To use GGPLAB you need Matlab 6.1 or later. CVX: A system for disciplined convex programming Introduction.
Both SuperSCS and SCS were compiled using gcc v4.8.5 and accessed via MATLAB® using a MEX interface. In contrast, the CVX specification isgiven bywhich asks for the norm to be maximized, you will get an error messagestating that a convex function cannot be maximized (at least indisciplined convex programming):With CVX, the same problem is specified as follows:The Huber penalty function is convex, and has been provided in theCVX function library. CVX is a Matlab-based modeling system for convex optimization. Powered by Wordpress and MathJax.Wordpress and MathJax. Setting maximum number of iterations in CVX (Matlab) Ask Question Asked 3 years, 4 months ago. Asz = bs z 2 K (2) where K is a cone (or a union of cones) Example of convex optimization problem – p. 5/7 We define the problem data as follows:We formulate the problem as follows:For \(\epsilon=10^{-3}\), SuperSCS converged after \(11.3\)s ( \(1355\) iterations) , while SCS required \(39\)s ( \(4911\) iters).Notice that \(A\) is a sparse matrix with density \(10\%\) and reciprocal condition number \(0.1\).SuperSCS converges after \(40.7\)s ( \(314\) iterations), whereas SCS converges after \(415\)s ( \(4748\) iterations)\begin{eqnarray*} &&\mathrm{Minimize}_{Z}\ \|Z-P\|_{\mathrm{fro}}\\ &&Z\geq 0\\ &&Z: \mathrm{Hermitian},\ \mathrm{Toeplitz} \end{eqnarray*}In both cases, SuperSCS is about \(2.5\) times faster.The problem is generated as follows:For a lower tolerance of \(\epsilon=10^{-3}\) , SuperSCS with memory equal to \(100\) terminates at \(162\) iterations after \(23.5\)s while SCS converges after \(43.9\)s at \(787\) iterations.For this example we chose a (stable) matrix \(A\) with its eigenvalues uniformly logarithmically spaced between \(-10^{-1}\) and \(-10^{1}\).\begin{eqnarray*} &&\mathrm{Minimize}_{P}\ \mathrm{trace}(P)\\ &&P=P'\\ &&P \geq I\\ &&A'P + PA \leq -I \end{eqnarray*}\begin{eqnarray*} &&\mathrm{Minimize}_x\ \|Ax-b\|\\ &&\|Gx\| \leq 1 \end{eqnarray*}And we need to solve it up to an accuracy of \(\epsilon=10^{-3}\).SuperSCS exhibits a speed-up of \(\times 9.4\).SuperSCS converges after \(37.1\)s ( \(116\) iterations).\[ \mathrm{Minimize}_x\ \frac{1}{2} \|Ax-b\| + \mu \|x\|_1, \]Let \(M\) be a given matrix whose elements \(\{(i,j)\}_{i\in I, j\in J}\) are missing.This is a preliminary study to show that SuperSCS outperforms SCS, but a more thorough analysis is necessary.with \(x\in\mathbb{R}^n\), \(A\in\mathbb{R}^{m\times n}\).SuperSCS: \(7.6\)s, \(95\) iterationsThis is the problem formulation in CVXIn what follows we give a few code snippets in MATLAB and compare SuperSCS with SCS.Here we solve the following PCA problem (using an \(\ell_1\)-regularization).Here, we formulate the matrix completion problem as a nuclear norm minimization problem.\begin{eqnarray*} &&\mathrm{Maximize}_X\ \mathrm{trace}(SX) - \lambda \|X\|_1\\ && \mathrm{trace}(X) = 1\\ && X = X' \succeq 0 \end{eqnarray*}with \(x\in\mathbb{R}^m\), \(A\in\mathbb{R}^{m\times n}\) and \(G\in\mathbb{R}^{2n\times n}\).\begin{eqnarray*} &&\mathrm{Minimize}_{X}\ \|X-M\|_* + \lambda \|X\|_{\mathrm{fro}}^2\\ &&X_{i',j'} = M_{i',j'},\ \forall i'\notin I,\ j'\notin J \end{eqnarray*}\begin{eqnarray*} &&\mathrm{Minimize}_x\ \|x\|_p\\ &&Gx = f \end{eqnarray*}Here we solve the following portfolio selection problem:The problem is solved using CVX as follows:The problem is formulated in CVX as follows:Here we solve a constrained problem of the formHere we solve the following \(\ell_1\)-regularized logistic regression problem:On the other hand, SCS did not converge after \(242s\) ( \(5000\) iterations).\begin{eqnarray*} &&\mathrm{Maximize}_z\ \mu'z - \gamma z'\Sigma z\\ &&1'z = 1\\ &&z \geq 0, \end{eqnarray*}The above problem is solved in \(46.7\)s and at \(292\) iterations with SuperSCS running with restarted Broyden directions with memory equal to \(100\).We solve a simple LASSO problem of the formFor \(\epsilon=10^{-4}\), SuperSCS requires \(179\)s ( \(183\) iterations), whereas SCS still requires as much as \(497\)s ( \(983\) iterations).SCS: \(65\)s, \(2291\) iterations\[ \mathrm{Minimize}_w\ \lambda \|w\|_1 - \sum_{i}\log(1+\exp(a' w_i + b)) \]The respective results for SCS are \(157\)s and \(3050\) iterations (approximately \(3.3\) times slower).SuperSCS solves this problem in \(218\)s ( \(103\) iterations), whereas SCS requires \(500\)s to terminate ( \(392\) iterations).For \(\epsilon=10^{-3}\), SuperSCS terminates after \(131\)s ( \(134\) iterations), while SCS requires \(329\)s ( \(675\) iterations).We then formulate the problem in CVX with accuracy \(\epsilon=10^{-4}\).SuperSCS with the above options terminates after \(169\) iterations and \(85.1\) s.Another interesting SDP problem is that of determining a symmetric positive definite matrix \(P\) which solvesFor \(\epsilon=10^{-4}\), SuperSCS terminated after 17s ( \(4909\) iterations), while SCS failed to terminate within \(10^4\) iterations.In what follows we compare the two algorithms using five different types of problems: (i) a LASSO-like problem ( \(\ell_1\)-regularized least squares), (ii) a semidefinite program (SDP) and, in particular, a minimum-norm problem and an LMI-constrained problem, (iii) a logistic regression problem, (iv) a minimum \(p\)-norm problem, (v) a 2-norm-constrained minimum-norm problem and, last , (vi) a matrix completion problem involving the nuclear norm.SuperSCS converges in \(18.4s\) ( \(102\) iterations), whereas SCS takes \(270s\) ( \(6061\) iterations).Instead, SCS requires \(3243\) iterations which corresponds to \(802\) s.Note that SuperSCS can attain much higher precision; for instance, it converges with \(\epsilon=10^{-4}\) in \(41.2\)s ( \(131\) iterations) and with \(\epsilon=10^{-6}\) it converges after \(58\)s ( \(194\) iterations).
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