A Filippov-type lemma for functions involving delays and its application to time-delayed optimal control problems

A Filippov-type lemma for functions involving delays is derived. This lemma is used to prove the existence of an optimal control for a class of nonlinear control processes with delays appearing in both state and control variables.The author wishes to express his gratitude to Dr. E. Noussair and Dr. K. L. Teo for their guidance in the preparation of this paper. Also, the author wishes to thank Professor L. Cesari for his valuable suggestion.     

Extensions of CGS algorithms: Generalized least-square solutions

The CGS (conjugate Gram-Schmidt) algorithms of Hestenes and Stiefel are formulated so as to obtain least-square solutions of a system of equationsg(x)=0 inn independent variables. Both the linear caseg(x)=Ax–h and the nonlinear case are discussed. In the linear case, a least-square solution is obtained in no more thann steps, and a method of obtaining the least-square solution of minimum length is given. In the nonlinear case, the CGS algorithm is combined with the Gauss-Newton process to minimize sums of squares of nonlinear functions. Results of numerical experiments with several versions of CGS on test functions indicate that the algorithms are effective.The author wishes to express appreciation and to acknowledge the ideas and help of Professor M. R. Hestenes which made this paper possible.     

Hybrid methods for large sparse nonlinear least squares

Hybrid methods are developed for improving the Gauss-Newton method in the case of large residual or ill-conditioned nonlinear least-square problems. These methods are used usually in a form suitable for dense problems. But some standard approaches are unsuitable, and some new possibilities appear in the sparse case. We propose efficient hybrid methods for various representations of the sparse problems. After describing the basic ideas that help deriving new hybrid methods, we are concerned with designing hybrid methods for sparse Jacobian and sparse Hessian representations of the least-square problems. The efficiency of hybrid methods is demonstrated by extensive numerical experiments.This work was supported by the Czech Republic Grant Agency, Grant 201/93/0129. The author is indebted to Jan Vlek for his comments on the first draft of this paper and to anonymous referees for many useful remarks.     

A note on the method of multipliers

Implementation of the penalty function method for constrained optimization poses numerical difficulties as the penalty parameter increases. To offset this problem, one often resorts to Newton's method. In this note, working in the context of the penalty function method, we establish an intimate connection between the second-order updating formulas which result from Newton's method on the primal problem and Newton's method on the dual problem.The author wishes to thank Professor R. A. Tapia for his careful review of this note. He has contributed significantly to its content through several crucial observations.     

Solving the nonlinear least square problem: Application of a general method

An algorithm for solving the general nonlinear least-square problem is developed. An estimate for the Hessian matrix is constructed as the sum of two matrices. The first matrix is the usual first-order estimate used by the Gauss method, while the second matrix is generated recursively using a rank-one formula. Test results indicate that the method is superior to the standard Gauss method and compares favorably with other methods, especially for problems with nonzero residuals at the solution.This work was supported by the US Air Force under Contract No. F04701-73-C-0074.The author expresses his appreciation to Dr. H. E. Pickett and Dr. J. L. Searcy for their continuing support in the theoretical and practical development of the algorithm. The recursive method for generating the estimate of the Hessian matrix was developed jointly with Drs. Pickett and Searcy and is included here with their permission. The author would also like to acknowledge the contribution made by the stimulating environment of an optimal control seminar held at The Aerospace Corporation since 1970. Principle members of the seminar have been H. E. Pickett, J. L. Searcy, R. W. Reid, and the author.     

A Globally Convergent Trust-Region Method for Large-Scale Symmetric Nonlinear Systems

This study presents a novel adaptive trust-region method for solving symmetric nonlinear systems of equations. The new method uses a derivative-free quasi-Newton formula in place of the exact Jacobian. The global convergence and local quadratic convergence of the new method are established without the nondegeneracy assumption of the exact Jacobian. Using the compact limited memory BFGS, we adapt a version of the new method for solving large-scale problems and develop the dogleg scheme for solving the associated trust-region subproblems. The sufficient decrease condition for the adapted dogleg scheme is established. While the efficiency of the present trust-region approach can be improved by using adaptive radius techniques, utilizing the compact limited memory BFGS adjusts this approach to handle large-scale symmetric nonlinear systems of equations. Preliminary numerical results for both medium- and large-scale problems are reported.     

A new view on some characterizations of simplices

The author wishes to thank Professor R. Schneider for helpful discussion, especially for referring his attention to the useful notionincluded double cone.     

Necessary conditions for optimality of singular controls in systems governed by partial differential equations

In this paper, we present a method to obtain necessary conditions for optimality of singular controls in systems governed by partial differential equations (distributed-parameter systems). The method is based on the one developed earlier by the author for singular control problems described by ordinary differential equations. As applications, we consider conditions for optimality of singular controls in a Darboux-Goursat system and in control systems that describe chemical processes.This research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.     

A one-sweep method in the solution of finite-element equations: An application to the vibration of structures in heavy fluids

A one-sweep method for the numerical solution of finite-element equations is presented. This procedure is especially efficient in computing time and storage when the solution is required at only a few nodes of the finite-element mesh. Furthermore, the method is particularly useful in dealing with problems on infinite or semi-infinite domains. Artificial boundaries must be introduced in such cases, and the one-sweep method affords an extremely efficient algorithm by which the dependence of the solution on the location of these boundaries can be assessed. An application of the method to the vibration of a half-submerged circular cylinder in a heavy fluid is presented.The second author wishes to express his thanks to Professor J. L. Sackman for reviewing this work.(deceased).     

