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testStreamLit.htm
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<html>
<head>
<meta http-equiv=Content-Type content="text/html; charset=windows-1252">
<meta name=Generator content="Microsoft Word 15 (filtered)">
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<div class=WordSection1>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><b><span style='font-size:18.0pt;font-family:Roboto-Bold'>Assignment 4</span></b></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><b><span style='font-size:18.0pt;font-family:Roboto-Bold'> </span></b></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><b><span style='font-size:14.0pt;font-family:Roboto-Bold'>Q1. Scaling (30
points)</span></b></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Consider the
following optimization problem:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal;text-autospace:none'><img width=356 height=78 id="Picture 1"
src="testStreamLit_files/image001.jpg"
alt="Diagram Description automatically generated"></p>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;text-indent:-.25in;
line-height:normal;text-autospace:none'><span style='font-size:12.0pt;
font-family:Roboto-Regular'>(a)<span style='font:7.0pt "Times New Roman"'>
</span></span><span style='font-size:12.0pt;font-family:Roboto-Regular'>Find
the optimal solution to the problem.</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>First
we find the jacobian of the lagragian of the problem and set it equal to zero:</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=388 height=80 src="testStreamLit_files/image002.png"></span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>By
observing the first three equations, and setting the inside of the parenthesis equal
to zero it will lead to 3 different local minimum as shown below:</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=426 height=58 src="testStreamLit_files/image003.png"></span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Clearly
the optimal solution is <b>X*</b> = [0,1,0], J(<b>X*</b>) = -0.001</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>The
hessian is shown below:</span></p>
<p class=MsoListParagraphCxSpMiddle align=center style='margin-bottom:0in;
text-align:center;line-height:normal;text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=360 height=77 src="testStreamLit_files/image004.png"></span></p>
<p class=MsoListParagraphCxSpMiddle align=center style='margin-bottom:0in;
text-align:center;line-height:normal;text-autospace:none'><span
style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>The
eigenvalues at X* are [2, -2, 2.002, 2000.002]</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Solving
this using newton’s method the result (<b>X*</b>) was achieved in 7 iterations
however the X* had slightly errors built in:</span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>X1
=1.38619e-26, X2 = 1, X3 = 4.44748e-47, </span><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;position:relative;
top:4.0pt'><img width=9 height=20 src="testStreamLit_files/image005.png"></span><span
style='font-size:12.0pt;font-family:Roboto-Regular'> = 0.001, J = -0.001</span></p>
<span style='font-size:12.0pt;line-height:107%;font-family:Roboto-Regular'><br
clear=all style='page-break-before:always'>
</span>
<p class=MsoNormal><span style='font-size:12.0pt;line-height:107%;font-family:
Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;text-indent:-.25in;
line-height:normal;text-autospace:none'><span style='font-size:12.0pt;
font-family:Roboto-Regular'>(b)<span style='font:7.0pt "Times New Roman"'>
</span></span><span style='font-size:12.0pt;font-family:Roboto-Regular'>Find a
non-singular transformation x=Ly such that the condition number of the Hessian
matrix of f is close to unity.</span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>First
lets define X as:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=73 height=53 src="testStreamLit_files/image006.png"></span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Transforming
the minimizer and the constraints we have:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=265 height=21 src="testStreamLit_files/image007.png"></span><br>
<span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=222 height=21 src="testStreamLit_files/image008.png"></span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Solving for the
Jacobian and Hessian:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=243 height=85 src="testStreamLit_files/image009.png"></span><br>
<span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=466 height=85 src="testStreamLit_files/image010.png"></span><span
style='font-size:12.0pt;font-family:Roboto-Regular'>clearly ‘a’ should be set
to 1/sqrt(2), ‘b’ and ‘c’ are a little more complicated to define. Since the magnitude
within the parenthesis in the diagonal will not change independently of ‘b’ or
‘c’, what it was done was to have both components with in the parenthesis to be
the same distance away from one. Since the units are separated in the 1
thousand, it was taken sqrt(1000) and since b is squared it was taken yet
another square root, finally it was divided by sqrt(2) to offset the 2 in front.
