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Update pset5.ipynb

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Steven G. Johnson 2023-04-09 16:56:38 -04:00 committed by GitHub
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@ -44,7 +44,7 @@
"$$\n",
"for some $r > 0$, $m \\times n$ matrix $A$ (of rank $n$), and $b \\in \\mathbb{R}^m$ — that is, least-squares optimization with the solution constrained to lie inside a sphere of radius $r$.\n",
"\n",
"**(a)** What is the Lagrange dual function $g(\\lambda)$? (You can give a closed-form expression. Hint: review Tikhonov-regularized least-squares.) Define a corresponding Julia function `g(λ; r=1.0)` for the sample parameters given below (this syntax defines an optional keyword argument `r` that defaults to $r=1$).\n",
"**(a)** What is the Lagrange dual function $g(\\lambda)$? (You can give a closed-form expression. Hint: review Tikhonov-regularized least-squares.) Define a corresponding Julia function `g(λ; r=1.0)` for the sample parameters given below (this syntax defines an optional keyword argument `r` that defaults to $r=1$). Make a plot of $$g(\\lambda)$$ for $$r=1$$ and $$r=0.5$$ to verify that it looks concave with a single maximum.\n",
"\n",
"**(b)** If the unconstrained least-square solution $\\hat{x} = (A^T A)^{-1} A^T b$ satisfies $\\Vert \\hat{x} \\Vert_2 < r$, then what must be true of the derivative $g'(0)$? What if $\\Vert \\hat{x} \\Vert_2 > r$?\n",
"\n",