lectures/15-juliaopt/juliaopt.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimization in Julia\n",
    "\n",
    "This lecture gives an overview of some optimization tools in Julia."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.6.2\n",
      "Commit 1b93d53fc4 (2021-07-14 15:36 UTC)\n",
      "Platform Info:\n",
      "  OS: macOS (x86_64-apple-darwin18.7.0)\n",
      "  CPU: Intel(R) Core(TM) i5-8279U CPU @ 2.40GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-11.0.1 (ORCJIT, skylake)\n"
     ]
    }
   ],
   "source": [
    "versioninfo()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Commercial solvers [MOSEK](https://www.mosek.com) and [Gurobi](https://www.gurobi.com) need to be installed for this session.\n",
    "\n",
    "The following Julia packages are also needed. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
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   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m  Activating\u001b[22m\u001b[39m environment at `~/Dropbox/class/M1399.000200/2021/M1399_000200-2021fall/Project.toml`\n"
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      "\u001b[32m\u001b[1m      Status\u001b[22m\u001b[39m `~/Dropbox/class/M1399.000200/2021/M1399_000200-2021fall/Project.toml`\n",
      " \u001b[90m [7d9fca2a] \u001b[39mArpack v0.4.0\n",
      " \u001b[90m [6e4b80f9] \u001b[39mBenchmarkTools v1.2.0\n",
      " \u001b[90m [1e616198] \u001b[39mCOSMO v0.8.1\n",
      " \u001b[90m [f65535da] \u001b[39mConvex v0.14.16\n",
      " \u001b[90m [a93c6f00] \u001b[39mDataFrames v1.2.2\n",
      " \u001b[90m [31a5f54b] \u001b[39mDebugger v0.6.8\n",
      " \u001b[90m [31c24e10] \u001b[39mDistributions v0.24.18\n",
      " \u001b[90m [e2685f51] \u001b[39mECOS v0.12.3\n",
      " \u001b[90m [f6369f11] \u001b[39mForwardDiff v0.10.21\n",
      " \u001b[90m [28b8d3ca] \u001b[39mGR v0.61.0\n",
      " \u001b[90m [c91e804a] \u001b[39mGadfly v1.3.3\n",
      " \u001b[90m [bd48cda9] \u001b[39mGraphRecipes v0.5.7\n",
      " \u001b[90m [2e9cd046] \u001b[39mGurobi v0.9.14\n",
      " \u001b[90m [f67ccb44] \u001b[39mHDF5 v0.14.3\n",
      " \u001b[90m [82e4d734] \u001b[39mImageIO v0.5.8\n",
      " \u001b[90m [6218d12a] \u001b[39mImageMagick v1.2.1\n",
      " \u001b[90m [916415d5] \u001b[39mImages v0.24.1\n",
      " \u001b[90m [b6b21f68] \u001b[39mIpopt v0.7.0\n",
      " \u001b[90m [42fd0dbc] \u001b[39mIterativeSolvers v0.9.1\n",
      " \u001b[90m [4076af6c] \u001b[39mJuMP v0.21.10\n",
      " \u001b[90m [b51810bb] \u001b[39mMatrixDepot v1.0.4\n",
      " \u001b[90m [6405355b] \u001b[39mMosek v1.2.1\n",
      " \u001b[90m [1ec41992] \u001b[39mMosekTools v0.9.4\n",
      " \u001b[90m [76087f3c] \u001b[39mNLopt v0.6.3\n",
      " \u001b[90m [47be7bcc] \u001b[39mORCA v0.5.0\n",
      " \u001b[90m [a03496cd] \u001b[39mPlotlyBase v0.4.3\n",
      " \u001b[90m [f0f68f2c] \u001b[39mPlotlyJS v0.15.0\n",
      " \u001b[90m [91a5bcdd] \u001b[39mPlots v1.22.6\n",
      " \u001b[90m [438e738f] \u001b[39mPyCall v1.92.3\n",
      " \u001b[90m [d330b81b] \u001b[39mPyPlot v2.10.0\n",
      " \u001b[90m [dca85d43] \u001b[39mQuartzImageIO v0.7.3\n",
      " \u001b[90m [6f49c342] \u001b[39mRCall v0.13.12\n",
      " \u001b[90m [ce6b1742] \u001b[39mRDatasets v0.7.5\n",
      " \u001b[90m [c946c3f1] \u001b[39mSCS v0.7.1\n",
      " \u001b[90m [276daf66] \u001b[39mSpecialFunctions v1.7.0\n",
      " \u001b[90m [2913bbd2] \u001b[39mStatsBase v0.33.12\n",
      " \u001b[90m [f3b207a7] \u001b[39mStatsPlots v0.14.28\n",
      " \u001b[90m [b8865327] \u001b[39mUnicodePlots v2.4.6\n",
      " \u001b[90m [0f1e0344] \u001b[39mWebIO v0.8.16\n",
      " \u001b[90m [8f399da3] \u001b[39mLibdl\n",
      " \u001b[90m [2f01184e] \u001b[39mSparseArrays\n",
      " \u001b[90m [10745b16] \u001b[39mStatistics\n"
     ]
    }
   ],
   "source": [
    "using Pkg\n",
    "Pkg.activate(\"../..\")\n",
    "Pkg.status()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Package `QuartzImageIO` is for macOS only."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Flowchart\n",
    "\n",
    "* Category of optimization problems:\n",
    "\n",
    "    1. Problems with analytical solutions: least squares, principle component analysis, canonical correlation analysis, ...\n",
    "    \n",
    "    2. Problems subject to Disciplined Convex Programming (DCP): linear programming (LP), quadratic programming (QP), second-order cone programming (SOCP), semidefinite programming (SDP), and geometric programming (GP).\n",
    "    \n",
    "    3. Nonlinear programming (NLP): Newton type algorithms, Fisher scoring algorithm, EM algorithm, MM algorithms. \n",
    "    \n",
    "    4. Large scale optimization: ADMM, SGD, ...\n",
    "    \n",
    "<img src=\"./optflow.png\" width=400>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modeling tools and solvers\n",
    "\n",
    "Getting familiar with **good** optimization softwares broadens the scope and scale of problems we are able to solve in statistics. Following table lists some of the best optimization softwares. \n",
    "\n",
    "\n",
    "|           |   | LP | MILP | SOCP |     MISOCP     | SDP | GP | NLP | MINLP |   | R | Matlab | Julia | Python |   | Cost |  \n",
    "|:---------:|:-:|:--:|:----:|:----:|:--------------:|:---:|:--:|:---:|:-----:|:-:|:-:|:------:|:-----:|:------:|:-:|:----:|  \n",
    "|   **modeling tools**   |   |    |      |      |                |     |    |     |       |   |   |        |       |        |   |      |  \n",
    "|    AMPL   |   |  x |   x  |   x  |        x       |  x  |  x |  x  |   x   |   | x |    x   |       |    x   |   |   \\$  |  \n",
    "|    cvx    |   |  x |   x  |   x  |        x       |  x  |  x |     |       |   |   |    x   |       |    x   |   |   A  |  \n",
    "| Convex.jl |   |  x |   x  |   x  |        x       |  x  |    |     |       |   |   |        |   x   |        |   |   O  |  \n",
    "|  JuMP.jl  |   |  x |   x  |   x  |        x       |     |    |  x  |   x   |   |   |        |   x   |        |   |   O  |  \n",
