Simplicial global optimization focuses on deterministic covering methods for global optimization partitioning the feasible region by simplices. Although rectangular partitioning is used most often in global optimization, simplicial covering has advantages shown in this book. The purpose of the book is to present global optimization methods based on simplicial partitioning in one volume. The book describes features of simplicial partitioning and demonstrates its advantages in global optimization. A simplex is a polyhedron in a multidimensional space, which has the minimal number of vertices. Therefore simplicial partitions are preferable in global optimization when the values of the objective function at all vertices of partitions are used to evaluate subregions. The feasible region defined by linear constraints may be covered by simplices and therefore simplicial optimization algorithms may cope with linear constraints in a delicate way by initial covering. This makes simplicial partitions very attractive for optimization problems with linear constraints. There are optimization problems where the objective functions have symmetries which may be taken into account for reducing the search space significantly by setting linear inequality constraints. The resulted search region may be covered by simplices. Applications benefiting from simplicial partitioning are examined in the book: nonlinear least squares regression, center-based clustering of data having one feature, and pile placement in grillage-type foundations. In the examples shown, the search region reduced taking into account symmetries of the objective functions is a simplex thus simplicial global optimization algorithms may use it as a starting partition. The book provides exhaustive experimental investigation and shows the impact of various bounds, types of subdivision, and strategies of candidate selection on the performance of global optimization algorithms. Researchers and engineers will benefit from simplicial partitioning algorithms presented in the book: Lipschitz branch-and-bound, Lipschitz optimization without the Lipschitz constant. We hope

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前言

单纯全局优化侧重于确定性覆盖方法，用于通过单纯划分可行区域的全局优化。尽管矩形分区最常用于全局优化，但简单覆盖在本书中具有优势。本书的目的是在一卷中介绍基于简单分区的全局优化方法。本书描述了简单分区的特点，并展示了它在全局优化中的优势。单纯形是多维空间中的多面体，它的顶点数最少。因此，当使用分区的所有顶点处的目标函数值来评估子区域时，在全局优化中最好使用简单分区。由线性约束定义的可行区域可能被单纯形覆盖，因此单纯优化算法可以通过初始覆盖以精细的方式处理线性约束。这使得简单分区对于具有线性约束的优化问题非常有吸引力。存在优化问题，其中目标函数具有对称性，可以通过设置线性不等式约束来显着减少搜索空间。结果搜索区域可能被单纯形覆盖。本书研究了受益于简单划分的应用程序：非线性最小二乘回归、具有一个特征的基于中心的数据聚类以及在格架型基础中的桩放置。在所示的示例中，考虑到目标函数的对称性而减少的搜索区域是一个单纯形，因此简单的全局优化算法可以将其用作起始分区。本书提供了详尽的实验研究，并展示了各种边界、细分类型和候选选择策略对全局优化算法性能的影响。研究人员和工程师将从书中介绍的简单划分算法中受益：Lipschitz 分支定界，没有 Lipschitz 常数的 Lipschitz 优化。我们希望 以及候选选择策略对全局优化算法性能的影响。研究人员和工程师将从书中介绍的简单划分算法中受益：Lipschitz 分支定界，没有 Lipschitz 常数的 Lipschitz 优化。我们希望 以及候选选择策略对全局优化算法性能的影响。研究人员和工程师将从书中介绍的简单划分算法中受益：Lipschitz 分支定界，没有 Lipschitz 常数的 Lipschitz 优化。我们希望