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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta
name="viewport"
content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"
/>
<title>APMA E2000 - Optimization</title>
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<style>
#lagframe {
width: auto;
height: 100%;
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border: 0;
max-width: 100vw;
max-height: 100vh;
}
</style>
</head>
<body>
<div class="reveal">
<div class="slides">
<section>
<section>
<h3 class="framelabel">Lecture 13</h3>
<h1>Optimization</h1>
<h2>APMA E2000</h2>
<div class="r-stretch"></div>
<p style="text-align: right">
Drew Youngren <code>[email protected]</code>
</p>
</section>
<section>
<!-- Set up standard LaTeX macros -->
$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$
$\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$
$\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$
<h2>Announcements</h2>
<ul>
<li class="fragment">
Quiz 4 (on HW 6) this week
<ul>
<li class="fragment"><strike>Chain Rule</strike></li>
<li class="fragment">Directional Derivative</li>
<li class="fragment">Properties of the Gradient</li>
</ul>
</li>
<li class="fragment">HW7 due Tues</li>
</ul>
</section>
</section>
<section>
<section>
<h1>1-minute review</h1>
</section>
<section>
<p>
<em>Critical points</em> of a function $f$ are those where $f$ is
not differentiable or $\nabla f = 0$.
</p>
<p class="fragment">
Local mins and maxes (on open sets) only occur at critical points.
</p>
<p class="fragment">
The
<a
href="https://drew.youngren.nyc/mvc-slides/lec12.html#/3/2/1"
target="_blank"
rel="noopener noreferrer"
>second derivative</a
>
test can classify some critical points of $f(x,y)$.
</p>
</section>
</section>
<section>
<section>
<h1>Preliminaries</h1>
</section>
<section>
<h3>Kinds of Optimization</h3>
<h6 class="framelabel fragment">Unconstrained</h6>
<p class="fragment">
On open sets
<span class="fragment"
>$\longrightarrow$ look for critical points</span
>
</p>
<h6 class="framelabel fragment">Constrained</h6>
<p class="fragment">
On boundary points
<span class="fragment"
>$\longrightarrow$ Lagrange multipliers</span
>
</p>
</section>
</section>
<section>
<section>
<h1>Framework</h1>
</section>
<section>
<p>
For an <strong>optimization problem</strong> one must identify:
</p>
<ul>
<li class="fragment">
The <em>target function</em> or <em>desideratum</em> $f$
</li>
<li class="fragment">
The <em>control variables</em> $x, y, \ldots$
<ul class="">
<li>Give description and units where appropriate</li>
</ul>
</li>
<li class="fragment">
The <em>domain</em> $D$ of admissible values
</li>
</ul>
</section>
<section>
<h6 class="framelabel">Example - Unconstrained</h6>
<p>
Find the closest point to the origin on the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJm9iajBfa2luZD1ncmFwaCZvYmowX2NvbG9yPSUyMzBkMDg4NyZvYmowX3BhcmFtc19hPS0yJm9iajBfcGFyYW1zX2I9MiZvYmowX3BhcmFtc19jPS0yJm9iajBfcGFyYW1zX2Q9MiZvYmowX3BhcmFtc196PXgrLSsyK3krJTJCKzMmb2JqMF9wYXJhbXNfdDA9MCZvYmowX3BhcmFtc190MT0x"
target="_blank"
rel="noopener noreferrer"
>plane</a
>
\[z = x - 2y + 3.\]
</p>
<p class="fragment">
<strong>Solution.</strong> We will choose $x$ and $y$ freely (so
$D = \RR^2$, open) and minimize the distance to $(x,y,x - 2y +
3)$. \[f(x,y) = x^2 + y^2 + (x - 2y + 3)^2\]
</p>
</section>
<section>
<p>
\[\nabla f = \bv{2x + 2(x - 2y + 3) \\ 2y - 4(x - 2y + 3)} = \bv{0
\\ 0}\] \[ = \bv{4x - 4y + 6 \\ - 4x + 10y - 12} = \bv{0 \\ 0}\]
</p>
<p class="fragment">\[ \implies y = 1, x = -\frac12 \]</p>
<p class="fragment">
which means the closest point is $(-1/2, 1, 1/2)$
</p>
</section>
</section>
<section>
<section>
<h1>Lagrange Multipliers</h1>
</section>
<section>
<h6 class="framelabel">Constraint Function</h6>
<p>
In most real applications, we cannot choose out control variables
freely, but rather they are subject to a
<strong>constraint</strong>.
