若干最优化方法的收敛性分析及应用研究
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摘要
最优化方法是运筹学的一个重要的组成部分，在自然科学、社会科学、生产实际、工程设计和现代化管理中有着重要的实用价值。
本文对近年来备受关注的几类最优化方法(共轭梯度算法、进化算法和目标规划法)的理论性质及应用进行了研究，主要研究成果如下：
1．给出了一个混合的HS-FR共轭梯度算法，在无充分下降条件下，得到了关于HS-FR算法的两个收敛性定理。讨论了一类共轭梯度算法的收敛性，推广了1992年Gilbert和Nocedal的收敛性结果。
2．讨论了进化策略的收敛性。对于有界闭集上的连续函数，证明了进化策略以概率为1收敛(几乎处处收敛)于优化问题的全局极小点。针对实值连续函数优化问题，提出了一种混合的EP-ES进化算法，典型数值实验表明，所提出的算法是可行的、有效的。
3．针对某省的公路交通的实际情况，建立了一个干线公路网等级结构优化模型，同时给出了算法，计算结果已经应用于该省“十五”时期公路网的实际规划当中。
Optimization is an important component of operations research, which has been applied to practical problems hi many scientific and engineering disciplines. This paper is devoted to some numerical optimization methods and optimization models for solving practical problems in real world. The methods we concern with are the conjugate gradient algorithms, evolutionary algorithms and goal programming. The main work of the paper can be summarized as follows:
1. A mixed HS-FR conjugate gradient algorithm is proposed. Two convergence theorems without the sufficient descent condition for the mixed HS-FR algorithm are given. The convergence theorem of a class of conjugate gradient algorithms is proven, which extend the main convergence theorem in Gilbert and Noceda (1992).
2. Evolution strategy procedures for real-valued function optimization for the purpose of analyzing its asymptotic convergence properties are described. Two convergence theorems, which show that under suitable conditions evolution strategy asymptotically converges to a global minimum point with probability one, are given. A mixed EP-ES evolutionary algorithm for real-valued function optimization is proposed. Numerical results illustrate that the proposed algorithm is efficient.
3. According to the demands of making the medium- and long-term highway networks planning in certain province, a goal-programming model for arterial highway network grade structure optimization is established, and an algorithm with example is given.
本文对近年来备受关注的几类最优化方法(共轭梯度算法、进化算法和目标规划法)的理论性质及应用进行了研究，主要研究成果如下：
1．给出了一个混合的HS-FR共轭梯度算法，在无充分下降条件下，得到了关于HS-FR算法的两个收敛性定理。讨论了一类共轭梯度算法的收敛性，推广了1992年Gilbert和Nocedal的收敛性结果。
2．讨论了进化策略的收敛性。对于有界闭集上的连续函数，证明了进化策略以概率为1收敛(几乎处处收敛)于优化问题的全局极小点。针对实值连续函数优化问题，提出了一种混合的EP-ES进化算法，典型数值实验表明，所提出的算法是可行的、有效的。
3．针对某省的公路交通的实际情况，建立了一个干线公路网等级结构优化模型，同时给出了算法，计算结果已经应用于该省“十五”时期公路网的实际规划当中。
Optimization is an important component of operations research, which has been applied to practical problems hi many scientific and engineering disciplines. This paper is devoted to some numerical optimization methods and optimization models for solving practical problems in real world. The methods we concern with are the conjugate gradient algorithms, evolutionary algorithms and goal programming. The main work of the paper can be summarized as follows:
1. A mixed HS-FR conjugate gradient algorithm is proposed. Two convergence theorems without the sufficient descent condition for the mixed HS-FR algorithm are given. The convergence theorem of a class of conjugate gradient algorithms is proven, which extend the main convergence theorem in Gilbert and Noceda (1992).
2. Evolution strategy procedures for real-valued function optimization for the purpose of analyzing its asymptotic convergence properties are described. Two convergence theorems, which show that under suitable conditions evolution strategy asymptotically converges to a global minimum point with probability one, are given. A mixed EP-ES evolutionary algorithm for real-valued function optimization is proposed. Numerical results illustrate that the proposed algorithm is efficient.
3. According to the demands of making the medium- and long-term highway networks planning in certain province, a goal-programming model for arterial highway network grade structure optimization is established, and an algorithm with example is given.
引文
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[14]. E.Polak, G.Ribi(?)re. Note sur la convergence de direction conjug-
ees. Rev. Francaise Informal Recherche Opertionelle, 3e Annee, 16(1969) , 35-43．
[15] . B.T.Polyak The conjugate gradient method in extreme problems. USSR Comp Math and Math Phys. 9(1969) , 94-112．
[16] . Y.H.Dai, Y.X.Yuan. A nonlinear conjugate gradient with a strong global convergence property. SIAM J. Optimization, 10(2000) , 177-182．
[17] . M.R.Hestenes. Conjugate direction methods in optimization. New York: Heidelberg Berlin, Springer-Verlag, 1980．
[18] . J.Nocedal. Theory of algorithm for unconstrained optimization. Acta Numerica, (1991) , 199-242．
[19] . J.C.Gilbert, J.Nocedal. Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optimization, 2:1(1992) , 21-42．
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