Morten Hjorth-Jensen
602 lines
47 KiB
HTML
602 lines
47 KiB
HTML
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('Thursday September 9', 2, None, 'thursday-september-9'),
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('More interpretations', 2, None, 'more-interpretations'),
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('Deriving the Lasso Regression Equations',
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('Simple example to illustrate Ordinary Least Squares, Ridge and '
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('Writing the Cost Function',
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('The first Case', 2, None, 'the-first-case'),
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('Simple code for solving the above problem',
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2,
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('With Lasso Regression', 2, None, 'with-lasso-regression'),
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('Another Example, now with a polynomial fit',
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2,
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('Using CVXOPT', 2, None, 'using-cvxopt'),
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("Example of Usage of Bayes' theorem",
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("Invoking Bayes' theorem", 2, None, 'invoking-bayes-theorem'),
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<a class="navbar-brand" href="week36-bs.html">Week 36: Statistical interpretation of Linear Regression and Resampling techniques</a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week36-bs001.html#plans-for-week-36" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs002.html#thursday-september-9" style="font-size: 80%;">Thursday September 9</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.html#summary-from-last-week-and-examples" style="font-size: 80%;">Summary from last Week and Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#linear-regression-and-the-svd" style="font-size: 80%;">Linear Regression and the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#what-does-it-mean" style="font-size: 80%;">What does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;">And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#code-for-svd-and-inversion-of-matrices" style="font-size: 80%;">Code for SVD and Inversion of Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#inverse-of-rectangular-matrix" style="font-size: 80%;">Inverse of Rectangular Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs009.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs010.html#from-ols-to-ridge-and-lasso" style="font-size: 80%;">From OLS to Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#note-on-scikit-learn" style="font-size: 80%;">Note on Scikit-Learn</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#comparison-with-ols" style="font-size: 80%;">Comparison with OLS</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression" style="font-size: 80%;">Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#writing-the-cost-function" style="font-size: 80%;">Writing the Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#simple-code-for-solving-the-above-problem" style="font-size: 80%;">Simple code for solving the above problem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#another-example-now-with-a-polynomial-fit" style="font-size: 80%;">Another Example, now with a polynomial fit</a></li>
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<!-- navigation toc: --> <li><a href="#using-cvxopt" style="font-size: 80%;">Using CVXOPT</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#the-simpler-example" style="font-size: 80%;">The simpler Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#friday-september-10" style="font-size: 80%;">Friday September 10</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#linking-the-regression-analysis-with-a-statistical-interpretation" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.html#expectation-value-and-variance" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs036.html#expectation-value-and-variance-for-boldsymbol-beta" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs037.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs038.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs039.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs040.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs041.html#more-basic-statistics-and-bayes-theorem" style="font-size: 80%;">More basic Statistics and Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs042.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs045.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs046.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs049.html#test-function-for-what-happens-with-ols-ridge-and-lasso" style="font-size: 80%;">Test Function for what happens with OLS, Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs051.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs052.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0030"></a>
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<!-- !split -->
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<h2 id="using-cvxopt" class="anchor">Using CVXOPT </h2>
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<p>As a small addendum, we note that you can also solve this problem
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using the convex optimization package
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<a href="https://cvxopt.org/examples/mlbook/l1regls.html" target="_self">CVXOPT</a>. This
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requires, in addition to having installed <b>CVXOPT</b>, you need to
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download the file <em>l1regl.py</em>. The following code example solves the
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simpler problem we discussed above, where we have added the latter
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python file.