</p>
<p class="fragment">
Often, we express this by specifying this as a
<strong>level set</strong>
\[g(x,y,\ldots) = k \]
</p>
<p class="fragment">
e.g., a fixed budget or a set proportion of ingredients for a
recipe.
</p>
</section>
<section>
<p class="smaller">
What does the gradient tell us at local extremes on constrained
sets?
</p>
<div class="r-stretch">
<iframe
src="https://3demos.ctl.columbia.edu/?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"
frameborder="0"
height="100%"
width="100%"
></iframe>
</div>
</section>
<section>
<h6 class="framelabel">Example</h6>
<p style="font-size: smaller">
Identify the local minima of the distance to the Sun from a body
restricted to the drawn "orbit".
</p>
<div class="r-stretch">
<iframe
src="./lagrange-2d.html"
id="lagframe"
frameborder="0"
></iframe>
</div>
<!-- <div
id="lagbox"
class="jxgbox"
style="width: 600px; height: 600px"
></div> -->
<!-- <script type="text/javascript" src="./js/attractor.js"></script> -->
</section>
<section>
<h6 class="framelabel">Theorem - Lagrange Multipliers</h6>
<p>
A local minimum (or maximum) of $f(\vec x)$ subject to the
constraint $g(\vec x) = k$ occurs at $\vec a$ only if $\nabla g(a)
= \vec 0$ or \[\nabla f(\vec a) = \lambda \nabla g(\vec a)\] for
some scalar $\lambda$.
</p>
</section>
<section>
<h6 class="framelabel">Example (Reprise)</h6>
<p>
Find the closest point to the origin on the
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SG93K1RvJm9iajBfa2luZD1maWVsZCZvYmowX2NvbG9yPSUyMzBkMDg4NyZvYmowX3BhcmFtc19wPXgmb2JqMF9wYXJhbXNfcT15Jm9iajBfcGFyYW1zX3I9eiZvYmowX3BhcmFtc19uVmVjPTYmb2JqMV9raW5kPWxldmVsJm9iajFfY29sb3I9JTIzZGM1ZTY2Jm9iajFfcGFyYW1zX2c9eistK3grJTJCKzIreSZvYmoxX3BhcmFtc19rPTMmb2JqMV9wYXJhbXNfYT0tMiZvYmoxX3BhcmFtc19iPTImb2JqMV9wYXJhbXNfYz0tMiZvYmoxX3BhcmFtc19kPTImb2JqMV9wYXJhbXNfZT0tMiZvYmoxX3BhcmFtc19mPTI="
target="_blank"
rel="noopener noreferrer"
>plane</a
>
\[z = x - 2y + 3.\]
</p>
<p class="r-stretch"></p>
</section>
<section>
<h6 class="framelabel">Solution</h6>
<p>
Viewed as a
<a
href="https://3demos.ctl.columbia.edu/?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"
target="_blank"
rel="noopener noreferrer"
>constrained problem</a
>
in $\mathbb{R}^3$, we target distance$^2$ and use the plane
equation as the constraint. \[ f(x,y,z) = x^2 + y^2 + z^2,\
g(x,y,z) = x - 2y - z + 3 = 0 \]
</p>
<p class="fragment">
\[ \nabla f = 2\bv{x \\ y \\ z} = \lambda \bv{1 \\ -2 \\ -1} =
\lambda \nabla g\]
</p>
</section>
<section>
<h6 class="framelabel">Example – Cobb-Douglass</h6>
<div class="container">
<div class="col">
<p style="font-size: smaller">
By investing $x$ units of labor and $y$ units of capital, a
low-end watch manufacturer can produce $x^{0.4}y^{0.6}$
watches. Find the maximum number of watches that can be
produced with a budget of $\$20000$ if labor costs $\$100$ per
unit and capital costs $\$200$ per unit.