In summary:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=243 height=68 src="testStreamLit_files/image011.png"></span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>The hessian
becomes:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=489 height=77 src="testStreamLit_files/image012.png"></span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<span style='font-size:12.0pt;line-height:107%;font-family:Roboto-Regular'><br
clear=all style='page-break-before:always'>
</span>
<p class=MsoNormal><span style='font-size:12.0pt;line-height:107%;font-family:
Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;text-indent:-.25in;
line-height:normal;text-autospace:none'><span style='font-size:12.0pt;
font-family:Roboto-Regular'>(c)<span style='font:7.0pt "Times New Roman"'>
</span></span><span style='font-size:12.0pt;font-family:Roboto-Regular'>Quantify
the effect of rescaling the problem, either on the number of iterations or function
evaluations required to find the solution (convergence), or the quality of the
optimal solution itself, or a combination of the two.</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Counter
to my intuition, rescaling the system did not alter the number of iteration
needed to converge. The yield result was the exact same J(X) = -0.001, and the
quality of the result remained unchanged and the values of X are remarkably near
to those before scaling.</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=366 height=52 src="testStreamLit_files/image013.png"></span><span
style='font-size:12.0pt;font-family:Roboto-Bold'><br>
<br clear=all style='page-break-before:always'>
</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><b><span style='font-size:14.0pt;font-family:Roboto-Bold'>Q2.
Isoperformance (30 points)</span></b></p>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>The
Jacobian matrix at a point x</span><span style='font-size:7.0pt;font-family:
Roboto-Regular'>0 </span><span style='font-size:12.0pt;font-family:Roboto-Regular'>is
given as:</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><img width=624 height=186 id="Picture 10"
src="testStreamLit_files/image014.jpg"
alt="Chart, scatter chart Description automatically generated"></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;text-indent:-.25in;
line-height:normal;text-autospace:none'><span style='font-size:12.0pt;
font-family:Roboto-Regular'>(a)<span style='font:7.0pt "Times New Roman"'>
</span></span><span style='font-size:12.0pt;font-family:Roboto-Regular'>What
are the performance invariant directions that we can step to from x</span><span
style='font-size:7.0pt;font-family:Roboto-Regular'>o</span><span
style='font-size:12.0pt;font-family:Roboto-Regular'>?</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>By
calculating the null space of the Jacobian of J we have the 2 orthonormal
vectors of isoperformance. In python it was used the scipy.linalg.null_space
library to calculate the null space and it can be seen below. As we have 6 variables
and 4 equations, therefore we have 2 DOF, so it is expected two vectors.</span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<div align=center>
<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width=192
style='width:2.0in;border-collapse:collapse'>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'> </span></p>
</td>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-left:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t1</span></p>
</td>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-left:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t2</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>X1</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.58988</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.27646</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>X2</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.24959</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.910574</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>X3</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.015811</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.06546</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>X4</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.607658</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.193909</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>X5</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.294589</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.226</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>X6</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.36534</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.03823</span></p>
</td>
</tr>
</table>
</div>
<p class=MsoListParagraph style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<span style='font-size:12.0pt;line-height:107%;font-family:Roboto-Regular'><br
clear=all style='page-break-before:always'>
</span>
<p class=MsoNormal><span style='font-size:12.0pt;line-height:107%;font-family:
Roboto-Regular'> </span></p>
<p class=MsoListParagraph style='margin-bottom:0in;text-indent:-.25in;
line-height:normal;text-autospace:none'><span style='font-size:12.0pt;
font-family:Roboto-Regular'>(b)<span style='font:7.0pt "Times New Roman"'>
</span></span><span style='font-size:12.0pt;font-family:Roboto-Regular'>Show an
example of a step direction (vector), Δx, such that J</span><span
style='font-size:12.0pt;font-family:MS-PGothic'>(x+</span><span
style='font-size:12.0pt;font-family:Roboto-Regular'>Δx) ≈ 10</span><span
style='font-size:7.0pt;font-family:Roboto-Regular'>-4</span><span