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</p>
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<p>We list first the needed file. Thereafter we set up the problem to solve.</p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="inner_cell">
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<div class="input_area">
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<div class="highlight" style="background: #f8f8f8">
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<pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> matrix, spdiag, mul, div, sqrt, normal, setseed
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> blas, lapack, solvers, sparse, spmatrix
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span>
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<span style="color: #008000; font-weight: bold">try</span>:
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">mosek</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sys</span>
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__MOSEK <span style="color: #666666">=</span> <span style="color: #008000; font-weight: bold">True</span>
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<span style="color: #008000; font-weight: bold">except</span>: __MOSEK <span style="color: #666666">=</span> <span style="color: #008000; font-weight: bold">False</span>
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<span style="color: #008000; font-weight: bold">if</span> __MOSEK:
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">l1regls_mosek</span>(A, b):
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<span style="color: #BA2121; font-style: italic">"""</span>
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<span style="color: #BA2121; font-style: italic"> Returns the solution of l1-norm regularized least-squares problem</span>
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<span style="color: #BA2121; font-style: italic"> minimize || A*x - b ||_2^2 + e'*u</span>
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<span style="color: #BA2121; font-style: italic"> subject to -u <= x <= u</span>
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<span style="color: #BA2121; font-style: italic"> """</span>
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m, n <span style="color: #666666">=</span> A<span style="color: #666666">.</span>size
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env <span style="color: #666666">=</span> mosek<span style="color: #666666">.</span>Env()
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task <span style="color: #666666">=</span> env<span style="color: #666666">.</span>Task(<span style="color: #666666">0</span>,<span style="color: #666666">0</span>)
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task<span style="color: #666666">.</span>set_Stream(mosek<span style="color: #666666">.</span>streamtype<span style="color: #666666">.</span>log, <span style="color: #008000; font-weight: bold">lambda</span> x: sys<span style="color: #666666">.</span>stdout<span style="color: #666666">.</span>write(x))
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task<span style="color: #666666">.</span>appendvars( <span style="color: #666666">2*</span>n) <span style="color: #408080; font-style: italic"># number of variables</span>
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task<span style="color: #666666">.</span>appendcons( <span style="color: #666666">2*</span>n) <span style="color: #408080; font-style: italic"># number of constraints</span>
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<span style="color: #408080; font-style: italic"># input quadratic objective</span>
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Q <span style="color: #666666">=</span> matrix(<span style="color: #666666">0.0</span>, (n,n))
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blas<span style="color: #666666">.</span>syrk(A, Q, alpha <span style="color: #666666">=</span> <span style="color: #666666">2.0</span>, trans<span style="color: #666666">=</span><span style="color: #BA2121">'T'</span>)
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I <span style="color: #666666">=</span> []
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
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I<span style="color: #666666">.</span>extend(<span style="color: #008000">range</span>(i,n))
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J <span style="color: #666666">=</span> []
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
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J<span style="color: #666666">.</span>extend((n<span style="color: #666666">-</span>i)<span style="color: #666666">*</span>[i])
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task<span style="color: #666666">.</span>putqobj(I, J, <span style="color: #008000">list</span>(Q[matrix(I) <span style="color: #666666">+</span> matrix(J)<span style="color: #666666">*</span>n]))
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task<span style="color: #666666">.</span>putclist(<span style="color: #008000">range</span>(<span style="color: #666666">2*</span>n), <span style="color: #008000">list</span>(<span style="color: #666666">-2*</span>A<span style="color: #666666">.</span>T<span style="color: #666666">*</span>b) <span style="color: #666666">+</span> n<span style="color: #666666">*</span>[<span style="color: #666666">1.0</span>]) <span style="color: #408080; font-style: italic"># setup linear objective</span>
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<span style="color: #408080; font-style: italic"># input constraint matrix row by row</span>
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
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task<span style="color: #666666">.</span>putarow( i, [i, n<span style="color: #666666">+</span>i], [<span style="color: #666666">1.0</span>, <span style="color: #666666">-1.0</span>])