    "| MathProgBase.jl |   |  x |   x  |   x  |        x       |     |    |  x  |   x   |   |   |        |   x   |        |   |   O  |  \n",
    "| MathOptInterface.jl |   |  x |   x  |   x  |        x       |     |    |  x  |   x   |   |   |        |   x   |        |   |   O  |  \n",
    "|   **convex solvers** |   |    |      |      |                |     |    |     |       |   |   |        |       |        |   |      |  \n",
    "|   Mosek   |   |  x |   x  |   x  |        x       |  x  |  x |  x  |       |   | x |    x   |   x   |    x   |   |   A  |  \n",
    "|   Gurobi  |   |  x |   x  |   x  |        x       |     |    |     |       |   | x |    x   |   x   |    x   |   |   A  |  \n",
    "|   CPLEX   |   |  x |   x  |   x  |        x       |     |    |     |       |   | x |    x   |   x   |    x   |   |   A  |  \n",
    "|    SCS    |   |  x |      |   x  |                |  x  |    |     |       |   |   |    x   |   x   |    x   |   |   O  |  \n",
    "|   **NLP solvers**  |   |    |      |      |                |     |    |     |       |   |   |        |       |        |   |      |  \n",
    "|   NLopt   |   |  x |      |      |                |     |    |  x  |       |   |   |    x   |   x   |    x   |   |   O  |  \n",
    "|   Ipopt   |   |  x |      |      |                |     |    |  x  |       |   |   |    x   |   x   |    x   |   |   O  |  \n",
    "|   KNITRO  |   |  x |   x  |      |                |     |    |  x  |   x   |   | x |    x   |   x   |    x   |   |   $  |  \n",
    "\n",
    "* O: open source  \n",
    "* A: free academic license  \n",
    "* $: commercial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Difference between **modeling tools** and **solvers**\n",
    "\n",
    "### Solvers\n",
    "\n",
    "* Mosek, Gurobi, Cplex, SCS, etc are *concrete* software implementation of optimization algorithms. \n",
    "    - Mosek/Gurobi/SCS for convex optimization\n",
    "    - Ipopt/NLopt for nonlinear programming\n",
    "    - Mosek and Gurobi are commercial software but free for academic use. SCS/Ipopt/NLopt are open source.  \n",
    "\n",
    "* Users need to implement the problem to solve using their application programming interface (API). [Example](https://docs.mosek.com/9.1/capi/_downloads/lo1.c)\n",
    "\n",
    "\n",
    "### Modeling tools\n",
    "\n",
    "* AMPL (comercial, <https://ampl.com>) is an algebraic modeling language that allows describe the optimization problem most close to its mathematical formulation. [Sample model](https://en.wikipedia.org/wiki/AMPL#A_sample_model)\n",
    "\n",
    "* [cvx](http://cvxopt.org) (for Matlab) and [Convex.jl](https://github.com/jump-dev/Convex.jl) (Julia) implement the disciplined convex programming (DCP) paradigm proposed by [Grant and Boyd (2008)](http://stanford.edu/~boyd/papers/disc_cvx_prog.html). DCP prescribes a set of simple rules from which users can construct convex optimization problems easily.\n",
    "\n",
    "* `Convex.jl` interfaces with actural problem solvers via [MathOptInterface](https://github.com/jump-dev/MathOptInterface.jl), an abstraction layer for mathematical optimization solvers in Julia.\n",
    "    - For example, [MOSEK](https://www.mosek.com) is an actual interior-point solver for convex  and mixed-integer programs. It provides APIs that can be accessed in low-level using [Mosek.jl](https://github.com/MOSEK/Mosek.jl), which is abstractized by [MosekTools.jl](https://github.com/jump-dev/MosekTools.jl), which in turn implements `MathOptInterface`.\n",
    "    \n",
    "* Modeling tools usually have the capability to use a variety of solvers. But modeling tools are solver agnostic so users do not have to worry about specific solver interface."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DCP Using Convex.jl"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standard convex problem classes like LP (linear programming), QP (quadratic programming), SOCP (second-order cone programming), SDP (semidefinite programming), and GP (geometric programming), are becoming a **technology**.\n",
    "\n",
    "<img src=\"./cvxoptdiagram.png\" width=300>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: microbiome regression analysis -- compositional data\n",
    "\n",
    "We illustrate optimization tools in Julia using microbiome analysis as an example.\n",
    "\n",
    "Diversity of gut microbiome is believed to be an important factor for human health or diseases such as obesity; respiratory microbiome affects pulmonary function in an HIV-infected population.\n",
    "\n",
    "16S microbiome sequencing techonology generates sequence counts of various organisms (operational taxonomic units, or OTUs) in samples. \n",
    "\n",
    "<img src=\"https://norgenbiotek.com/sites/default/files/services/16s-rrna-sq.jpg\" width=\"300\" align=\"center\"/>    \n",
    "\n",
    "The major interest is the **composition** of the microbiome in the population, hence \n",
    "for statistical analysis, OTU counts are normalized into **proportions** for each sample. \n",
    "Thus if there are $p$ OTUs, each observation $\\mathbf{z}_i$ lies in the positive probability simplex $\\Delta_{p-1,+}=\\{z_1, \\dotsc, z_p : z_j > 0, \\sum_{j=1}^p z_j=1\\}$. This unit-sum constraint introduces dependency between the $p$ variables, causing intrinsic difficulties in providing sensible interpretations for the regression parameters. A well-known resolution of this difficulty is the log-ratio transfomation due to Aitchson, which constructs the data matrix $\\tilde{\\mathbf{X}}=(\\log\\frac{z_{ij}}{z_{ip}}) \\in \\mathbb{R}^{n\\times (p-1)}$. Thus the regression model is\n",
    "$$\n",
    "    \\mathbf{y} = \\tilde{\\mathbf{X}}\\tilde{\\boldsymbol\\beta} + \\boldsymbol{\\varepsilon}, \\quad \\tilde{\\boldsymbol\\beta} \\in \\mathbb{R}^{p-1}.\n",
    "$$\n",
    "\n",
    "By introducing an extra coefficient $\\beta_p = -\\sum_{j=1}^{p-1}\\beta_j$, and letting $\\mathbf{X}=(\\log z_{ij})$, the above, assymetric model becomes symmetric:\n",
    "$$\n",
