积分型总极值方法理论的发展及其并行算法
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摘要
最优化问题从产生到现在，众多的学者和数学家已经提出和总结了许多的最优化方法。但应该指出，目前大多数的算法求得的都是局部极小点，仅当问题具有某种凸性时，局部极小点才是全局极小点。一般来说求全局极小点是一个相当困难的任务，其中的难点又在于最优性条件的给定。
我们讨论如下形式的最优化问题：
设X是拓扑空间，S是X的非空子集，实值函数f：X→R。求全局最优值c~*=(?)f(x) (0.0.1)及其全局最优点集H~*={x∈S|f(x)=c~*} (0．0．2)
我们对讨论的问题作如下基本假设：
(A)：函数f是下半连续的，集合S是闭集，而且存在一个实数b使得集合H_b={x∈S|f(x)≤b}是非空紧集；
(R)：函数f在S上是上丰满的，即对于所有的c，集合{x∈S|f(x) (M)：(X，Ω，μ)是Q-测度空间，即对于所有的非空开集G，都满足μ(G)> 0，且对于所有的紧集K，都有μ(K)<∞。
我们推广水平均值和修正方差的概念：
m：R~1→R~1是给定的连续的严格递增函数。
假设(A)、(M)、(R)成立，c>c~*=(?)f(x)。函数f在其水平集H_c∩S上的m-均值：
函数v：R~1→R~1被称作v-函数，如果它满足下面的条件：
1．v(y)是非负函数而且v(y)=0当且仅当y=0。
2．v(-y)=v(y)。
3．当y≥0，函数v连续严格递增
函数f在其水平集H_c∩S上的v-方差：
最后给出最优化问题的最优性条件：
在(A)、(M)、(R)的假设成立之下，点x~*∈S是函数f在集合S上的全局最小点且c~*=f(x~*)是全局最小值当且仅当下面两个条件中的一个成立：
i)m-均值条件(m-Mean Value Condition)：M_1(f，c~*；S)=m(c~*)；
ii)v-方差条件(v-Variance Condition)：V_1(f，c~*，S)=0．
整个论文的结构如下：
第一章，我们简要回顾了全局最优化问题的提出和发展以及目前国内外关于这个问题的研究工作状况。给出了全局最优解的定义。介绍了郑权教授提出的求解全局最优解的积分总极值法。
第二章，我们引入了丰满集、丰满函数和Q-测度空间等概念。通过丰满分析，我们知道，在积分型算法中，目标函数f不一定要是连续的。
第三章，为了能够使积分总极值方法能够更有效地应用在处理最优化问题的情况，我们引进积分总极值中m-均值和v-方差等概念，发展了积分型总极值的最优性条件并给出了算法。
第四章，对求解有约束的最优化问题。我们借鉴罚函数的概念，利用不连续精确罚函数的概念对积分总极值方法进行了推广。给出了处理有约束最优化问题的积分总极值罚函数最优性条件及其算法。
第五章，我们给出了变测度积分总极值方法。讨论了在无限维空间中，如何用有限维子空间的全局最优值去逼近无限维空间中的全局最优值。引用了Q-测度收敛和变测度的概念，定义了在此概念下的m-均值和v-方差，并推导出变测度意义下的最优性条件。同时还给出了算法，并验证了其收敛性。
第六章，给出了变测度的罚函数积分型方法。对于无限维空间中的有约束的问题，我们引入不连续罚函数的方法，使有约束问题化为无约束问题求解。
第七章，大规模并行计算则成为研究科学与工程技术的一种崭新的手段和方式。在前面的研究中，我们发现积分水平集算法在应用并行计算得以实现时，对求解大规模问题具有独特的优势，为此在这一章中我们在这个方面进行一些研究和探讨。
The problem considered in this dissertation is how to characterize and find the global minimum value and global minimizers of an objective function in R~n and functional space.
Let X be a topological space and f : X→R a real-valued function. Consider the following minimization problem:c~* = (?) f(ar) (0.0.5)In general, minimizers of (0.0.5) may not exist. Under the assumption
(A): f is lower semi continuous, X is inf-compact.minimizers of (0.0.5) exist.
The problem of minimizing a function has been investigated since the seventeenth century with the concepts of derivative and Lagrangian multiplier. The gradient-based approach to optimization is the mainstream of that research. However , the requirement of differentiability restricts its application to many practical problems. Moreover, it can only be utilized to characterize and find a local solution of a general optimization problem.
In this dissertation, we apply the integral global optimization technique to investigate a minimization problem with discontinuous objective function. Integral global optimization technique is a very powerful yet flexible tool to treat various optimization problems.
We first recall basic concepts of robust sets, functions and the integral approach to global minimization. With the integral approach to global optimization , several new optimality conditions for global minimization are studied. Using m-mean value condition one can design algorithms for finding unconstrained global minimizers. With u-variance condition, one can set stopping criterion. A class of discontinuous penalty functions is proposed to solve constrained minimization problems with the integral approach to global optimization. Optimality conditions of a penalized minimization problem and a non sequential algorithm is proposed. New optimality conditions of the integral global minimization are applied to characterize global minimum in functional space as a sequence of approximating solutions in finite-dimensional spaces. A variable measure algorithm is used to find such solutions. For a constrained problem, a discontinuous penalty method is proposed to convert it to unconstrained ones. The integral algorithm can be implemented by a properly designed Monte Carlo method. Numerical examples are given to illustrate the effectiveness of the algorithm.