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task<span style="color: #666666">.</span>putarow( n<span style="color: #666666">+</span>i, [i, n<span style="color: #666666">+</span>i], [<span style="color: #666666">1.0</span>, <span style="color: #666666">1.0</span>])
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<span style="color: #408080; font-style: italic"># setup bounds on constraints</span>
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task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>con,
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<span style="color: #666666">0</span>, n, n<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>up], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>])
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task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>con,
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n, <span style="color: #666666">2*</span>n, n<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>lo], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>])
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<span style="color: #408080; font-style: italic"># setup variable bounds</span>
|
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task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>var,
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<span style="color: #666666">0</span>, <span style="color: #666666">2*</span>n, <span style="color: #666666">2*</span>n<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>fr], <span style="color: #666666">2*</span>n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>], <span style="color: #666666">2*</span>n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>])
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<span style="color: #408080; font-style: italic"># optimize the task</span>
|
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task<span style="color: #666666">.</span>putobjsense(mosek<span style="color: #666666">.</span>objsense<span style="color: #666666">.</span>minimize)
|
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task<span style="color: #666666">.</span>optimize()
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task<span style="color: #666666">.</span>solutionsummary(mosek<span style="color: #666666">.</span>streamtype<span style="color: #666666">.</span>log)
|
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x <span style="color: #666666">=</span> n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>]
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task<span style="color: #666666">.</span>getsolutionslice(mosek<span style="color: #666666">.</span>soltype<span style="color: #666666">.</span>itr, mosek<span style="color: #666666">.</span>solitem<span style="color: #666666">.</span>xx, <span style="color: #666666">0</span>, n, x)
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<span style="color: #008000; font-weight: bold">return</span> matrix(x)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">l1regls_mosek2</span>(A, b):
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<span style="color: #BA2121; font-style: italic">"""</span>
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<span style="color: #BA2121; font-style: italic"> Returns the solution of l1-norm regularized least-squares problem</span>
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<span style="color: #BA2121; font-style: italic"> minimize w'*w + e'*u</span>
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<span style="color: #BA2121; font-style: italic"> subject to -u <= x <= u</span>
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<span style="color: #BA2121; font-style: italic"> A*x - w = b</span>
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<span style="color: #BA2121; font-style: italic"> """</span>
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m, n <span style="color: #666666">=</span> A<span style="color: #666666">.</span>size
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env <span style="color: #666666">=</span> mosek<span style="color: #666666">.</span>Env()
|
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task <span style="color: #666666">=</span> env<span style="color: #666666">.</span>Task(<span style="color: #666666">0</span>,<span style="color: #666666">0</span>)
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task<span style="color: #666666">.</span>set_Stream(mosek<span style="color: #666666">.</span>streamtype<span style="color: #666666">.</span>log, <span style="color: #008000; font-weight: bold">lambda</span> x: sys<span style="color: #666666">.</span>stdout<span style="color: #666666">.</span>write(x))
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task<span style="color: #666666">.</span>appendvars(<span style="color: #666666">2*</span>n <span style="color: #666666">+</span> m) <span style="color: #408080; font-style: italic"># number of variables</span>
|
|
task<span style="color: #666666">.</span>appendcons(<span style="color: #666666">2*</span>n <span style="color: #666666">+</span> m) <span style="color: #408080; font-style: italic"># number of constraints</span>
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<span style="color: #408080; font-style: italic"># input quadratic objective</span>
|
|
task<span style="color: #666666">.</span>putqobj(<span style="color: #008000">range</span>(<span style="color: #666666">2*</span>n,<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m), <span style="color: #008000">range</span>(<span style="color: #666666">2*</span>n,<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m), m<span style="color: #666666">*</span>[<span style="color: #666666">2.0</span>])