    "    \\mathbf{y} = \\mathbf{X}\\boldsymbol{\\beta} + \\boldsymbol{\\varepsilon}, \\quad \\boldsymbol{\\beta} \\in \\mathbb{R}^{p}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Zero-sum regression\n",
    "\n",
    "In other words, we need to solve a **zero-sum least squares problem**  \n",
    "$$\n",
    "    \\begin{array}{ll}\n",
    "    \\text{minimize} & \\frac{1}{2} \\|\\mathbf{y} - \\mathbf{X} \\boldsymbol{\\beta}\\|_2^2 \\\\\n",
    "    \\text{subject to} & \\mathbf{1}^T\\boldsymbol{\\beta} = 0.\n",
    "    \\end{array}\n",
    "$$\n",
    "For details, see\n",
    "\n",
    "> Lin, W., Shi, P., Feng, R. and Li, H., 2014. Variable selection in regression with compositional covariates. Biometrika, 101(4), pp.785-797. <https://doi.org/10.1093/biomet/asu031>\n",
    "\n",
    "The sum-to-zero contrained least squares is a standard quadratic programming (QP) problem so should be solved easily by any QP solver.\n",
    "\n",
    "For simplicity we ignore intercept and non-OTU covariates in this presentation.\n",
    "\n",
    "Let's first generate an artifical dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Random, LinearAlgebra, SparseArrays\n",
    "\n",
    "Random.seed!(123) # seed\n",
    "\n",
    "n, p = 100, 50\n",
    "X = rand(n, p)\n",
    "lmul!(Diagonal(1 ./ vec(sum(X, dims=2))), X)\n",
    "β = sprandn(p, 0.1) # sparse vector with about 10% non-zero entries\n",
    "y = X * β + randn(n);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Modeling using Convex.jl\n",
    "\n",
    "We use the [Convex.jl](https://github.com/jump-dev/Convex.jl) package to model this QP problem. For a complete list of operations supported by Convex.jl, see how close the code is to the mathematical formulation above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Convex\n",
    "\n",
    "β̂cls = Variable(size(X, 2))\n",
    "problem = minimize(0.5sumsquares(y - X * β̂cls)) # objective\n",
    "problem.constraints += sum(β̂cls) == 0; # sum-to-zero constraint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Mosek\n",
    "\n",
    "We first use the Mosek solver to solve this QP."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Problem\n",
      "  Name                   :                 \n",
      "  Objective sense        : min             \n",
      "  Type                   : CONIC (conic optimization problem)\n",
      "  Constraints            : 107             \n",
      "  Cones                  : 2               \n",
      "  Scalar variables       : 157             \n",
      "  Matrix variables       : 0               \n",
      "  Integer variables      : 0               \n",
      "\n",
      "Optimizer started.\n",
      "Presolve started.\n",
      "Linear dependency checker started.\n",
      "Linear dependency checker terminated.\n",
      "Eliminator started.\n",
      "Freed constraints in eliminator : 0\n",
      "Eliminator terminated.\n",
      "Eliminator - tries                  : 1                 time                   : 0.00            \n",
      "Lin. dep.  - tries                  : 1                 time                   : 0.00            \n",
      "Lin. dep.  - number                 : 0               \n",
      "Presolve terminated. Time: 0.00    \n",
      "Problem\n",
      "  Name                   :                 \n",
      "  Objective sense        : min             \n",
      "  Type                   : CONIC (conic optimization problem)\n",
      "  Constraints            : 107             \n",
      "  Cones                  : 2               \n",
      "  Scalar variables       : 157             \n",
      "  Matrix variables       : 0               \n",
      "  Integer variables      : 0               \n",
      "\n",
      "Optimizer  - threads                : 8               \n",
      "Optimizer  - solved problem         : the dual        \n",
      "Optimizer  - Constraints            : 52\n",
      "Optimizer  - Cones                  : 3\n",
      "Optimizer  - Scalar variables       : 106               conic                  : 106             \n",
      "Optimizer  - Semi-definite variables: 0                 scalarized             : 0               \n",
      "Factor     - setup time             : 0.00              dense det. time        : 0.00            \n",
      "Factor     - ML order time          : 0.00              GP order time          : 0.00            \n",
      "Factor     - nonzeros before factor : 1328              after factor           : 1328            \n",
      "Factor     - dense dim.             : 0                 flops                  : 3.08e+05        \n",
      "ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  \n",
      "0   3.0e+00  5.0e-01  2.0e+00  0.00e+00   0.000000000e+00   -1.000000000e+00  1.0e+00  0.00  \n",
      "1   2.9e-01  4.8e-02  5.7e-01  -9.26e-01  2.915150182e+00   9.703344707e+00   9.7e-02  0.01  \n",
      "2   4.4e-02  7.4e-03  1.3e-01  -7.10e-01  1.845835780e+01   3.574733771e+01   1.5e-02  0.01  \n",
      "3   2.1e-03  3.5e-04  4.6e-04  7.65e-01   2.901642971e+01   2.908435432e+01   7.0e-04  0.02  \n",
      "4   1.7e-06  2.9e-07  1.3e-08  1.00e+00   2.899788909e+01   2.899798941e+01   5.7e-07  0.02  \n",
      "5   2.5e-10  4.2e-11  2.8e-14  1.00e+00   2.899829742e+01   2.899829744e+01   8.4e-11  0.02  \n",
      "Optimizer terminated. Time: 0.02    \n",
      "\n",
      " 22.001013 seconds (50.39 M allocations: 2.889 GiB, 7.04% gc time)\n"
     ]
    }
   ],
   "source": [
    "using Mosek, MosekTools\n",
    "opt = () -> Mosek.Optimizer(LOG=1)  # anonymous function\n",
    "@time solve!(problem, opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(MathOptInterface.OPTIMAL, 28.998297421260087, [13.668687461115228; -9.772741354380681; … ; 21.133785591984804; -15.115087762560165])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the status, optimal value, and minimizer of the problem\n",
    "problem.status, problem.optval, β̂cls.value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Gurobi\n",
    "\n",
    "Switch to [Gurobi](https://www.gurobi.com) solver:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"/Users/jhwon/gurobi/gurobi.lic\""