The integral global minimization algorithm is also implemented on parallel computer, the results show the great advantage of the integral approach.
我们讨论如下形式的最优化问题：
设X是拓扑空间，S是X的非空子集，实值函数f：X→R。求全局最优值c~*=(?)f(x) (0.0.1)及其全局最优点集H~*={x∈S|f(x)=c~*} (0．0．2)
我们对讨论的问题作如下基本假设：
(A)：函数f是下半连续的，集合S是闭集，而且存在一个实数b使得集合H_b={x∈S|f(x)≤b}是非空紧集；
(R)：函数f在S上是上丰满的，即对于所有的c，集合{x∈S|f(x)
我们推广水平均值和修正方差的概念：
m：R~1→R~1是给定的连续的严格递增函数。
假设(A)、(M)、(R)成立，c>c~*=(?)f(x)。函数f在其水平集H_c∩S上的m-均值：
函数v：R~1→R~1被称作v-函数，如果它满足下面的条件：
1．v(y)是非负函数而且v(y)=0当且仅当y=0。
2．v(-y)=v(y)。
3．当y≥0，函数v连续严格递增
函数f在其水平集H_c∩S上的v-方差：
最后给出最优化问题的最优性条件：
在(A)、(M)、(R)的假设成立之下，点x~*∈S是函数f在集合S上的全局最小点且c~*=f(x~*)是全局最小值当且仅当下面两个条件中的一个成立：
i)m-均值条件(m-Mean Value Condition)：M_1(f，c~*；S)=m(c~*)；
ii)v-方差条件(v-Variance Condition)：V_1(f，c~*，S)=0．
整个论文的结构如下：
第一章，我们简要回顾了全局最优化问题的提出和发展以及目前国内外关于这个问题的研究工作状况。给出了全局最优解的定义。介绍了郑权教授提出的求解全局最优解的积分总极值法。
第二章，我们引入了丰满集、丰满函数和Q-测度空间等概念。通过丰满分析，我们知道，在积分型算法中，目标函数f不一定要是连续的。
第三章，为了能够使积分总极值方法能够更有效地应用在处理最优化问题的情况，我们引进积分总极值中m-均值和v-方差等概念，发展了积分型总极值的最优性条件并给出了算法。
第四章，对求解有约束的最优化问题。我们借鉴罚函数的概念，利用不连续精确罚函数的概念对积分总极值方法进行了推广。给出了处理有约束最优化问题的积分总极值罚函数最优性条件及其算法。
第五章，我们给出了变测度积分总极值方法。讨论了在无限维空间中，如何用有限维子空间的全局最优值去逼近无限维空间中的全局最优值。引用了Q-测度收敛和变测度的概念，定义了在此概念下的m-均值和v-方差，并推导出变测度意义下的最优性条件。同时还给出了算法，并验证了其收敛性。
第六章，给出了变测度的罚函数积分型方法。对于无限维空间中的有约束的问题，我们引入不连续罚函数的方法，使有约束问题化为无约束问题求解。
第七章，大规模并行计算则成为研究科学与工程技术的一种崭新的手段和方式。在前面的研究中，我们发现积分水平集算法在应用并行计算得以实现时，对求解大规模问题具有独特的优势，为此在这一章中我们在这个方面进行一些研究和探讨。
The problem considered in this dissertation is how to characterize and find the global minimum value and global minimizers of an objective function in R~n and functional space.
Let X be a topological space and f : X→R a real-valued function. Consider the following minimization problem:c~* = (?) f(ar) (0.0.5)In general, minimizers of (0.0.5) may not exist. Under the assumption
(A): f is lower semi continuous, X is inf-compact.minimizers of (0.0.5) exist.
The problem of minimizing a function has been investigated since the seventeenth century with the concepts of derivative and Lagrangian multiplier. The gradient-based approach to optimization is the mainstream of that research. However , the requirement of differentiability restricts its application to many practical problems. Moreover, it can only be utilized to characterize and find a local solution of a general optimization problem.
In this dissertation, we apply the integral global optimization technique to investigate a minimization problem with discontinuous objective function. Integral global optimization technique is a very powerful yet flexible tool to treat various optimization problems.
We first recall basic concepts of robust sets, functions and the integral approach to global minimization. With the integral approach to global optimization , several new optimality conditions for global minimization are studied. Using m-mean value condition one can design algorithms for finding unconstrained global minimizers. With u-variance condition, one can set stopping criterion. A class of discontinuous penalty functions is proposed to solve constrained minimization problems with the integral approach to global optimization. Optimality conditions of a penalized minimization problem and a non sequential algorithm is proposed. New optimality conditions of the integral global minimization are applied to characterize global minimum in functional space as a sequence of approximating solutions in finite-dimensional spaces. A variable measure algorithm is used to find such solutions. For a constrained problem, a discontinuous penalty method is proposed to convert it to unconstrained ones. The integral algorithm can be implemented by a properly designed Monte Carlo method. Numerical examples are given to illustrate the effectiveness of the algorithm.
The integral global minimization algorithm is also implemented on parallel computer, the results show the great advantage of the integral approach.
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