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task<span style="color: #666666">.</span>putclist(<span style="color: #008000">range</span>(<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m), n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>] <span style="color: #666666">+</span> n<span style="color: #666666">*</span>[<span style="color: #666666">1.0</span>] <span style="color: #666666">+</span> m<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>]) <span style="color: #408080; font-style: italic"># setup linear objective</span>
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|
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<span style="color: #408080; font-style: italic"># input constraint matrix row by row</span>
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
|
|
task<span style="color: #666666">.</span>putarow( i, [i, n<span style="color: #666666">+</span>i], [<span style="color: #666666">1.0</span>, <span style="color: #666666">-1.0</span>])
|
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task<span style="color: #666666">.</span>putarow( n<span style="color: #666666">+</span>i, [i, n<span style="color: #666666">+</span>i], [<span style="color: #666666">1.0</span>, <span style="color: #666666">1.0</span>])
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
|
|
task<span style="color: #666666">.</span>putarow( <span style="color: #666666">2*</span>n<span style="color: #666666">+</span>i, <span style="color: #008000">range</span>(n) <span style="color: #666666">+</span> [<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>i], <span style="color: #008000">list</span>(A[i,:]) <span style="color: #666666">+</span> [<span style="color: #666666">-1.0</span>])
|
|
|
|
<span style="color: #408080; font-style: italic"># setup bounds on constraints</span>
|
|
task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>con,
|
|
<span style="color: #666666">0</span>, n, n<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>up], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>])
|
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task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>con,
|
|
n, <span style="color: #666666">2*</span>n, n<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>lo], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>], n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>])
|
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task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>con,
|
|
<span style="color: #666666">2*</span>n, <span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m, m<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>fx], <span style="color: #008000">list</span>(b), <span style="color: #008000">list</span>(b))
|
|
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|
<span style="color: #408080; font-style: italic"># setup variable bounds</span>
|
|
task<span style="color: #666666">.</span>putboundslice(mosek<span style="color: #666666">.</span>accmode<span style="color: #666666">.</span>var, <span style="color: #666666">0</span>, <span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m, (<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m)<span style="color: #666666">*</span>[mosek<span style="color: #666666">.</span>boundkey<span style="color: #666666">.</span>fr],
|
|
(<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m)<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>], (<span style="color: #666666">2*</span>n<span style="color: #666666">+</span>m)<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>])
|
|
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<span style="color: #408080; font-style: italic"># optimize the task</span>
|
|
task<span style="color: #666666">.</span>putobjsense(mosek<span style="color: #666666">.</span>objsense<span style="color: #666666">.</span>minimize)
|
|
task<span style="color: #666666">.</span>optimize()
|
|
task<span style="color: #666666">.</span>solutionsummary(mosek<span style="color: #666666">.</span>streamtype<span style="color: #666666">.</span>log)
|
|
x <span style="color: #666666">=</span> n<span style="color: #666666">*</span>[<span style="color: #666666">0.0</span>]
|
|
task<span style="color: #666666">.</span>getsolutionslice(mosek<span style="color: #666666">.</span>soltype<span style="color: #666666">.</span>itr, mosek<span style="color: #666666">.</span>solitem<span style="color: #666666">.</span>xx, <span style="color: #666666">0</span>, n, x)
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<span style="color: #008000; font-weight: bold">return</span> matrix(x)
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|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">l1regls</span>(A, b):
|
|
<span style="color: #BA2121; font-style: italic">"""</span>
|
|
<span style="color: #BA2121; font-style: italic"> </span>
|
|
<span style="color: #BA2121; font-style: italic"> Returns the solution of l1-norm regularized least-squares problem</span>
|
|
<span style="color: #BA2121; font-style: italic"> </span>
|
|
<span style="color: #BA2121; font-style: italic"> minimize || A*x - b ||_2^2 + || x ||_1.</span>
|
|
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|
<span style="color: #BA2121; font-style: italic"> """</span>
|
|
|
|
m, n <span style="color: #666666">=</span> A<span style="color: #666666">.</span>size
|
|
q <span style="color: #666666">=</span> matrix(<span style="color: #666666">1.0</span>, (<span style="color: #666666">2*</span>n,<span style="color: #666666">1</span>))
|
|
q[:n] <span style="color: #666666">=</span> <span style="color: #666666">-2.0</span> <span style="color: #666666">*</span> A<span style="color: #666666">.</span>T <span style="color: #666666">*</span> b
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">P</span>(u, v, alpha <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>, beta <span style="color: #666666">=</span> <span style="color: #666666">0.0</span> ):
|
|
<span style="color: #BA2121; font-style: italic">"""</span>
|
|
<span style="color: #BA2121; font-style: italic"> v := alpha * 2.0 * [ A'*A, 0; 0, 0 ] * u + beta * v </span>