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ENV[\"GRB_LICENSE_FILE\"]=\"/Users/jhwon/gurobi/gurobi.lic\"  # set as YOUR path to license file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only - expires 2021-12-30\n",
      "Gurobi Optimizer version 9.1.2 build v9.1.2rc0 (mac64)\n",
      "Thread count: 4 physical cores, 8 logical processors, using up to 8 threads\n",
      "Optimize a model with 107 rows, 157 columns and 5160 nonzeros\n",
      "Model fingerprint: 0xab10ce97\n",
      "Model has 2 quadratic constraints\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e-05, 2e+00]\n",
      "  QMatrix range    [1e+00, 1e+00]\n",
      "  Objective range  [1e+00, 1e+00]\n",
      "  Bounds range     [0e+00, 0e+00]\n",
      "  RHS range        [2e-03, 3e+00]\n",
      "Presolve removed 2 rows and 1 columns\n",
      "Presolve time: 0.00s\n",
      "Presolved: 105 rows, 156 columns, 5158 nonzeros\n",
      "Presolved model has 2 second-order cone constraints\n",
      "Ordering time: 0.00s\n",
      "\n",
      "Barrier statistics:\n",
      " Free vars  : 50\n",
      " AA' NZ     : 5.154e+03\n",
      " Factor NZ  : 5.262e+03\n",
      " Factor Ops : 3.590e+05 (less than 1 second per iteration)\n",
      " Threads    : 1\n",
      "\n",
      "                  Objective                Residual\n",
      "Iter       Primal          Dual         Primal    Dual     Compl     Time\n",
      "   0   1.12826689e+01 -5.01000000e-01  2.29e+01 1.00e-01  2.00e-01     0s\n",
      "   1   3.22840813e+00  6.27715458e-02  4.49e+00 3.72e-05  3.88e-02     0s\n",
      "   2   4.08178592e+00  5.62149847e+00  3.32e+00 3.97e-05  2.14e-02     0s\n",
      "   3   1.64813094e+01  8.57926093e+00  9.91e-01 3.54e-05  9.22e-02     0s\n",
      "   4   1.75521420e+01  1.58590777e+01  7.37e-01 4.78e-06  4.66e-02     0s\n",
      "   5   1.92008663e+01  2.62514505e+01  6.28e-01 3.87e-06  1.63e-02     0s\n",
      "   6   3.03618788e+01  2.79564255e+01  9.67e-03 1.02e-07  2.45e-02     0s\n",
      "   7   2.91004528e+01  2.89455416e+01  3.83e-04 4.40e-08  1.54e-03     0s\n",
      "   8   2.90061906e+01  2.89953287e+01  2.12e-05 3.54e-09  1.07e-04     0s\n",
      "   9   2.89983319e+01  2.89982844e+01  2.40e-07 1.52e-09  4.70e-07     0s\n",
      "  10   2.89982975e+01  2.89982974e+01  4.99e-09 3.80e-11  2.65e-09     0s\n",
      "\n",
      "Barrier solved model in 10 iterations and 0.02 seconds\n",
      "Optimal objective 2.89982975e+01\n",
      "\n",
      "\n",
      "User-callback calls 63, time in user-callback 0.00 sec\n",
      "  4.380080 seconds (8.73 M allocations: 511.805 MiB, 5.60% gc time)\n"
     ]
    }
   ],
   "source": [
    "using Gurobi\n",
    "const GRB_ENV = Gurobi.Env()\n",
    "opt = () -> Gurobi.Optimizer(GRB_ENV)  # anonymous function\n",
    "@time solve!(problem, opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(MathOptInterface.OPTIMAL, 28.998297507778627, [13.668678309342742; -9.7727402141536; … ; 21.133769207305104; -15.115075595252181])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the status, optimal value, and minimizer of the problem\n",
    "problem.status, problem.optval, β̂cls.value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### SCS\n",
    "\n",
    "Switch to the open source [SCS](https://github.com/jump-dev/SCS.jl) solver:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  5.340976 seconds (10.35 M allocations: 607.867 MiB, 4.62% gc time, 5.22% compilation time)\n",
      "----------------------------------------------------------------------------\n",
      "\tSCS v2.1.4 - Splitting Conic Solver\n",
      "\t(c) Brendan O'Donoghue, Stanford University, 2012\n",
      "----------------------------------------------------------------------------\n",
      "Lin-sys: sparse-indirect, nnz in A = 5056, CG tol ~ 1/iter^(2.00)\n",
      "eps = 1.00e-05, alpha = 1.50, max_iters = 5000, normalize = 1, scale = 1.00\n",
      "acceleration_lookback = 10, rho_x = 1.00e-03\n",
      "Variables n = 53, constraints m = 107\n",
      "Cones:\tprimal zero / dual free vars: 2\n",
      "\tlinear vars: 1\n",
      "\tsoc vars: 104, soc blks: 2\n",
      "Setup time: 3.35e-04s\n",
      "----------------------------------------------------------------------------\n",
      " Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)\n",
      "----------------------------------------------------------------------------\n",
      "     0| 9.17e+19  5.76e+19  1.00e+00 -1.04e+21  8.59e+20  1.59e+21  5.64e-04 \n",
      "   100| 1.44e-05  9.44e-06  5.09e-06  2.90e+01  2.90e+01  7.55e-16  1.01e-02 \n",
      "   120| 3.60e-06  2.64e-06  1.23e-06  2.90e+01  2.90e+01  3.93e-16  1.10e-02 \n",
      "----------------------------------------------------------------------------\n",
      "Status: Solved\n",
      "Timing: Solve time: 1.10e-02s\n",
      "\tLin-sys: avg # CG iterations: 6.79, avg solve time: 7.90e-05s\n",
      "\tCones: avg projection time: 1.69e-07s\n",
      "\tAcceleration: avg step time: 9.57e-06s\n",
      "----------------------------------------------------------------------------\n",
      "Error metrics:\n",
      "dist(s, K) = 3.5527e-15, dist(y, K*) = 4.4409e-16, s'y/|s||y| = -8.9856e-17\n",
      "primal res: |Ax + s - b|_2 / (1 + |b|_2) = 3.6046e-06\n",
      "dual res:   |A'y + c|_2 / (1 + |c|_2) = 2.6431e-06\n",
      "rel gap:    |c'x + b'y| / (1 + |c'x| + |b'y|) = 1.2313e-06\n",
      "----------------------------------------------------------------------------\n",
      "c'x = 28.9983, -b'y = 28.9983\n",
      "============================================================================\n"
     ]
    }
   ],
   "source": [
    "# Use SCS solver\n",
    "using SCS\n",
    "opt = () -> SCS.Optimizer(verbose=1)  # anonymous function\n",
    "@time solve!(problem, opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(MathOptInterface.OPTIMAL, 28.99834367864642, [13.668719778517216; -9.772627374792503; … ; 21.13380901845421; -15.115084607474534])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the status, optimal value, and minimizer of the problem\n",
