Optimization, as a powerful solution approach, finds wide applications in extremely broad spectra of fields and human activities including engineering, science, finance, etc. The existence of multiple local minima of a general nonconvex objective function, f, makes global optimization (GO) a great challenge. The main purpose of this research is to develop some high performance methods for solving global optimization problems (GOP) over continuous or discrete variables. This research makes several contributions to advance the state-of-the-art of GO. These contributions include both theoretical and numerical aspects as stated below.; In solving continuous GOP, we move from one minimizer to another better one with the help of some auxiliary functions, namely the filled functions (FF). We have proposed four novel FF's in Part I of this dissertation. The proposed FF's not only preserve the promising theoretical properties of the traditional FF but also overcome the difficulties in its computational implementation. Furthermore, the final one is superior to all the existing FF's, since it has a minimizer over the problem domain instead of a line. This property ensures that a better minimizer of f can be found by some classical local search methods. Extensive numerical experiments on several test problems with up to 1000 variables are reported. These results indicate that the proposed solution algorithms are efficient.; In Part II of this dissertation, we have developed two discrete filled function (DFF) methods to tackle discrete GOP. These methods are probably the first in discrete optimization that are capable of solving nonlinear, inseparable and nonconvex large scale integer optimization problems. The first DFF has a discrete minimizer on a “discrete path” between the current discrete minimizer and a better discrete minimizer of f. The second one not only guarantees to have a discrete minimizer over the problem domain, but also ensures that its discrete minimizers coincide with the better discrete minimizers of f. This property assures that a better discrete minimizer of f can be found by some classical local search methods. Numerical experiments on several test problems with up to 100 variables have demonstrated the applicability and efficiency of the proposed methods.
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机译：作为一种强大的解决方案，优化在广泛的领域和人类活动（包括工程，科学，金融等）中得到了广泛的应用。一般非凸目标函数 f 的多个局部最小值的存在，使全局优化（GO）面临巨大挑战。这项研究的主要目的是开发一些高性能的方法来解决连续或离散变量上的全局优化问题（GOP）。这项研究为推进GO技术的发展做出了一些贡献。这些贡献包括理论和数值方面，如下所述。在求解连续GOP时，我们借助一些辅助功能（即填充功能（FF））从一个最小化器移到另一个更好的化解器。在本论文的第一部分中，我们提出了四个新颖的FF。提出的FF不仅保留了传统FF的有希望的理论特性，而且克服了其计算实现上的困难。此外，最后一个优于所有现有FF，因为它在问题域而不是线路上具有最小化器。此属性确保可以通过某些经典的局部搜索方法找到更好的 f 最小值。报告了对多达1000个变量的几个测试问题的广泛数值实验。这些结果表明所提出的解决方案算法是有效的。在本文的第二部分中，我们开发了两种离散填充函数（DFF）方法来解决离散GOP。这些方法可能是离散优化中的第一个方法，能够解决非线性，不可分割和非凸的大规模整数优化问题。第一个DFF在当前离散最小化器和更好的 f 离散最小化器之间的“离散路径”上具有离散最小化器。第二个方法不仅保证在问题域上具有离散的最小化器，而且还确保其离散的最小化器与 f 更好的离散的最小化器一致。此属性确保可以通过某些经典的局部搜索方法找到更好的 f 离散最小值。通过对多达100个变量的几个测试问题进行数值实验，证明了所提方法的适用性和有效性。
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