re-add line after intro

Author: karanveerm

ee364a/approx.html

@@ -73,6 +73,7 @@ <h1>Approximation</h1>
 \]
 with variable $x \in \mathrm{R}^n$, where $A\in \mathrm{R}^{m \times n}$,
 with $m \ll n$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 For $p=2$, we would expect to see many components of $x^\star$ equal to zero.</div>

ee364a/duality.html

@@ -80,6 +80,7 @@ <h1>Duality</h1>
 and then solve the modified problem.
 We are interested in the optimal objective value of this modified problem,
 compared to the original one above.</div>
+		<hr/>
 		<div>
 			<div class="description">
 If $\lambda^\star$ is large, then decreasing $s$</div>
@@ -174,6 +175,7 @@ <h1>Duality</h1>
 \end{array}
 \]
 that satisfies Slater's constraint qualification.</div>
+		<hr/>
 		<div>
 			<div class="description">
 The primal and dual problems have the same objective value.</div>

ee364a/functions.html

@@ -174,6 +174,7 @@ <h1>Convex functions</h1>
 	<br/>
 	<fieldset>
 		<div class="intro">$f(x) = (x^2 + 2)/(x+2)$, with $\mathbf{dom}f = (-\infty, -2)$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $f$ is convex.</div>
@@ -219,6 +220,7 @@ <h1>Convex functions</h1>
 	<br/>
 	<fieldset>
 		<div class="intro">$f(x) = 1/(1-x^2)$, with $\mathbf{dom} f = (-1, 1)$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $f$ is convex.</div>
@@ -287,6 +289,7 @@ <h1>Convex functions</h1>
 	<br/>
 	<fieldset>
 		<div class="intro">$f(x) = \cosh x = (e^x+e^{-x})/2$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $f$ is convex.</div>
@@ -404,6 +407,7 @@ <h1>Convex functions</h1>
 	<fieldset>
 		<div class="intro">We define $(x)_-$ to be $\max\{0,-x\}$,
 such that $x = (x)_+ - (x)_-$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 The constraint $\mathbf{1}^T(x)_- \leq (1/2) \mathbf{1}^T(x)_+$

ee364a/geom.html

@@ -68,6 +68,7 @@ <h1>Geometrical problems</h1>
 linear inequalities, and $a$ a point in $\mathbf{R}^n$.
 Are the following problems easy or hard?  (Easy means the solution can
 be found by solving one or a modest number of convex optimization problems.)</div>
+		<hr/>
 		<div>
 			<div class="description">
 Find a point in $\mathcal P$ that is closest to $a$

ee364a/intro.html

@@ -166,6 +166,7 @@ <h1>Introduction to convex optimization</h1>
 {\rm D} &amp; 11 &amp; 50 \\ \hline
 \end{array}
 \]</div>
+		<hr/>
 		<div>
 			<div class="description">
 Design B is better than design A.</div>

ee364a/num_lin_alg.html

@@ -203,6 +203,7 @@ <h1>Numerical linear algebra</h1>
 	<br/>
 	<fieldset>
 		<div class="intro">Suppose $A\in \mathbf{R}^{n \times n}$ is lower triangular.</div>
+		<hr/>
 		<div>
 			<div class="description">
 The flop count for computing $Ab$ is
@@ -229,6 +230,7 @@ <h1>Numerical linear algebra</h1>
 	<fieldset>
 		<div class="intro">Suppose $A\in \mathbf{R}^{m \times n}$, and we need to compute $x$ that
 minimizes $\|Ax-b\|^2_2 + (\rho/2)\|x\|_2^2$, where $\rho &gt;0$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 For $m \geq n$, the flop count (order) is</div>

ee364a/problems.html

@@ -72,6 +72,7 @@ <h1>Convex problems</h1>
 \end{array}
 \]
 with variable $x=(x_1,x_2)$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 The point $(-1,1)$ is a solution.</div>

ee364a/sets.html

@@ -67,6 +67,7 @@ <h1>Convex sets</h1>
 		<div class="intro">Define the square (in $\mathbf{R}^2$)
 $S = \{ x \in \mathbf{R}^2 \mid 0 \leq x_i \leq 1,~i=1,2 \}$,
 and the disk $D = \{ x \in \mathbf{R}^2 \mid \|x\|_2 \leq 1 \}$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $S \cap D$ is convex.</div>
@@ -133,6 +134,7 @@ <h1>Convex sets</h1>
 	<br/>
 	<fieldset>
 		<div class="intro">$C = \{ (1,0), (1,1), (-1,-1), (0,0) \}$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $(0,-1/3) \in \mathbb{conv}\; C$.</div>
@@ -199,6 +201,7 @@ <h1>Convex sets</h1>
 	<br/>
 	<fieldset>
 		<div class="intro">Consider the set $S = \{ (0,2),~ (1,1),~ (2,3),~ (1,2),~ (4,0) \}$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $(0,2)$ is the minimum element of $S$.</div>
@@ -310,6 +313,7 @@ <h1>Convex sets</h1>
 	<fieldset>
 		<div class="intro">$S = \{ \alpha \in \mathbf{R}^3 \mid \alpha_1 + \alpha_2e^{-t} +
 \alpha_3 e^{-2t} \leq 1.1 \mbox{ for } t\geq 1\}$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $S$ is affine.</div>
@@ -377,6 +381,7 @@ <h1>Convex sets</h1>
 	<fieldset>
 		<legend>Generalized inequality.</legend>
 		<div class="intro">$K = \{(x_1,x_2) \mid 0 \leq x_1 \leq x_2 \}$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 $(1,3) \preceq_K (3,4)$.</div>

ee364a/stat.html

@@ -69,6 +69,7 @@ <h1>Statistical estimation</h1>
 $\Sigma^\mathrm{min} \preceq \Sigma \preceq \Sigma^\mathrm{max}$,
 where $\Sigma^\mathrm{min},~ \Sigma^\mathrm{max}$
 are given positive definite matrices.</div>
+		<hr/>
 		<div>
 			<div class="description">
 The associated log-likelihood function $\ell(\mu,\Sigma)$ (including

ee364a/unconstrained.html

@@ -311,6 +311,7 @@ <h1>Algorithms for unconstrained optimization</h1>
 f(x) = (c^Tx)^4 + \sum_{i=1}^n w_i \exp x_i,
 \]
 over $x \in \mathbf{R}^n$, where $w \succ 0$.</div>
+		<hr/>
 		<div>
 			<div class="description">
 Newton's method would probably require fewer iterations than the gradient