformly bounded. ... 

Some process models have more than one feasible solution. Most numerical methods have local convergence, so the solution obtained is dependent upon the initial guess for the solution before the first iteration. There is an ongoing effort to develop techniques that have global convergence or to find all solutions to multisolution problems. 

Very recently Sargent 88 has shown that, although the Lagrangian function is not an exact penalty function, it can indeed be used in a descent test to force global convergence. Moreover the test is satisfied without step-reduction in the final stages, so the convergence is superlinear. Thus the penalty parameter and its associated problems are eliminated. 

The last two formulas are generally preferred in practice, though the first has better theoretical global convergence properties. In fact, very recent research has focused on combining these practical and theoretical properties for construction of more efficient schemes.77 78 The simple modification of... 

Unfortunately, there is a disparity between this theoretical convergence result and the practical behavior of the method in general. Thus, modifications of the classic Newton iteration are essential for guaranteeing global convergence, with quadratic convergence rate near the solution. 

M. Al-Baali, Inst. Math. Appl. J. Numer. Anal., 5,121 (1985). Descent Property and Global Convergence of the Fletcher-Reeves Method with Inexact Linear Search. 

R. H. Byrd, J. Nocedal, and Y. Yuan, SIAM ]. Numer. Anal., 24, 1171 (1987). Global Convergence of a Class of Quasi-Newton Methods on Convex Problems. 

A. R. Conn, N. I. M. Gould, and Ph. L. Toint, SIAM J. Numer. Anal., 25, 433 (1988). Global Convergence of a Class of Trust Region Algorithms for Optimization with Simple Bounds. 

S. C. Eisenstat and H. F. Walker, preprint, 1992. Globally Convergent Inexact Newton Methods. 

Reich P. B., Walters M. B., and Ellsworth D. S. (1997) From tropics to tundra global convergence in plant functioning. Proc. Natl. Acad. Sci. USA. 94, 13730-13734. 

For the optimization of Hartree-Fock wave functions, it is usually sufficient to apply the SCF scheme described in Sec. 3.1. By contrast, the optimization of MCSCF wave functions requires more advanced methods (e.g., the quasi-Newton method or some globally convergent modification of Newton s method, which involves, directly or indirectly, the calculation of the electronic Hessian as well as the electronic gradient at each iteration) [45]. 

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