|
|
<span style="color: #BA2121; font-style: italic"> """</span>
|
|
v <span style="color: #666666">*=</span> beta
|
|
v[:n] <span style="color: #666666">+=</span> alpha <span style="color: #666666">*</span> <span style="color: #666666">2.0</span> <span style="color: #666666">*</span> A<span style="color: #666666">.</span>T <span style="color: #666666">*</span> (A <span style="color: #666666">*</span> u[:n])
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|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">G</span>(u, v, alpha<span style="color: #666666">=1.0</span>, beta<span style="color: #666666">=0.0</span>, trans<span style="color: #666666">=</span><span style="color: #BA2121">'N'</span>):
|
|
<span style="color: #BA2121; font-style: italic">"""</span>
|
|
<span style="color: #BA2121; font-style: italic"> v := alpha*[I, -I; -I, -I] * u + beta * v (trans = 'N' or 'T')</span>
|
|
<span style="color: #BA2121; font-style: italic"> """</span>
|
|
|
|
v <span style="color: #666666">*=</span> beta
|
|
v[:n] <span style="color: #666666">+=</span> alpha<span style="color: #666666">*</span>(u[:n] <span style="color: #666666">-</span> u[n:])
|
|
v[n:] <span style="color: #666666">+=</span> alpha<span style="color: #666666">*</span>(<span style="color: #666666">-</span>u[:n] <span style="color: #666666">-</span> u[n:])
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h <span style="color: #666666">=</span> matrix(<span style="color: #666666">0.0</span>, (<span style="color: #666666">2*</span>n,<span style="color: #666666">1</span>))
|
|
|
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<span style="color: #408080; font-style: italic"># Customized solver for the KKT system </span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># [ 2.0*A'*A 0 I -I ] [x[:n] ] [bx[:n] ]</span>
|
|
<span style="color: #408080; font-style: italic"># [ 0 0 -I -I ] [x[n:] ] = [bx[n:] ].</span>
|
|
<span style="color: #408080; font-style: italic"># [ I -I -D1^-1 0 ] [zl[:n]] [bzl[:n]]</span>
|
|
<span style="color: #408080; font-style: italic"># [ -I -I 0 -D2^-1 ] [zl[n:]] [bzl[n:]]</span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># where D1 = W['di'][:n]**2, D2 = W['di'][:n]**2.</span>
|
|
<span style="color: #408080; font-style: italic"># </span>
|
|
<span style="color: #408080; font-style: italic"># We first eliminate zl and x[n:]:</span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># ( 2*A'*A + 4*D1*D2*(D1+D2)^-1 ) * x[:n] = </span>
|
|
<span style="color: #408080; font-style: italic"># bx[:n] - (D2-D1)*(D1+D2)^-1 * bx[n:] + </span>
|
|
<span style="color: #408080; font-style: italic"># D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] - </span>
|
|
<span style="color: #408080; font-style: italic"># D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:] </span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] ) </span>
|
|
<span style="color: #408080; font-style: italic"># - (D2-D1)*(D1+D2)^-1 * x[:n] </span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># zl[:n] = D1 * ( x[:n] - x[n:] - bzl[:n] )</span>
|
|
<span style="color: #408080; font-style: italic"># zl[n:] = D2 * (-x[:n] - x[n:] - bzl[n:] ).</span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># The first equation has the form</span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># (A'*A + D)*x[:n] = rhs</span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># and is equivalent to</span>
|
|
<span style="color: #408080; font-style: italic">#</span>
|
|
<span style="color: #408080; font-style: italic"># [ D A' ] [ x:n] ] = [ rhs ]</span>
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<span style="color: #408080; font-style: italic"># [ A -I ] [ v ] [ 0 ].</span>
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<span style="color: #408080; font-style: italic">#</span>
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<span style="color: #408080; font-style: italic"># It can be solved as </span>
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<span style="color: #408080; font-style: italic">#</span>
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<span style="color: #408080; font-style: italic"># ( A*D^-1*A' + I ) * v = A * D^-1 * rhs</span>
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<span style="color: #408080; font-style: italic"># x[:n] = D^-1 * ( rhs - A'*v ).</span>
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S <span style="color: #666666">=</span> matrix(<span style="color: #666666">0.0</span>, (m,m))
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Asc <span style="color: #666666">=</span> matrix(<span style="color: #666666">0.0</span>, (m,n))
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v <span style="color: #666666">=</span> matrix(<span style="color: #666666">0.0</span>, (m,<span style="color: #666666">1</span>))
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">Fkkt</span>(W):
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<span style="color: #408080; font-style: italic"># Factor </span>
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<span style="color: #408080; font-style: italic">#</span>
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<span style="color: #408080; font-style: italic"># S = A*D^-1*A' + I </span>
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<span style="color: #408080; font-style: italic">#</span>
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<span style="color: #408080; font-style: italic"># where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**-2, D2 = d[n:]**-2.</span>
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d1, d2 <span style="color: #666666">=</span> W[<span style="color: #BA2121">'di'</span>][:n]<span style="color: #666666">**2</span>, W[<span style="color: #BA2121">'di'</span>][n:]<span style="color: #666666">**2</span>