    "problem.status, problem.optval, β̂cls.value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### COSMO\n",
    "\n",
    "Switch to the open source [COSMO](https://oxfordcontrol.github.io/COSMO.jl/stable/) solver:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------------\n",
      "          COSMO v0.8.1 - A Quadratic Objective Conic Solver\n",
      "                         Michael Garstka\n",
      "                University of Oxford, 2017 - 2021\n",
      "------------------------------------------------------------------\n",
      "\n",
      "Problem:  x ∈ R^{53},\n",
      "          constraints: A ∈ R^{107x53} (5056 nnz),\n",
      "          matrix size to factor: 160x160,\n",
      "          Floating-point precision: Float64\n",
      "Sets:     SecondOrderCone) of dim: 101\n",
      "          SecondOrderCone) of dim: 3\n",
      "          ZeroSet) of dim: 2\n",
      "          Nonnegatives) of dim: 1\n",
      "Settings: ϵ_abs = 1.0e-05, ϵ_rel = 1.0e-05,\n",
      "          ϵ_prim_inf = 1.0e-04, ϵ_dual_inf = 1.0e-04,\n",
      "          ρ = 0.1, σ = 1e-06, α = 1.6,\n",
      "          max_iter = 5000,\n",
      "          scaling iter = 10 (on),\n",
      "          check termination every 25 iter,\n",
      "          check infeasibility every 40 iter,\n",
      "          KKT system solver: QDLDL\n",
      "Acc:      Anderson Type2{QRDecomp},\n",
      "          Memory size = 15, RestartedMemory,\t\n",
      "          Safeguarded: true, tol: 2.0\n",
      "Setup Time: 998.06ms\n",
      "\n",
      "Iter:\tObjective:\tPrimal Res:\tDual Res:\tRho:\n",
      "1\t-2.4722e+00\t6.0889e+00\t3.8576e+02\t1.0000e-01\n",
      "25\t2.6959e+01\t1.5279e-01\t5.7669e+00\t1.0000e-01\n",
      "50\t2.8666e+01\t1.4083e-02\t5.0905e-03\t1.2134e-02\n",
      "75\t2.8998e+01\t4.2267e-09\t4.8880e-11\t1.2134e-02\n",
      "\n",
      "------------------------------------------------------------------\n",
      ">>> Results\n",
      "Status: \u001b[32mSolved\u001b[39m\n",
      "Iterations: 85 (incl. 10 safeguarding iter)\n",
      "Optimal objective: 29\n",
      "Runtime: 2.924s (2923.52ms)\n",
      "\n",
      " 18.050220 seconds (51.06 M allocations: 2.824 GiB, 8.01% gc time, 1.91% compilation time)\n"
     ]
    }
   ],
   "source": [
    "# Use COSMO solver\n",
    "using COSMO\n",
    "opt = () -> COSMO.Optimizer(max_iter=5000) # anonymous function\n",
    "@time solve!(problem, opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(MathOptInterface.OPTIMAL, 28.998297502384982, [13.668687496059547; -9.772741376125092; … ; 21.133785644856946; -15.115087801073152])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the status, optimal value, and minimizer of the problem\n",
    "problem.status, problem.optval, β̂cls.value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Zero-sum lasso\n",
    "\n",
    "Suppose we want to know which organisms (OTUs) are associated with the response. We can answer this question using a zero-sum contrained lasso\n",
    "$$\n",
    "    \\begin{array}{ll}\n",
    "    \\text{minimize} & \\frac 12 \\|\\mathbf{y} - \\mathbf{X} \\boldsymbol{\\beta}\\|_2^2 + \\lambda \\|\\boldsymbol{\\beta}\\|_1 \\\\\n",
    "    \\text{subject to} & \\mathbf{1}^T\\boldsymbol{\\beta} = 0.\n",
    "    \\end{array}\n",
    "$$\n",
    "\n",
    "Varying $\\lambda$ from small to large values will generate a solution path."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  1.631235 seconds (3.64 M allocations: 264.189 MiB, 4.66% gc time, 50.18% compilation time)\n"
     ]
    }
   ],
   "source": [
    "# Use Mosek solver\n",
    "#opt = () -> Mosek.Optimizer(LOG=0)\n",
    "\n",
    "## Use Gurobi solver\n",
    "##solver = Gurobi.Optimizer(GRB_ENV)\n",
    "#using JuMP\n",
    "#opt = () -> Gurobi.Optimizer(GRB_ENV)\n",
    "##model = direct_model(Gurobi.Optimizer(GRB_ENV))\n",
    "##MOI.set(model, MOI.RawParameter(\"OutputFlag\"), 0)\n",
    "##set_optimizer_attribute(JuMP.Model(opt), \"OutputFlag\", 0)\n",
    "\n",
    "# Use SCS solver\n",
    "opt = () -> SCS.Optimizer(verbose=0)\n",
    "\n",
    "## Use COSMO solver\n",
    "#opt = () -> COSMO.Optimizer(max_iter=10000, verbose=false)\n",
    "\n",
    "# solve at a grid of λ\n",
    "λgrid = 0:0.01:0.35\n",
    "β̂path = zeros(length(λgrid), size(X, 2)) # each row is β̂ at a λ\n",
    "β̂classo = Variable(size(X, 2))\n",
    "@time for i in 1:length(λgrid)\n",
    "    λ = λgrid[i]\n",
    "    # objective\n",
    "    problem = minimize(0.5sumsquares(y - X * β̂classo) + λ * sum(abs, β̂classo))\n",
    "    # constraint\n",
    "    problem.constraints += sum(β̂classo) == 0 # constraint\n",
    "    solve!(problem, opt)\n",
    "    β̂path[i, :] = β̂classo.value\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots; gr()\n",
    "\n",
    "p = plot(collect(λgrid), β̂path, legend=:none)\n",
    "xlabel!(p, \"lambda\")\n",
    "ylabel!(p, \"beta_hat\")\n",
    "title!(p, \"Zero-sum Lasso\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Zero-sum group lasso\n",
    "\n",
    "Suppose we want to do variable selection not at the OTU level, but at the Phylum level. OTUs are clustered into various Phyla. We can answer this question using a sum-to-zero contrained group lasso\n",
    "$$\n",
    "    \\begin{array}{ll}\n",
    "    \\text{minimize} & \\frac 12 \\|\\mathbf{y} - \\mathbf{X} \\boldsymbol{\\beta}\\|_2^2 + \\lambda \\sum_j \\|\\boldsymbol{\\beta}_j\\|_2 \\\\\n",
    "    \\text{subject to} & \\mathbf{1}^T\\boldsymbol{\\beta} = 0,\n",
    "    \\end{array}\n",
    "$$\n",
    "where $\\boldsymbol{\\beta}_j$ is the $j$th partition of the regression coefficients corresponding to the $j$th phylum. This is a **second-order cone programming (SOCP)** problem readily modeled by Convex.jl.\n",
    "\n",
    "Let's assume each 10 contiguous OTUs belong to one Phylum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.747849 seconds (3.03 M allocations: 249.958 MiB, 3.58% gc time, 31.95% compilation time)\n"
     ]
    }
   ],
   "source": [
    "# Use Mosek solver\n",