</p>
</div>
<div class="col fragment">
<img
src="./assets/cobb-doug-contours.png"
alt=""
class="fragment"
/>
</div>
</div>
</section>
<section>
<h6 class="framelabel">Example – 3D</h6>
<p>
Find the minimum surface area of a lidless shoebox with volume $32
\text{ L}$.
</p>
<div class="r-stretch"></div>
</section>
</section>
<section>
<section>
<h1>XVT</h1>
<h3 class="fragment">Extreme Value Theorem</h3>
</section>
<section>
<div class="r-frame">
<h6 class="framelabel">Theorem</h6>
<p>
A
<a
href="https://3demos.ctl.columbia.edu/?Y3VycmVudENoYXB0ZXI9SW50cm8mc2NhbGU9MCZzaG93UGFuZWw9ZmFsc2Umb2JqMF9raW5kPWN1cnZlJm9iajBfY29sb3I9JTIzZmYzMzMzJm9iajBfcGFyYW1zX2E9MCZvYmowX3BhcmFtc19iPTIqcGkmb2JqMF9wYXJhbXNfeD0xJTJGMislMkIrMyUyRjIlMjh0KyUzRStwaSUyRjIrJTI2K3QrJTNDK3BpKyUzRislMjgyJTJGMyUyQjElMkYzKmNvcyUyODIqdCUyOSU1RTIlMjkrJTNBKzElMjkqK2NvcyUyOHQlMjkmb2JqMF9wYXJhbXNfeT0zJTJGMiUyOHQrJTNFK3BpJTJGMislMjYrdCslM0MrcGkrJTNGKyUyODIlMkYzJTJCMSUyRjMqY29zJTI4Mip0JTI5JTVFMiUyOSslM0ErMSUyOSorc2luJTI4dCUyOSZvYmowX3BhcmFtc196PTAmb2JqMF9wYXJhbXNfdGF1PTAmb2JqMV9raW5kPWN1cnZlJm9iajFfY29sb3I9JTIzZmYzMzMzJm9iajFfcGFyYW1zX2E9MCZvYmoxX3BhcmFtc19iPTIqcGkmb2JqMV9wYXJhbXNfeD0xJTJGMislMkIrMyUyRjIqJTI4dCslM0UrcGklMkYyKyUyNit0KyUzQytwaSslM0YrJTI4MiUyRjMlMkIxJTJGMypjb3MlMjgyKnQlMjklNUUyJTI5KyUzQSsxJTI5Kitjb3MlMjh0JTI5Jm9iajFfcGFyYW1zX3k9MyUyRjIqJTI4dCslM0UrcGklMkYyKyUyNit0KyUzQytwaSslM0YrJTI4MiUyRjMlMkIxJTJGMypjb3MlMjgyKnQlMjklNUUyJTI5KyUzQSsxJTI5KitzaW4lMjh0JTI5Jm9iajFfcGFyYW1zX3o9LSUyOCUyODElMkYyKyUyQiszJTJGMiolMjh0KyUzRStwaSUyRjIrJTI2K3QrJTNDK3BpKyUzRislMjgyJTJGMyUyQjElMkYzKmNvcyUyODIqdCUyOSU1RTIlMjkrJTNBKzElMjkqK2NvcyUyOHQlMjklMjklNUU0KyUyQislMjgzJTJGMiolMjh0KyUzRStwaSUyRjIrJTI2K3QrJTNDK3BpKyUzRislMjgyJTJGMyUyQjElMkYzKmNvcyUyODIqdCUyOSU1RTIlMjkrJTNBKzElMjkqK3NpbiUyOHQlMjklMjklNUU0Ky0rNColMjgxJTJGMislMkIrMyUyRjIqJTI4dCslM0UrcGklMkYyKyUyNit0KyUzQytwaSslM0YrJTI4MiUyRjMlMkIxJTJGMypjb3MlMjgyKnQlMjklNUUyJTI5KyUzQSsxJTI5Kitjb3MlMjh0JTI5JTI5KiUyODMlMkYyKiUyOHQrJTNFK3BpJTJGMislMjYrdCslM0MrcGkrJTNGKyUyODIlMkYzJTJCMSUyRjMqY29zJTI4Mip0JTI5JTVFMiUyOSslM0ErMSUyOSorc2luJTI4dCUyOSUyOSstKzElMjkrJTJGJTI4MTArJTJCKzEwKyUyOCUyODElMkYyKyUyQiszJTJGMiolMjh0KyUzRStwaSUyRjIrJTI2K3QrJTNDK3BpKyUzRislMjgyJTJGMyUyQjElMkYzKmNvcyUyODIqdCUyOSU1RTIlMjkrJTNBKzElMjkqK2NvcyUyOHQlMjklMjkrLSslMjgzJTJGMiolMjh0KyUzRStwaSUyRjIrJTI2K3QrJTNDK3BpKyUzRislMjgyJTJGMyUyQjElMkYzKmNvcyUyODIqdCUyOSU1RTIlMjkrJTNBKzElMjkqK3NpbiUyOHQlMjklMjklMjklNUUyJTI5Ky41KyUyQisxJm9iajFfcGFyYW1zX3RhdT0wJm9iajJfa2luZD1ncmFwaCZvYmoyX2NvbG9yPSUyMzMyMzJmZiZvYmoyX3BhcmFtc19hPS0xJm9iajJfcGFyYW1zX2I9MiZvYmoyX3BhcmFtc19jPS0zJTJGMiZvYmoyX3BhcmFtc19kPTMlMkYyJm9iajJfcGFyYW1zX3o9LSUyOHglNUU0KyUyQit5JTVFNCstKzQqeCp5Ky0rMSUyOSslMkYlMjgxMCslMkIrMTArJTI4eCstK3klMjklNUUyJTI5Ky41KyUyQisxJm9iajJfcGFyYW1zX3RhdT0wJm9iajJfcGFyYW1zX3QwPTAmb2JqMl9wYXJhbXNfdDE9MQ=="
target="_blank"
rel="noopener noreferrer"
>continuous function</a
>
on a <em>closed and bounded</em> set $D$ must achieve its
absolute minimum and maximum.
</p>
</div>
<div class="fragment">
The upshot is:
<ul>
<li class="fragment">
Find critical points on interior of $D$.
</li>
<li class="fragment">
Find Lagrange points on boundary of $D$.
</li>
<li class="fragment">
Choose least/greatest value of $f$ on combined list.
</li>
</ul>
</div>
</section>
<section>
<h6 class="framelabel">Example</h6>
<div class="container">
<div class="col">
<p>
Suppose the temperature distribution on the closed half-disk
$0 \leq y \leq \sqrt{16-x^2}$ is given by \[u(x,y) = x^2 - 6x
+ 4y^2 - 8y. \] Find the hottest and coldest points.
</p>
</div>
<div class="col">
<img src="assets/half-disk-lagrange.png" alt="" />
</div>
</div>
<p></p>
</section>
</section>
<section>
<section><h1>Learning Outcomes</h1></section>
<section id="learning-outcomes">
<h6 class="framelabel">You should be able to...</h6>
<ul>
<li>
Model an optimization problem by identifying:
<ul>
<li>the target function.</li>
<li>the variables and domain of feasibility.</li>
<li>any constraint functions.</li>
</ul>
</li>
<li>
Identify whether a domain is open, closed, or neither, and where
to look for extrema.
</li>
<li>
Set up and solve systems of equations for the method of Lagrange
multipliers.
</li>
<li>Interpret the Lagrange equations geometrically.</li>
<li>
Utilize the Extreme Value Theorem to find global minima and
maxima.
</li>
</ul>
</section>
</section>
</div>
</div>
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