The monoid of strong endomorphisms of a graph

We explicitly find all the idempotents in eachL(R)-class and all the inverses of each element of the strong endomorphism monoid of a graph. The number of these idempotents and inverses is also obtained. The author is deeply indebted to Professor Dr. T. E. Hall for his stimulating questions about this theme and much improvement made to an earlier version of this paper. The author would like to thank Professor Dr. U. Knauer and Dr. E. Wilkeit for helpful comments.     

Local convergence of the diagonalized method of multipliers

In this study, we consider a modification of the method of multipliers of Hestenes and Powell in which the iteration is diagonalized, that is, only a fixed finite number of iterations of Newton's method are taken in the primal minimization stage. Conditions are obtained for quadratic convergence of the standard method, and it is shown that a diagonalization where two Newton steps are taken preserves the quadratic convergence for all multipler update formulas satisfying these conditions.This work constitutes part of the author's doctoral dissertation in the Department of Mathematical Sciences, Rice University, under the direction of Professor R. A. Tapia and was supported in part by ERDA Contract No. E-(40-1)-5046.The author would like to thank Professor Richard Tapia for his comments, suggestions, and discussions on this material.     

On the decomposition map of Grothendieck groups

This article is a part of the doctoral dissertation of the author at the University of Minnesota under the guidance of Professor Morton E. Harris. The author wishes to express his gratitude to Professor Harris for his advice and suggestions     

Convergence of the BFGS-SQP Method for Degenerate Problems

In this paper, we propose a BFGS (Broyden–Fletcher–Goldfarb–Shanno)-SQP (sequential quadratic programming) method for nonlinear inequality constrained optimization. At each step, the method generates a direction by solving a quadratic programming subproblem. A good feature of this subproblem is that it is always consistent. Moreover, we propose a practical update formula for the quasi-Newton matrix. Under mild conditions, we prove the global and superlinear convergence of the method. We also present some numerical results.     

The connection between normalizable and spectral operators

A necessary and sufficient condition for an operator to be normalizable is given in terms of Dunford’s spectral theory. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his indebtedness to Professor S. R. Foguel for his guidance and kind encouragement.     

Second-order characterizations of pseudoconvex quadratic functions

The paper deals with strictly pseudoconvex quadratic functions on open convex sets. It presents second-order conditions in terms of an extended Hessian and in terms of bordered determinants. All of the conditions are both necessary and sufficient.The author wishes to thank Professor M. Avriel, visiting Stanford University, for helpful discussions. He is also grateful to Professor R. W. Cottle, Stanford University, for his suggestions.     

Benders' decomposition for the constrainedl 1-problem

A constrainedl ₁-problem, involving linear functions only, is considered, and the application of the Benders decomposition method to the solution of the same is discussed. This approach, in principle, seems to be promising and is also applicable to the unconstrained case. Certain small illustrative examples are also presented.This work was done while the first author was visiting the Numerical Optimisation Centre at the Hatfield Polytechnic under the auspices of the British Council and he wishes to acknowledge the excellent research facilities provided by the Polytechnic during the period of his stay. Thanks are also due to Dr. M. C. Bartholomew-Biggs for his valuable comments at various stages of the preparation of this report.     

Differential games without pure strategy saddle-point solutions

In some two-player, zero-sum differential games, pure strategy saddle-point solutions do not exist. For such games, the concept of a minmax strategy is examined, and sufficient conditions for a control to be a minmax control are presented. Both the open-loop and the closed-loop cases are considered.The research was partially supported by ONR under Contract No. N00014-69-A-0200-12. An earlier version of this paper was presented at the Eleventh Annual Allerton Conference on Circuit and System Theory, Monticello, Illinois, 1973.The author wishes to acknowledge his many valuable discussions of this problem with Professor G. Leitmann and also to thank one of the reviewers for his suggestions for simplifying the proof of Theorem 2.1.     

Necessary conditions for optimality of singular controls

The present paper is concerned with the study of controls which are singular in the sense of the maximum principle. We obtain necessary conditions for optimality of singular controls in systems governed by ordinary differential equations. A useful feature of the method considered here is that it can be applied to optimal control problems with distributed parameters.this research was supported in part by the National Science Foundation under Grant No. NSF-MCS-80-02337 at the University of Michigan.The author wishes to express his deep gratitude to Professor L. Cesari for his valuable guidance and constant encouragement during the preparation of this paper.     

An approximate solution of linear multicriteria control problems through the multicriteria simplex method

This paper deals with multicriteria, linear, continuous optimal control problems through the application of the multicriteria simplex method. Since the system is linear, the state variables at a given time can be expressed in terms of definite integrals. Upon using standard numerical integration formulas, the definite integrals are approximately expressed by means of weighted sums of the integrands. Then, after introducing some suitable auxiliary variables, approximate linear multicriteria programming problems are formulated. Illustrative numerical examples are provided to indicate the efficiency of the proposed method.The author is indebted to Professor Y. Sawaragi of Kyoto University for his constant encouragement. The author also wishes to thank T. Suwa for his cooperation in preparing the programming for the digital computer.     

Continuity of the feasible solution sets of probabilistic constrained programs

In this paper, we show the continuity of the feasibility set with respect to the reliability levels and with respect to the distribution of the random elements of a stochastic program with probabilistic constraints. Continuity is then used to obtain stability results for this type of stochastic program. An easy criterion is given for checking the conditions which guarantee the continuity of the feasibility set.The author wishes to thank Professor R. Wets for his comments.