【协作算法】2 凸性分析与方法学精要

智能计算优化理论:凸性分析与方法学精要

1 优化问题的核心框架

1.1 决策变量与目标函数

1.1.1.1 优化的本质

优化问题的本质在于寻找一组决策参数,使得某个评价指标达到极值。设目标函数将决策向量映射为单一实数值,决策向量的维度决定了问题的复杂程度。当维度为一维时,问题退化为单变量优化;当维度大于等于二维时,决策变量构成向量,问题升格为多变量优化。在机器学习场景中,神经网络的权重矩阵、支持向量机的超平面系数本质上均为高维决策变量。

1.1.1.2 代价函数与适应度函数

目标函数依优化方向不同而具有不同称谓。当问题指向最小化时,通常称为代价函数或损失函数;当指向最大化时,称为适应度函数。二者可通过严格的数学变换相互转化。符号反转法指出,最小化一个函数等价于最大化其相反数。倒数法指出,在函数值为正的前提下,最小化一个函数等价于最大化其倒数。若仅关注最优解的位置而非最优值本身,则无需后续修正。在深度学习实践中,极大似然估计要求最大化数据似然函数,而神经网络框架通常设计为最小化目标,因此训练目标常设为负对数似然,即通过符号反转将最大化问题嵌入最小化框架。

1.2 优化问题的分类体系

1.2.1.1 无约束优化

无约束优化问题仅包含目标函数本身。此类问题在微积分中已有成熟解法:对可微函数求梯度并令其为零,解得的驻点通过二阶信息判别为极小值、极大值

凸壳串行算法 关于凸壳的串行算法,可以说有好多种,有时间复杂度O(n^2)的,也有O(nlogn)的,下面依次介绍几种算法:1,我认为最土的一种方法,时间复杂度为O(n^2) 叫做 卷包裹法由Chand 和 Kapur 于1970年提出,基本思想:首先过y坐标最小的点p1画一条水平直线L,显然该点是凸壳的一个顶点。然后L绕p1按逆时针方向旋转,碰到S(顶点集合)中的第二个点p2时,直线绕p2按逆时针旋转 阅读详情

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