回顾: 支撑函数 \sigma_C(y)=\max_{x\in C}\langle y,x\rangle 通俗来说是集合C的y法向切平面最大截距(集合的闭凸包可以由集合的超平面包络得到)支撑函数是次线性函数 次梯度和次微分 f(y)\ge f(x)+\langle g,y…
邻近点映射 proximal mapping邻近点映射在非光滑凸优化的很多算法中至关重要,第一个研究它的人是Moreau,该映射定义为 \text{prox}_f(x)=\arg\min_{u\in\mathbb{E}}\{f(u)+\frac{1}{2}\|u-x\|^2\},\forall x\in\m…
对偶空间与双重对偶空间：向量空间上的线性泛函构成对偶空间，用符号表示。给定向量空间，其内积与对偶空间内积保持一致。对偶范数定义为线性泛函在单位球面上的最大内积值。引理证明了广义不等式成立。在有限维空间，双重对偶空间与原始空间同构，双重对偶范数与原始空间范数一致。伴随变换定义：给定向量空间与线...
套丛书还有 《Introduction to the Scenario Approach》《Algebraic and Geometric Ideas in the Theory of Discrete Optimization》《Evaluation Complexity of Algorithms for Nonconvex Optimization》《Introduction to Derivative-Free Optimization》《A Mathematical View of Interior-point Methods in Convex Optimization》...
AI小助手 测试版 记笔记 《机器学习中的一阶优化算法 1》 This course provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. We first delve into first-order optimization methods in convex optimization, off...
Springer Series in the Data Sciences(共11册),这套丛书还有 《Data Science for Public Policy》《Deep Learning Architectures》《Statistics with Julia: Fundamentals for Data Science, Machine Learning and Artificial Intelligence》《Statistics in the Public Interest》《Mathematical Foundations for Data Analysis...
aFinally, the coordination of the foregoing optimization efforts at a supervisory level is discussed. The trade-off between pumping aid compressor power is discussed to arrive at the optimum temperature settings of chilled water and condenser water. The implementation of the optimization methods in dis...
\begin{equation*} S_{+}^n = \{A \in R^{n \times n}| A \ge O\} \end{equation*} (3)全体n \times n正定(positive \quad define)矩阵的集合S_{++}^n 所有n \times n维正定矩阵的全体构成的集合,记作S_{++}^n: \begin{equation*} S_{++}^n = \{A \in R^{n \times n}| A...
故,(x_n,\alpha) \in epif,又因为f是闭函数,因此epif是闭集,设x_n \rightarrow \bar{x}(n\rightarrow \infty),因此可得:(x,\alpha) \in epif,即f(x) \le \alpha,故x \in Lev(f,\alpha),因此命题(Ⅲ)成立。 (Ⅲ)\Rightarrow(Ⅰ):利用反证法:即假设命题(Ⅲ)成立时,命题(Ⅰ)不成立。则f...
(h_1 \square h_2)(x) \equiv \underset{u \in E}{min}\{h_1(u) + h_2(x-u)\} 定理2.18的一个直接应用可以对应于如下命题,即一个正常凸函数和一个实值凸函数的下确界卷积总是凸的。 定理2.19\quad(下确界卷积的凸性)设h_1:E \rightarrow (-\infty,\infty]是一个正常凸函数,h_2: E ...