update on jupyter-book
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@@ -77,3 +77,13 @@ found info about 5 exercises
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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output in chapter1.ipynb
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Translating doconce text in chapter1.do.txt to ipynb
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*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
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found info about 5 exercises
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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output in chapter1.ipynb
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@@ -660,7 +660,7 @@ import os
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "datafiles/"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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@@ -1102,7 +1102,7 @@ import os
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "datafiles/"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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@@ -1607,7 +1607,7 @@ from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "datafiles/"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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@@ -1771,7 +1771,7 @@ from sklearn.model_selection import train_test_split
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "datafiles/"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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@@ -300,25 +300,22 @@ ensures convexity of a function $f$. We write $D_f$ to denote the
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domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more
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details and proofs we refer to: "S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press":"http://stanford.edu/boyd/cvxbook/, 2004".
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!bblock First order condition
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Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for
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_First order condition_: Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for
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all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$
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is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
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is a convex set and $f(y) \geq f(x) + \nabla f(x)^T (y-x)$ holds
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for all $x,y \in D_f$. This condition means that for a convex function
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the first order Taylor expansion (right hand side above) at any point
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is a global under estimator of the function. To convince yourself you can
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make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and
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note that it is always below the graph.
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!eblock
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!bblock Second order condition
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Assume that $f$ is twice
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_Second order condition_: Assume that $f$ is twice
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differentiable, i.e the Hessian matrix exists at each point in
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$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its
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Hessian is positive semi-definite for all $x\in D_f$. For a
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single-variable function this reduces to $f''(x) \geq 0$. Geometrically this means that $f$ has nonnegative curvature
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everywhere.
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!eblock
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This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
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@@ -1705,6 +1702,7 @@ Autograd supports many features. However, there are some functions that is not s
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Assigning a value to the variable being differentiated with respect to
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!bc pycod
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"""
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import autograd.numpy as np
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from autograd import grad
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def f8(x): # Assume x is an array
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@@ -1716,6 +1714,7 @@ f8_grad = grad(f8)
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x = 8.4
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print("The derivative of f8 is:",f8_grad(x))
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"""
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!ec
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Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
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