From 9e44694388b29f1861c834a84295363057575a4e Mon Sep 17 00:00:00 2001
From: Akshay '
+print('
')
diff --git a/html/chu2013socp-codegen.html b/html/chu2013socp-codegen.html
index 76acd0a..5d02444 100644
--- a/html/chu2013socp-codegen.html
+++ b/html/chu2013socp-codegen.html
@@ -52,7 +52,7 @@ ')
for list_item in list_items:
- print '
'
+ print('Novel contributions
JuMP is a
Julia-embedded modeling language for mathematical optimization that was written
-with performance in mind; its salient features include automatic differentation
+with performance in mind; its salient features include automatic differentiation
of user-defined functions, efficient parsing of updated problems, support for
user-defined problems and solve methods, and solver callbacks. JuMP targets
linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear
@@ -60,7 +60,7 @@
Background
a particular standard form, and it is the responsibility of the client to
express her problem in an acceptable fashion. It is often not
at all obvious to clients how to encode their problems using the solving
-substrate's standard form; this encoding proces may require
+substrate's standard form; this encoding proces may require
clever re-expressions of the original problem or onerous stuffing of
the problem's constraints into a rigid matrix structure.
Gatys et al. present a way to repurpose a deep network trained for image -classification to redraw images in the style of some reference image. +
Gatys et al. present a way to repurpose a deep network trained for image +classification to redraw images in the style of some reference image. For example, this method can be used to render arbitrary photographs in the style of Van Gogh’s The Starry Night.
@@ -75,7 +75,7 @@where is fixed, is a weight that incorporates the the size +
where is fixed, is a weight that incorporates the size of layer (see the paper for details), and is the optimization variable. Gram matrices are used above to capture correlations between the features within each layer. The -weighted expression measures the similarity diff --git a/html/gubin1966projections.html b/html/gubin1966projections.html index bb82a50..35ca033 100644 --- a/html/gubin1966projections.html +++ b/html/gubin1966projections.html @@ -199,12 +199,12 @@
John Von Neumann. The Geometry of Orthogonal Spaces. Functional Operators (AM-22), Vol. II. Princeton University Press, 1950. Reprint of lecture -notes originally compiled in 1933. ↩
+notes originally compiled in 1933. ↩Boyd, S., & Dattorro, J. (2003). Alternating projections. EE392o, Stanford -University. ↩
+University. ↩- $$ - \newcommand{\pmi}{\operatorname{pmi}} - \newcommand{\inner}[2]{\langle{#1}, {#2}\rangle} - \newcommand{\Pb}{\operatorname{Pr}} - \newcommand{\E}{\mathbb{E}} - \newcommand{\argmin}[2]{\underset{#1}{\operatorname{argmin}} {#2}} - \newcommand{\optmin}[3]{ - \begin{align*} - & \underset{#1}{\text{minimize}} & & #2 \\ - & \text{subject to} & & #3 - \end{align*} - } - \newcommand{\optmax}[3]{ - \begin{align*} - & \underset{#1}{\text{maximize}} & & #2 \\ - & \text{subject to} & & #3 - \end{align*} - } - \newcommand{\optfind}[2]{ - \begin{align*} - & {\text{find}} & & #1 \\ - & \text{subject to} & & #2 - \end{align*} - } - $$ -
- -Section Breakdown
-I. Methods of successive projection [4.5 pp]
-II. Proofs of [convergence results] theorems 1 and 2 [8.5 pp]
-III. Rate of convergence [3.5 pp]
-IV. Examples and Applications [7.5 pp]
This paper reviews methods of successive projections for finding a point in -the intersection of an arbitrary number of convex sets; this problem is known -as the convex feasibility problem. The family of methods under consideration -traces back to the alternating projections method of the 1930s, due to Johnn -von Neumann1. Though von Neumann cast it as a routine for finding a point -in the intersection of two subspaces, his method was later generalized to the -convex feasibility problem.
- -Convex cone programs that admit strong duality can be reduced to instances of -the convex feasibility -problem via the -KKT system or the self-dual homogeneous embedding thereof.
- -(A) Strong convergence results for the method of projections, -proving that under certain conditions the sequences in question converge -linearly (presented in section II of the paper).
- -(B) A cheap way to accelerate projection methods (presented in section -III of the paper).
- -Simple examples show that projection methods are often painfully slow to -converge, so convergence results are not that interesting. But, because, as we -shall see, such methods are so simple, accelerating them with minor alterations -is an attractive prospect.
- -A projection of the point onto a set of a normed space is a -point that minimizes the distance between and , in the -following sense:
- - - -Let the sets convex, for . The convex -feasibility problem is the problem of finding a common point .
- -The method of projections constructs a sequence of points where - is arbitrary and is chosen from by selecting a -set and stepping in the direction of projecting onto -the selected set, i.e.,
- - - -Different policies for selecting the index yield different -instantiations of the method of projections. A simple, classical policy is -the cyclic one that sets as .
- -Cyclic projections is an extension of von Neumman’s alternating -projections method: alternating projections is cyclic projections when the -number of sets . Perhaps surprisingly, the case of is not any -more general than : any convex feasibility problem that involves greater -than 2 sets can be reduced to a higher-dimensional problem with exactly two -sets: one set is the cartesian product of all the sets from the original -problem, and the other set is an affine set that enforces block-wise equality -of the vector.
- -A proof of convergence for the alternating projections method can be found in -in [2]2, and a more general proof can be found in the reviewed paper. Here,
- -lemmas about projection properties of projections, just for my own review / other’s reference
-no convergence proofs, can mention some2
-the accelerated method
-