    "#opt = () -> Mosek.Optimizer(LOG=0)\n",
    "\n",
    "## Use Gurobi solver\n",
    "##solver = Gurobi.Optimizer(GRB_ENV)\n",
    "#using JuMP\n",
    "#opt = () -> Gurobi.Optimizer(GRB_ENV)\n",
    "##model = direct_model(Gurobi.Optimizer(GRB_ENV))\n",
    "##MOI.set(model, MOI.RawParameter(\"OutputFlag\"), 0)\n",
    "##set_optimizer_attribute(JuMP.Model(opt), \"OutputFlag\", 0)\n",
    "\n",
    "# Use SCS solver\n",
    "opt = () -> SCS.Optimizer(verbose=0)\n",
    "\n",
    "## Use COSMO solver\n",
    "#opt = () -> COSMO.Optimizer(max_iter=5000, verbose=false)\n",
    "\n",
    "\n",
    "# solve at a grid of λ\n",
    "λgrid = 0.1:0.005:0.5\n",
    "β̂pathgrp = zeros(length(λgrid), size(X, 2)) # each row is β̂ at a λ\n",
    "β̂classo = Variable(size(X, 2))\n",
    "@time for i in 1:length(λgrid)\n",
    "    λ = λgrid[i]\n",
    "    # loss\n",
    "    obj = 0.5sumsquares(y - X * β̂classo)\n",
    "    # group lasso penalty term\n",
    "    for j in 1:(size(X, 2)/10)\n",
    "        βj = β̂classo[(10(j-1)+1):10j]\n",
    "        obj = obj + λ * norm(βj)\n",
    "    end\n",
    "    problem = minimize(obj)\n",
    "    # constraint\n",
    "    problem.constraints += sum(β̂classo) == 0 # constraint\n",
    "    solve!(problem, opt)\n",
    "    β̂pathgrp[i, :] = β̂classo.value\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It took Mosek <1 second to solve this seemingly hard optimization problem at **80** different $\\lambda$ values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p2 = plot(collect(λgrid), β̂pathgrp, legend=:none)\n",
    "xlabel!(p2, \"lambda\")\n",
    "ylabel!(p2, \"beta_hat\")\n",
    "title!(p2, \"Zero-Sum Group Lasso\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: matrix completion\n",
    "\n",
    "Load the $128 \\times 128$ [Lena](./lena128missing.png) image with missing pixels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "128×128 Array{Gray{N0f8},2} with eltype Gray{N0f8}:\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)    …  Gray{N0f8}(0.627)\n",
       " Gray{N0f8}(0.627)  Gray{N0f8}(0.624)     Gray{N0f8}(0.388)\n",
       " Gray{N0f8}(0.612)  Gray{N0f8}(0.612)     Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)       Gray{N0f8}(0.192)\n",
       " Gray{N0f8}(0.612)  Gray{N0f8}(0.0)       Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)    …  Gray{N0f8}(0.2)\n",
       " Gray{N0f8}(0.608)  Gray{N0f8}(0.0)       Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)       Gray{N0f8}(0.216)\n",
       " Gray{N0f8}(0.62)   Gray{N0f8}(0.62)      Gray{N0f8}(0.208)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)       Gray{N0f8}(0.188)\n",
       " Gray{N0f8}(0.635)  Gray{N0f8}(0.0)    …  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.631)  Gray{N0f8}(0.0)       Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.627)     Gray{N0f8}(0.184)\n",
       " ⋮                                     ⋱  \n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.129)     Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.149)  Gray{N0f8}(0.129)     Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.216)  Gray{N0f8}(0.0)       Gray{N0f8}(0.208)\n",
       " Gray{N0f8}(0.345)  Gray{N0f8}(0.341)     Gray{N0f8}(0.231)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)    …  Gray{N0f8}(0.259)\n",
       " Gray{N0f8}(0.298)  Gray{N0f8}(0.416)     Gray{N0f8}(0.259)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.369)     Gray{N0f8}(0.235)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.0)       Gray{N0f8}(0.208)\n",
       " Gray{N0f8}(0.22)   Gray{N0f8}(0.0)       Gray{N0f8}(0.2)\n",
       " Gray{N0f8}(0.0)    Gray{N0f8}(0.22)   …  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.196)  Gray{N0f8}(0.208)     Gray{N0f8}(0.345)\n",
       " Gray{N0f8}(0.192)  Gray{N0f8}(0.0)       Gray{N0f8}(0.0)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Images\n",
    "\n",
    "lena = load(\"lena128missing.png\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "128×128 Matrix{Float64}:\n",
       " 0.0       0.0       0.635294  0.0       …  0.0       0.0       0.627451\n",
       " 0.627451  0.623529  0.0       0.611765     0.0       0.0       0.388235\n",
       " 0.611765  0.611765  0.0       0.0          0.403922  0.219608  0.0\n",
       " 0.0       0.0       0.611765  0.0          0.223529  0.176471  0.192157\n",
       " 0.611765  0.0       0.615686  0.615686     0.0       0.0       0.0\n",
       " 0.0       0.0       0.0       0.619608  …  0.0       0.0       0.2\n",
       " 0.607843  0.0       0.623529  0.0          0.176471  0.192157  0.0\n",
       " 0.0       0.0       0.623529  0.0          0.0       0.0       0.215686\n",
       " 0.619608  0.619608  0.0       0.0          0.2       0.0       0.207843\n",
       " 0.0       0.0       0.635294  0.635294     0.2       0.192157  0.188235\n",
       " 0.635294  0.0       0.0       0.0       …  0.192157  0.180392  0.0\n",
       " 0.631373  0.0       0.0       0.0          0.0       0.0       0.0\n",
       " 0.0       0.627451  0.635294  0.666667     0.172549  0.0       0.184314\n",
       " ⋮                                       ⋱  ⋮                   \n",
       " 0.0       0.129412  0.0       0.541176     0.0       0.286275  0.0\n",
       " 0.14902   0.129412  0.196078  0.537255     0.345098  0.0       0.0\n",
       " 0.215686  0.0       0.262745  0.0          0.301961  0.0       0.207843\n",
       " 0.345098  0.341176  0.356863  0.513725     0.0       0.0       0.231373\n",
       " 0.0       0.0       0.0       0.0       …  0.0       0.243137  0.258824\n",
       " 0.298039  0.415686  0.458824  0.0          0.0       0.0       0.258824\n",
       " 0.0       0.368627  0.4       0.0          0.0       0.0       0.235294\n",
       " 0.0       0.0       0.34902   0.0          0.0       0.239216  0.207843\n",
       " 0.219608  0.0       0.0       0.0          0.0       0.0       0.2\n",
       " 0.0       0.219608  0.235294  0.356863  …  0.0       0.0       0.0\n",
       " 0.196078  0.207843  0.211765  0.0          0.0       0.270588  0.345098\n",
