A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS Ⅱ. LOCAL CONVERGENCE RATE AND NUMERICAL TESTS

In this paper we prove that a class of trust region methods presented in part Ⅰ is superlinearly convergent. Numerical tests are reported thereafter. Results by solving a set of typical problems selected from literatures have demonstrated that our algorithm is effective.     

A parallel variable-metric dynamic programming algorithm

This paper presents a potentially parallel iterative algorithm for the solution of the unconstrainedN-stage decision problem of dynamic programming. The basis of the algorithm is the use of variable-metric minimization techniques to develop a quadratic approximation to the cost function at each stage. The algorithm is applied to various problems, and comparisons with other algorithms are made.This research forms part of the author's PhD program, and is supported by the Department of Scientific and Industrial Research of the New Zealand Government. The author is indebted to Dr. B. A. Murtagh, PhD supervisor, for his encouragement and support during the preparation of this paper.     

Properties of updating methods for the multipliers in augmented Lagrangians

The convergence properties of different updating methods for the multipliers in augmented Lagrangians are considered. It is assumed that the updating of the multipliers takes place after each line search of a quasi-Newton method. Two of the updating methods are shown to be linearly convergent locally, while a third method has superlinear convergence locally. Modifications of the algorithms to ensure global convergence are considered. The results of a computational comparison with other methods are presented.This work was supported by the Swedish Institute of Applied Mathematics.     

A geometric method in nonlinear programming

A differential geometric approach to the constrained function maximization problem is presented. The continuous analogue of the Newton-Raphson method due to Branin for solving a system of nonlinear equations is extended to the case where the system is under-determined. The method is combined with the continuous analogue of the gradient-projection method to obtain a constrained maximization method with enforced constraint restoration. Detailed analysis of the global behavior of both methods is provided. It is shown that the conjugate-gradient algorithm can take advantage of the sparse structure of the problem in the computation of a vector field, which constitutes the main computational task in the methods.This is part of a paper issued as Stanford University, Computer Science Department Report No. STAN-CS-77-643 (Ref. 45), which was presented at the Gatlinburg VII Conference, Asilomar, California, 1977. This work was supported in part by NSF Grant No. NAT BUR OF ECON RES/PO No. 4369 and by Department of Energy Contract No. EY-76-C-02-0016.The main part of this work was presented at the Japan-France Seminar on Functional Analysis and Numerical Analysis, Tokyo, Japan, 1976. The paper was prepared in part while the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 1976–77, and was completed while he was a visitor at the Computer Science Department, Stanford University, Stanford, California, 1977. He acknowledges the hospitality and stimulating environment provided by Professor G. H. Golub, Stanford University, and Professors N. J. Rose and C. D. Meyer, North Carolina State University.     

Cofolga: a genetic algorithm for finding the common folding of two RNAs

In order to predict non-coding RNA genes and functions on the basis of genome sequences, accurate secondary structure prediction is useful. Although single-sequence folding programs such as mfold have been successful, it is of great importance to develop a novel approach for further improvement of the prediction performance. In the present paper, a secondary structure prediction method based on genetic algorithm, Cofolga, is proposed. The program developed performs folding and alignment of two homologous RNAs simultaneously. Cofolga was tested with a dataset composed of 13 tRNAs, seven 5S rRNAs, five RNase P RNAs, and five SRP RNAs; as a result, it turned out that the average prediction accuracies for the tRNAs, 5S rRNAs, RNase P RNAs, and SRP RNAs obtained by Cofolga with an optimal weight factor and default parameters were 83.6, 81.8, 73.5, and 67.7%, respectively. These results were superior to those obtained by a single-sequence folding based on free-energy minimization in which corresponding average prediction accuracies were 52.4, 47.4, 57.7, and 52.3%, respectively. Cofolga has a post-processing in which a single-sequence folding is performed after fixation of a predicted common structure; this post-processing enables Cofolga to predict a structure that is present in one of two RNAs alone. The executable files of Cofolga (for Windows/Unix/Mac) can be obtained by an e-mail request.     

Efficient and stereoselective synthesis of novel cis-4-substituted proline analogues

A series of novel cis-4-substituted proline analogues were designed and synthesized. Highly stereoselective alkylations at the γ-position of glutamic ester 2 were achieved, followed by reduction, mesylation, and cyclization to afford the title compounds 1 in good yields and high diastereoselectivity.     

Convex Separable Minimization Subject to Bounded Variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.     

On the Geometry of Inner Approximations to the Set of Null Controllability for Constrained Linear Systems

The problem of obtaining inner approximations to the set of null controllability for linear systems is considered, and the geometry of certain sets arising in the approximation of using linear feedback is explored. A two-dimensional example is worked in detail in order to delineate the limitations to the use of linear feedback.     

On Convergence of Minimization Methods: Attraction, Repulsion, and Selection

In this paper, we revisit the convergence properties of the iterationprocess x^k+1=x^k–(x^k)B(x^k)^–1f(x^k)for minimizing a function f(x). After reviewing some classic results andintroducing the notion of strong attraction, we give necessary andsufficient conditions for a stationary point of f(x) to be a point of strongattraction for the iteration process. Not only this result gives a newalgorithmic interpretation to the classic Ostrowski theorem, but alsoprovides insight into the interesting phenomenon called selectiveminimization. We present also illustrative numerical examples for nonlinearleast squares problems.