From 752d51fc21be0c049d4d54d0ce87271cde462251 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Sun, 9 Apr 2023 16:59:34 -0400 Subject: [PATCH] typo --- psets/pset5.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/psets/pset5.ipynb b/psets/pset5.ipynb index 3e29253..0e5f30b 100644 --- a/psets/pset5.ipynb +++ b/psets/pset5.ipynb @@ -44,7 +44,7 @@ "$$\n", "for some $r > 0$, $m \\times n$ matrix $A$ (of rank $n$), and $b \\in \\mathbb{R}^m$ — that is, least-squares optimization with the solution constrained to lie inside a sphere of radius $r$.\n", "\n", - "**(a)** What is the Lagrange dual function $g(\\lambda)$? (You can give a closed-form expression. Hint: review Tikhonov-regularized least-squares.) Define a corresponding Julia function `g(λ; r=1.0)` for the sample parameters given below (this syntax defines an optional keyword argument `r` that defaults to $r=1$). Make a plot of $g(\\lambda)$ for $r=1$ and $r=0.5$ for $\\lambda \\ge 0$$ to verify that it looks concave with a single maximum.\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 $r=0.5$ for $\\lambda \\ge 0$ 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",