       " 0.192157  0.0       0.196078  0.309804     0.266667  0.356863  0.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# convert to real matrices\n",
    "Y = Float64.(lena)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We fill out the missing pixels uisng a **matrix completion** technique developed by Candes and Tao\n",
    "$$\n",
    "    \\begin{array}{ll}\n",
    "    \\text{minimize} & \\|\\mathbf{X}\\|_* \\\\\n",
    "    \\text{subject to} & x_{ij} = y_{ij} \\text{ for all observed entries } (i, j).\n",
    "    \\end{array}\n",
    "$$\n",
    "\n",
    "Here $\\|\\mathbf{X}\\|_* = \\sum_{i=1}^{\\min(m,n)} \\sigma_i(\\mathbf{X})$ is the nuclear norm. It can be shown that \n",
    "$$\n",
    "\\|\\mathbf{X}\\|_* = \\sup_{\\|\\mathbf{Y}\\|_2 \\le 1} \\langle \\mathbf{X}, \\mathbf{Y} \\rangle,\n",
    "$$ \n",
    "where $\\|\\mathbf{Y}\\|_2=\\sigma_{\\max}(\\mathbf{Y})$ is the spectral (operator 2-) norm, and $\\langle \\mathbf{X}, \\mathbf{Y} \\rangle = \\text{tr}(\\mathbf{X}^T\\mathbf{Y})$. That is, $\\|\\cdot\\|_*$ is the dual norm of $\\|\\cdot\\|_2$.\n",
    "\n",
    "The nuclear norm can be considered as the best convex approximation to $\\text{rank}(M)$, just like $\\|\\mathbf{x}\\|_1$ is the best convex approximation to $\\|\\mathbf{x}\\|_0$.\n",
    "We want the matrix with the lowest rank that agrees with the observed entries, but instead seek one with the minimal nuclear norm as a convex relaxation.\n",
    "\n",
    "This is a **semidefinite programming (SDP)** problem readily modeled by Convex.jl.\n",
    "\n",
    "This example takes **long** because of high dimensionality."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------------\n",
      "          COSMO v0.8.1 - A Quadratic Objective Conic Solver\n",
      "                         Michael Garstka\n",
      "                University of Oxford, 2017 - 2021\n",
      "------------------------------------------------------------------\n",
      "\n",
      "Problem:  x ∈ R^{49153},\n",
      "          constraints: A ∈ R^{73665x49153} (73793 nnz),\n",
      "          matrix size to factor: 122818x122818,\n",
      "          Floating-point precision: Float64\n",
      "Sets:     ZeroSet) of dim: 40769\n",
      "          DensePsdConeTriangle) of dim: 32896\n",
      "Settings: ϵ_abs = 1.0e-05, ϵ_rel = 1.0e-05,\n",
      "          ϵ_prim_inf = 1.0e-04, ϵ_dual_inf = 1.0e-04,\n",
      "          ρ = 0.1, σ = 1e-06, α = 1.6,\n",
      "          max_iter = 5000,\n",
      "          scaling iter = 10 (on),\n",
      "          check termination every 25 iter,\n",
      "          check infeasibility every 40 iter,\n",
      "          KKT system solver: QDLDL\n",
      "Acc:      Anderson Type2{QRDecomp},\n",
      "          Memory size = 15, RestartedMemory,\t\n",
      "          Safeguarded: true, tol: 2.0\n",
      "Setup Time: 112.44ms\n",
      "\n",
      "Iter:\tObjective:\tPrimal Res:\tDual Res:\tRho:\n",
      "1\t-1.4426e+03\t1.1678e+01\t5.9856e-01\t1.0000e-01\n",
      "25\t1.4495e+02\t5.5360e-02\t1.1033e-03\t1.0000e-01\n",
      "50\t1.4754e+02\t1.2369e-02\t1.6744e-03\t7.4179e-01\n",
      "75\t1.4797e+02\t4.9490e-04\t5.5696e-05\t7.4179e-01\n",
      "100\t1.4797e+02\t1.4115e-05\t1.3438e-06\t7.4179e-01\n",
      "\n",
      "------------------------------------------------------------------\n",
      ">>> Results\n",
      "Status: \u001b[32mSolved\u001b[39m\n",
      "Iterations: 100\n",
      "Optimal objective: 148\n",
      "Runtime: 2.103s (2103.34ms)\n",
      "\n",
      " 10.198321 seconds (18.36 M allocations: 1.704 GiB, 7.96% gc time, 70.20% compilation time)\n"
     ]
    }
   ],
   "source": [
    "# Use COSMO solver\n",
    "using COSMO\n",
    "opt = () -> COSMO.Optimizer()\n",
    "## Use SCS solver\n",
    "#using SCS\n",
    "#opt = () -> SCS.Optimizer()\n",
    "## Use Mosek solver\n",
    "#using Mosek\n",
    "#opt = () -> Mosek.Optimizer()\n",
    "\n",
    "# Linear indices of obs. entries\n",
    "obsidx = findall(Y[:] .≠ 0.0)\n",
    "# Create optimization variables\n",
    "X = Convex.Variable(size(Y))\n",
    "# Set up optmization problem\n",
    "problem = minimize(nuclearnorm(X))\n",
    "problem.constraints += X[obsidx] == Y[obsidx]\n",
    "# Solve the problem by calling solve\n",
    "@time solve!(problem, opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "128×128 reinterpret(reshape, Gray{Float64}, ::Matrix{Float64}) with eltype Gray{Float64}:\n",
       " Gray{Float64}(0.510131)  Gray{Float64}(0.541841)  …  Gray{Float64}(0.627451)\n",
       " Gray{Float64}(0.627451)  Gray{Float64}(0.623529)     Gray{Float64}(0.388235)\n",
       " Gray{Float64}(0.611765)  Gray{Float64}(0.611765)     Gray{Float64}(0.337454)\n",
       " Gray{Float64}(0.58598)   Gray{Float64}(0.581621)     Gray{Float64}(0.192157)\n",
       " Gray{Float64}(0.611765)  Gray{Float64}(0.581892)     Gray{Float64}(0.277461)\n",
       " Gray{Float64}(0.557055)  Gray{Float64}(0.520127)  …  Gray{Float64}(0.2)\n",
       " Gray{Float64}(0.607843)  Gray{Float64}(0.582895)     Gray{Float64}(0.197009)\n",
       " Gray{Float64}(0.611246)  Gray{Float64}(0.637081)     Gray{Float64}(0.215686)\n",
       " Gray{Float64}(0.619608)  Gray{Float64}(0.619608)     Gray{Float64}(0.207843)\n",
       " Gray{Float64}(0.617212)  Gray{Float64}(0.588932)     Gray{Float64}(0.188235)\n",
       " Gray{Float64}(0.635294)  Gray{Float64}(0.594275)  …  Gray{Float64}(0.206487)\n",
       " Gray{Float64}(0.631373)  Gray{Float64}(0.592409)     Gray{Float64}(0.184318)\n",
       " Gray{Float64}(0.588057)  Gray{Float64}(0.627451)     Gray{Float64}(0.184314)\n",
       " ⋮                                                 ⋱  \n",
       " Gray{Float64}(0.149174)  Gray{Float64}(0.129412)     Gray{Float64}(0.241792)\n",
       " Gray{Float64}(0.14902)   Gray{Float64}(0.129412)     Gray{Float64}(0.265217)\n",
       " Gray{Float64}(0.215686)  Gray{Float64}(0.222662)     Gray{Float64}(0.207843)\n",
       " Gray{Float64}(0.345098)  Gray{Float64}(0.341176)     Gray{Float64}(0.231373)\n",
