View source for QE2013 AC-3 ECE580-1 - Rhea

[[Category:ECE]]
[[Category:QE]]
[[Category:CNSIP]]
[[Category:problem solving]]
[[Category:automatic control]]
[[Category:optimization]]

=QE2013_AC-3_ECE580-1=

:[[QE2013_AC-3_ECE580-1|Part 1]],[[QE2013_AC-3_ECE580-2|2]],[[QE2013_AC-3_ECE580-3|3]],[[QE2013_AC-3_ECE580-4|4]],[[QE2013_AC-3_ECE580-5|5]]

'''(i)'''
<br> '''Solution 1: ''' <br>
<math>\alpha_k</math> is the solution to <math>{d \over d\alpha}f(x^{(k)} + \alpha d^{(k)}) = 0 </math>

<math>{d \over d\alpha}f(x^{(k)} + \alpha d^{(k)}) = (x^{(k)T} + \alpha d^{(k)T}) Q d^{(k)} - d^{(k)T} b = 0</math>

<math>\therefore \alpha d^{(k)T} Q d^{(k)} = -x^{(k)T} Q d^{(k)} + d^{(k)T} b = (b - Qx^{(k)})^T d^{(k)} = - g^{(k)T} d^{(k)}</math>

<math>\therefore \alpha_k = - \frac {g^{(k)T} d^{(k)}} {d^{(k)T} Q d^{(k)}} </math>

<br> '''Solution 2: ''' <br>
Let <math> \Phi (\alpha) = f(x^{(k)} + \alpha d^{(k)}) </math>

then <math> {d \over d\alpha} \Phi (\alpha) = d^{(k)T} \nabla f(x^{(k)} + \alpha d^{(k)}) </math>

<math>f(x) = {1 \over 2} x^T Q x - x^T b + c, Q = Q^T > 0</math>

<math>\nabla f(x) = Qx - b</math>

so 

<math> \begin{align} 
{d \over d\alpha} \Phi (\alpha) & = d^{(k)T} (Q(x^{(k)} + \alpha d^{(k)}) - b) \\
& = d^{(k)T} Q x^{(k)} + \alpha d^{(k)T} Q d^{(k)} - d^{(k)T} b \\
& = d^{(k)T} (Q x^{(k)} - b) + \alpha d^{(k)T} Q d^{(k)} \\
& = d^{(k)T} g^{(k)} + \alpha d^{(k)T} Q d^{(k)} = 0 \\
\end{align} </math>

<math>\alpha_k = - \frac {d^{(k)T} g^{(k)}} {d^{(k)T} Q d^{(k)}} = - \frac {g^{(k)T} d^{(k)}} {d^{(k)T} Q d^{(k)}} = \frac {g^{(k)T} H_k g^{(k)}} {d^{(k)T} Q d^{(k)}}</math>

'''(ii)'''
<br> '''Solution 1: ''' <br>
<math>\because Q > 0,\ \therefore d^{(k)T} Q d^{(k)} > 0</math>

<math>\therefore \alpha_k > 0 \Leftrightarrow -g^{(k)T} d^{(k)} = g^{(k)T} H_k g^{(k)} > 0</math>

Therefore a sufficient condition is <math>H_k</math> is positive definite.

<br> '''Solution 2: ''' <br>

<math>\alpha_k > 0 \Rightarrow H_k = H_k^T > 0</math>

<br> '''Comment: ''' <br> Both solutions are similar and have the same result. <br>
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