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The application of Hahn-Banach Theorem 01
Suppose $X$ is a normed space and $X_0$ is a closed subspace of $X$ and $x_0 \in X \setminus X_0$. Then we can find $f \in X’$ such that $f(x_0) = 1$ and $f(x) = 0$, $\forall x \in X_0$. Proof: Since $x_0 \notin X_0$, we can find $\delta > 0$ such that $|x_0 - x| \geq \delta$, $\forall x \in X_0$, which is equivalent to $1 \leq \dfrac{|x_0 - x|}{\delta}$, $\forall x \in X_0$.
The application of Hahn-Banach Theorem 02
$X'$ = $\{ f: X \to \mathbb{K} \}$ where $f$ is is linear and continuous and $X$ is a Banach space over $\mathbb{K}$. Prove that $X' \neq {0}$, in fact, for every $x \neq 0 \in X$, we can find $f \in X’$ such that $f(x) = |x|$ and $|f| = 1$. Proof: Pick $x_0 \in X$. Define $X_0 = x_0 \cdot \mathbb{K}$, a subspace of $X$, and $g: X_0 \to \mathbb{K}$, $g(x) = x$, which is linear. Since $g$ and $|\cdot|$ satisfy the conditions of the Hahn-Banach theorem, we can find $f: X \to \mathbb{K}$ such that $f|_{X_0} = g$, $f$ is linear and $f(x) \leq |x|$, $\forall x \in X$. Therefore $f(x_0) = g(x_0) = |x_0|$ and $|f| \leq 1$. The equality $f(x_0) = |x_0|$ guarantees that $|f| = 1$.
Optimization Papers in JMLR Volume 26
Optimization Research Papers in JMLR Volume 25
Optimization Research Papers in JMLR Volume 25 (2024) # This document lists papers from JMLR Volume 25 (2024) that focus on optimization research, categorized by their primary themes. Each paper is numbered starting from 1 within its subsection, with a brief description of its key contributions to optimization theory, algorithms, or applications. Convex Optimization # Papers addressing convex optimization problems, including sparse NMF, differential privacy, and sparse regression. Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction Authors: Yuze Han, Guangzeng Xie, Zhihua Zhang Description: Investigates lower complexity bounds for finite-sum optimization problems in convex settings.
Ebooks & related papers on Convex Optimizations
Ebooks # Boris Mordukhovich , Nguyen Mau Nam. An Easy Path to Convex Analysis and Applications. 2023 Yurii Nesterov. Lectures on Convex Optimization. 2018 Sébastien Bubeck. Convex Optimization: Algorithms and Complexity. 2015 Dimitri Bertsekas. Nonlinear Programming. 2016 Boris Teodorovich Polyak. Introduction to Optimization. 1987 R. T. Rockafellar. Convex Analysis. 1970 H. H. Bauschke & P. L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. 2011 Lieven Vandenberghe and Stephen P. Boyd. Convex Optimization. 2004 Papers # Yu. E. Nesterov. A method of solving a convex programming problem with convergence rate. 1983