       " Gray{Float64}(0.263514)  Gray{Float64}(0.334634)  …  Gray{Float64}(0.258824)\n",
       " Gray{Float64}(0.298039)  Gray{Float64}(0.415686)     Gray{Float64}(0.258824)\n",
       " Gray{Float64}(0.268503)  Gray{Float64}(0.368627)     Gray{Float64}(0.235294)\n",
       " Gray{Float64}(0.249179)  Gray{Float64}(0.289787)     Gray{Float64}(0.207843)\n",
       " Gray{Float64}(0.219608)  Gray{Float64}(0.251989)     Gray{Float64}(0.2)\n",
       " Gray{Float64}(0.229015)  Gray{Float64}(0.219608)  …  Gray{Float64}(0.203569)\n",
       " Gray{Float64}(0.196078)  Gray{Float64}(0.207843)     Gray{Float64}(0.345098)\n",
       " Gray{Float64}(0.192157)  Gray{Float64}(0.202596)     Gray{Float64}(0.304798)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "colorview(Gray, X.value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nonlinear programming (NLP)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use MLE of Gamma distribution to illustrate some rudiments of nonlinear programming (NLP) in Julia. \n",
    "\n",
    "Let $x_1,\\ldots,x_m$ be a random sample from the gamma density with shape parameter $\\alpha$ and rate parameter $\\beta$:\n",
    "$$\n",
    "f(x) = \\Gamma(\\alpha)^{-1} \\beta^{\\alpha} x^{\\alpha-1} e^{-\\beta x}\n",
    "$$\n",
    "on $(0,\\infty)$. The log likelihood function is\n",
    "$$\n",
    "    L(\\alpha, \\beta) = m [- \\ln \\Gamma(\\alpha) + \\alpha \\ln \\beta + (\\alpha - 1)\\overline{\\ln x} - \\beta \\bar x],\n",
    "$$\n",
    "where $\\overline{x} = \\frac{1}{m} \\sum_{i=1}^m x_i$ and \n",
    "$\\overline{\\ln x} = \\frac{1}{m} \\sum_{i=1}^m \\ln x_i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.7886541275400365"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Random, Statistics, SpecialFunctions\n",
    "Random.seed!(280)\n",
    "\n",
    "function gamma_logpdf(x::Vector, α::Real, β::Real)\n",
    "    m = length(x)\n",
    "    avg = mean(x)\n",
    "    logavg = sum(log, x) / m\n",
    "    m * (- loggamma(α) + α * log(β) + (α - 1) * logavg - β * avg)\n",
    "end\n",
    "\n",
    "x = rand(5)\n",
    "gamma_logpdf(x, 1.0, 1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Many optimization algorithms involve taking derivatives of the objective function. The `ForwardDiff.jl` package implements automatic differentiation. For example, to compute the derivative and Hessian of the log-likelihood with data `x` at `α=1.0` and `β=1.0`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " -1.058800554530917\n",
       "  2.2113458724599635"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using ForwardDiff\n",
    "ForwardDiff.gradient(θ -> gamma_logpdf(x, θ...), [1.0; 1.0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Matrix{Float64}:\n",
       " -8.22467   5.0\n",
       "  5.0      -5.0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ForwardDiff.hessian(θ -> gamma_logpdf(x, θ...), [1.0; 1.0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True parameter values:\n",
      "α = 0.5535947086407722, β = 4.637963827225865\n"
     ]
    }
   ],
   "source": [
    "using Distributions, Random\n",
    "\n",
    "Random.seed!(280)\n",
    "(n, p) = (1000, 2)\n",
    "(α, β) = 5.0 * rand(p)\n",
    "x = rand(Gamma(α, β), n)\n",
    "println(\"True parameter values:\")\n",
    "println(\"α = \", α, \", β = \", β)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use [JuMP.jl](https://github.com/jump-dev/JuMP.jl) to define and solve our NLP problem.\n",
    "\n",
    "Recall that we want to solve the optimization problem:\n",
    "$$\n",
    "\\begin{array}{ll}\n",
    "    \\text{maximize} & L(\\alpha, \\beta)= m [- \\ln \\Gamma(\\alpha) + \\alpha \\ln \\beta + (\\alpha - 1)\\overline{\\ln x} - \\beta \\bar x] \\\\\n",
    "    \\text{subject to} & \\alpha \\ge 0 \\\\\n",
    "    ~ & \\beta \\ge 0\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "Observe the similarity and difference in modeling with [Convex.jl](https://github.com/jump-dev/Convex.jl):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{aligned}\n",
       "\\max\\quad & myf(α, β)\\\\\n",
       "\\text{Subject to} \\quad & α \\geq 1.0e-8\\\\\n",
       " & β \\geq 1.0e-8\\\\\n",
       "\\end{aligned} $$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MLE (JuMP):\n",
      "α = 0.5489638225054433, β = 0.20038521795829256\n",
      "Objective value: -1853.384817360246\n"
     ]
    }
   ],
   "source": [
    "using JuMP, Ipopt, NLopt\n",
    "\n",
    "#m = Model(with_optimizer(Ipopt.Optimizer, print_level=3))\n",
    "m = Model(\n",
    "           optimizer_with_attributes(\n",
    "               NLopt.Optimizer, \"algorithm\" => :LD_MMA\n",
    "           )\n",
    "       )\n",
    "\n",
    "myf(a, b) = gamma_logpdf(x, a, b)\n",
    "JuMP.register(m, :myf, 2, myf, autodiff=true)\n",
    "@variable(m, α >= 1e-8)\n",
    "@variable(m, β >= 1e-8)\n",
    "@NLobjective(m, Max, myf(α, β))\n",
    "\n",
    "print(m)\n",
    "status = JuMP.optimize!(m)\n",
    "\n",
    "println(\"MLE (JuMP):\")\n",
    "println(\"α = \", JuMP.value(α), \", β = \", JuMP.value(β))\n",
    "println(\"Objective value: \", JuMP.objective_value(m))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then convert the rate parameter to the scale parameter to compare the result with `fit_mle()` in the `Distribution` package:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "α = 0.5489638225054433, θ = 4.990388064493541\n",
      "MLE (Distribution package):\n",
      "Gamma{Float64}(α=0.5489317142213413, θ=4.990963923701522)\n"
     ]
    }
   ],
   "source": [
    "println(\"α = \", JuMP.value(α), \", θ = \", 1 / JuMP.value(β))\n",
    "println(\"MLE (Distribution package):\")\n",
    "println(fit_mle(Gamma, x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Acknowledgment\n",
    "\n",
    "Many parts of this lecture note is based on [Dr. Hua Zhou](http://hua-zhou.github.io)'s 2019 Spring Statistical Computing course notes available at <http://hua-zhou.github.io/teaching/biostatm280-2019spring/index.html>."
   ]
  }
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