多目标规划的整体解

本文在较弱的广义凸性假定下讨论多目标规划的几种整体有效性．给出的定理统一了目前已有的一些关于多目标规划局部解为整体解的充分条件．     

关于一类连续最优控制问题的一种可实现的离散方法

本文对具有状态终端约束、控制受限的非线性连续最优控制问题给出一种新的可实现的离散方法，此方法通过求解非线最小二乘问题避免这类问题离散后出现的不可行现象，文中给出这种做法的理论证明和实现方案。     

Extended well-posedness of optimization problems

The well-posedness concept introduced in Ref. 1 for global optimization problems with a unique solution is generalized here to problems with many minimizers, under the name of extended well-posedness. It is shown that this new property can be characterized by metric criteria, which parallel to some extent those known about generalized Tikhonov well-posedness.This work was partially supported by MURST, Fondi 40%, Rome, Italy.     

Test examples for nonlinear programming codes

The increasing importance of nonlinear programming software requires an enlarged set of test examples. The purpose of this note is to point out how an interested mathematical programmer could obtain computer programs of more than 120 constrained nonlinear programming problems which have been used in the past to test and compare optimization codes.     

Optimal control of systems with multiple criteria when disturbances are present

Optimal control problems with a vector performance index and uncertainty in the state equations are investigated. Nature chooses the uncertainty, subject to magnitude bounds. For these problems, a definition of optimality is presented. This definition reduces to that of a minimax control in the case of a scalar cost and to Pareto optimality when there is no uncertainty or disturbance present. Sufficient conditions for a control to satisfy this definition of optimality are derived. These conditions are in terms of a related two-player zero-sum differential game and suggest a technique for determining the optimal control. The results are illustrated with an example.This research was supported by AFOSR under Grant No. 76-2923.     

Deterministic optimal control,given only a partial observation of the initial state

The deterministic linear-system, quadratic-cost optimal control problem is considered when the only state information available is a partial linear observation of the initial statex ₀. Thus, it is only known that the initial condition belongs to a particular linear variety. A control function is found which is optimal, in the sense (roughly) that (i) it can be computed using available information aboutx ₀ and (ii) no other control function which can be found using that information gives lower cost than it does for every initial condition that could have given rise to the information. The optimal control can be found easily from the conventional Riccati equation of optimal control. Applications are considered in the presence of unknown exponential disturbances and to the case with a sequence of partial state observations.     

Computational experience with the Chow—Yorke algorithm

The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent. It is efficient and accurate for fixed point problems. L.T. Watson, T.Y. Li, and C.Y. Wang have adapted the method for zero finding problems, the nonlinear complementarity problem, and nonlinear two-point boundary value problems. Here theoretical justification is given for applying the method to some mathematical programming problems, and computational results are presented.This work was partially supported by NSF Grant MCS 7821337.     

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.