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<span style="color: #408080; font-style: italic"># ds is square root of diagonal of D</span>
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ds <span style="color: #666666">=</span> math<span style="color: #666666">.</span>sqrt(<span style="color: #666666">2.0</span>) <span style="color: #666666">*</span> div( mul( W[<span style="color: #BA2121">'di'</span>][:n], W[<span style="color: #BA2121">'di'</span>][n:]),
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sqrt(d1<span style="color: #666666">+</span>d2) )
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d3 <span style="color: #666666">=</span> div(d2 <span style="color: #666666">-</span> d1, d1 <span style="color: #666666">+</span> d2)
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<span style="color: #408080; font-style: italic"># Asc = A*diag(d)^-1/2</span>
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Asc <span style="color: #666666">=</span> A <span style="color: #666666">*</span> spdiag(ds<span style="color: #666666">**-1</span>)
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<span style="color: #408080; font-style: italic"># S = I + A * D^-1 * A'</span>
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blas<span style="color: #666666">.</span>syrk(Asc, S)
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S[::m<span style="color: #666666">+1</span>] <span style="color: #666666">+=</span> <span style="color: #666666">1.0</span>
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lapack<span style="color: #666666">.</span>potrf(S)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g</span>(x, y, z):
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x[:n] <span style="color: #666666">=</span> <span style="color: #666666">0.5</span> <span style="color: #666666">*</span> ( x[:n] <span style="color: #666666">-</span> mul(d3, x[n:]) <span style="color: #666666">+</span>
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mul(d1, z[:n] <span style="color: #666666">+</span> mul(d3, z[:n])) <span style="color: #666666">-</span> mul(d2, z[n:] <span style="color: #666666">-</span>
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mul(d3, z[n:])) )
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x[:n] <span style="color: #666666">=</span> div( x[:n], ds)
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<span style="color: #408080; font-style: italic"># Solve</span>
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<span style="color: #408080; font-style: italic">#</span>
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<span style="color: #408080; font-style: italic"># S * v = 0.5 * A * D^-1 * ( bx[:n] - </span>
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<span style="color: #408080; font-style: italic"># (D2-D1)*(D1+D2)^-1 * bx[n:] + </span>
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<span style="color: #408080; font-style: italic"># D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] - </span>
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<span style="color: #408080; font-style: italic"># D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:] )</span>
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blas<span style="color: #666666">.</span>gemv(Asc, x, v)
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lapack<span style="color: #666666">.</span>potrs(S, v)
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<span style="color: #408080; font-style: italic"># x[:n] = D^-1 * ( rhs - A'*v ).</span>
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blas<span style="color: #666666">.</span>gemv(Asc, v, x, alpha<span style="color: #666666">=-1.0</span>, beta<span style="color: #666666">=1.0</span>, trans<span style="color: #666666">=</span><span style="color: #BA2121">'T'</span>)
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x[:n] <span style="color: #666666">=</span> div(x[:n], ds)
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<span style="color: #408080; font-style: italic"># x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] ) </span>
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<span style="color: #408080; font-style: italic"># - (D2-D1)*(D1+D2)^-1 * x[:n] </span>
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x[n:] <span style="color: #666666">=</span> div( x[n:] <span style="color: #666666">-</span> mul(d1, z[:n]) <span style="color: #666666">-</span> mul(d2, z[n:]), d1<span style="color: #666666">+</span>d2 )\
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<span style="color: #666666">-</span> mul( d3, x[:n] )
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<span style="color: #408080; font-style: italic"># zl[:n] = D1^1/2 * ( x[:n] - x[n:] - bzl[:n] )</span>
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<span style="color: #408080; font-style: italic"># zl[n:] = D2^1/2 * ( -x[:n] - x[n:] - bzl[n:] ).</span>
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z[:n] <span style="color: #666666">=</span> mul( W[<span style="color: #BA2121">'di'</span>][:n], x[:n] <span style="color: #666666">-</span> x[n:] <span style="color: #666666">-</span> z[:n] )
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z[n:] <span style="color: #666666">=</span> mul( W[<span style="color: #BA2121">'di'</span>][n:], <span style="color: #666666">-</span>x[:n] <span style="color: #666666">-</span> x[n:] <span style="color: #666666">-</span> z[n:] )
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<span style="color: #008000; font-weight: bold">return</span> g
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<span style="color: #008000; font-weight: bold">return</span> solvers<span style="color: #666666">.</span>coneqp(P, q, G, h, kktsolver <span style="color: #666666">=</span> Fkkt)[<span style="color: #BA2121">'x'</span>][:n]
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</pre>
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