style='font-size:12.0pt;font-family:Roboto-Regular'>.</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>To
calculate Δx, we can use the equation below:</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=148 height=20 src="testStreamLit_files/image015.png"></span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Where
</span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:4.0pt'><img width=15 height=21
src="testStreamLit_files/image016.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'> is the direction in which we are moving and </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:4.0pt'><img width=18 height=21
src="testStreamLit_files/image017.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'> is the step size, the direction, </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:4.0pt'><img width=15 height=21
src="testStreamLit_files/image016.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'>, was defined in the previous question using the null
space.</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>It
is possible to re-arrange the equation below:</span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=274 height=37 src="testStreamLit_files/image018.png"></span><span
lang=PT-BR style='font-size:12.0pt;font-family:Roboto-Regular'><br>
Assuming:</span></p>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><span lang=PT-BR style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpFirst align=center style='margin-bottom:0in;
text-align:center;line-height:normal;text-autospace:none'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=79 height=20 src="testStreamLit_files/image019.png"></span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Then:</span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=285 height=37 src="testStreamLit_files/image020.png"></span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>Assuming
the Hessian is properly scaled and approximating it as the identity matrix. It
is possible to identify many step sizes </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:4.0pt'><img width=18 height=21
src="testStreamLit_files/image017.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'> such that </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:7.0pt'><img width=57 height=28
src="testStreamLit_files/image021.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'> ≈ 10</span><span style='font-size:7.0pt;
font-family:Roboto-Regular'>-4</span><span style='font-size:12.0pt;font-family:
Roboto-Regular'>. 2 of those can be seen in the table below:</span></p>
<div align=center>
<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width=199
style='width:149.0pt;border-collapse:collapse'>
<tr style='height:15.0pt'>
<td width=71 nowrap valign=bottom style='width:53.0pt;padding:0in 5.4pt 0in 5.4pt;
height:15.0pt'></td>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t1</span></p>
</td>
<td width=64 nowrap style='width:48.0pt;border:solid windowtext 1.0pt;
border-left:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t2</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=71 nowrap style='width:53.0pt;border:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>ΔX1</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.0059</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.00276</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=71 nowrap style='width:53.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>ΔX2</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.0025</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.009106</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=71 nowrap style='width:53.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>ΔX3</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.000158</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.00065</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=71 nowrap style='width:53.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>ΔX4</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.006077</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.001939</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=71 nowrap style='width:53.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>ΔX5</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>0.002946</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.00226</span></p>
</td>
</tr>
<tr style='height:15.0pt'>
<td width=71 nowrap style='width:53.0pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>ΔX6</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.00365</span></p>
</td>
<td width=64 nowrap valign=bottom style='width:48.0pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=right style='margin-bottom:0in;text-align:right;
line-height:normal'><span style='color:black'>-0.00038</span></p>
</td>
</tr>
</table>
</div>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>To
verify the assumption </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:4.0pt'><img width=79 height=20
src="testStreamLit_files/image019.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'> these were calculated, and shown in the table
below. The values are either zero or below machine precision. </span></p>
<div align=center>
<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width=253
style='width:189.85pt;border-collapse:collapse'>
<tr style='height:15.0pt'>
<td width=96 nowrap valign=bottom style='width:71.75pt;padding:0in 5.4pt 0in 5.4pt;
height:15.0pt'></td>
<td width=79 nowrap style='width:59.05pt;border:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t1</span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border:solid windowtext 1.0pt;
border-left:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t2</span></p>
</td>
</tr>
<tr style='height:15.75pt'>
<td width=96 nowrap valign=bottom style='width:71.75pt;border:solid windowtext 1.0pt;
border-right:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal'><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>∇</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>J1</span><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>・</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>Δx </span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>0.00E+00</span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border-top:none;border-left:none;
border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>8.67E-19</span></p>
</td>
</tr>
<tr style='height:15.75pt'>
<td width=96 nowrap valign=bottom style='width:71.75pt;border-top:none;
border-left:solid windowtext 1.0pt;border-bottom:solid windowtext 1.0pt;
border-right:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal'><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>∇</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>J2</span><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>・</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>Δx </span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>0</span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border-top:none;border-left:none;
border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>-2.04E-17</span></p>
</td>
</tr>
<tr style='height:15.75pt'>
<td width=96 nowrap valign=bottom style='width:71.75pt;border-top:none;
border-left:solid windowtext 1.0pt;border-bottom:solid windowtext 1.0pt;
border-right:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal'><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>∇</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>J3</span><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>・</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>Δx </span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>1.39E-17</span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border-top:none;border-left:none;
border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>-1.52E-17</span></p>
</td>
</tr>
<tr style='height:15.75pt'>
<td width=96 nowrap valign=bottom style='width:71.75pt;border-top:none;
border-left:solid windowtext 1.0pt;border-bottom:solid windowtext 1.0pt;
border-right:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal'><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>∇</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>J4</span><span
style='font-size:12.0pt;font-family:MS-PGothic;color:black'>・</span><span
style='font-size:12.0pt;font-family:Roboto-Regular;color:black'>Δx </span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>-6.51E-18</span></p>
</td>
<td width=79 nowrap style='width:59.05pt;border-top:none;border-left:none;
border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>-8.08E-18</span></p>
</td>
</tr>
</table>
</div>
<p class=MsoListParagraphCxSpFirst style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'> </span></p>
<p class=MsoListParagraphCxSpMiddle style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'>For
each of these vectors a different step size, </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:4.0pt'><img width=18 height=21
src="testStreamLit_files/image017.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'>, was determined starting at 1 and dividing by 10
each iteration until </span><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif;
position:relative;top:7.0pt'><img width=57 height=28
src="testStreamLit_files/image021.png"></span><span style='font-size:12.0pt;
font-family:Roboto-Regular'>term was smaller than 10e-4.</span></p>
<p class=MsoListParagraphCxSpLast style='margin-bottom:0in;line-height:normal;
text-autospace:none'><span style='font-size:12.0pt;font-family:Roboto-Regular'><br>
<br>
</span></p>
<div align=center>
<table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width=208
style='width:155.9pt;border-collapse:collapse'>
<tr style='height:15.0pt'>
<td width=56 nowrap style='width:42.1pt;border:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'> </span></p>
</td>
<td width=74 nowrap style='width:55.7pt;border:solid windowtext 1.0pt;
border-left:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t1</span></p>
</td>
<td width=77 nowrap style='width:58.1pt;border:solid windowtext 1.0pt;
border-left:none;padding:0in 5.4pt 0in 5.4pt;height:15.0pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>t2</span></p>
</td>
</tr>
<tr style='height:15.75pt'>
<td width=56 nowrap valign=bottom style='width:42.1pt;border:solid windowtext 1.0pt;
border-top:none;padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal'><span
style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif'><img
width=18 height=21 src="testStreamLit_files/image017.png"></span></p>
</td>
<td width=74 nowrap valign=bottom style='width:55.7pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>0.01</span></p>
</td>
<td width=77 nowrap valign=bottom style='width:58.1pt;border-top:none;
border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;
padding:0in 5.4pt 0in 5.4pt;height:15.75pt'>
<p class=MsoNormal align=center style='margin-bottom:0in;text-align:center;
line-height:normal'><span style='color:black'>0.01</span></p>
</td>
</tr>
</table>
</div>
<p class=MsoNormal style='margin-bottom:0in;line-height:normal;text-autospace:
none'><b><span style='font-size:18.0pt;font-family:Roboto-Bold'> </span></b></p>
</div>
</body>
</html>