examples/3-stick.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3-point stick \n",
    "\n",
    "In this notebook, we will obtain bounds on the logarithmic Sobolev constant of the following Markov chain on 3 points, for $p\\in(0,1/2)$.\n",
    "\n",
    "<img src=\"img/3stick.png\" width=500 />\n",
    "\n",
    "The kernel of this chain is\n",
    "$$ K = \\begin{pmatrix}0&1&0\\\\p&0&1-p\\\\0&1&0 \\end{pmatrix}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LogSobolevRelaxations\n",
    "using SDPAFamily\n",
    "using Plots\n",
    "import IJulia"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When $p$ is small, the problem is poorly condition. For this reason, we need a solver that is capable of solving SDPs using high-precision arithmetic.\n",
    "\n",
    "We choose [SDPA-GMP](http://sdpa.sourceforge.net)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SDPA_GMPFactory (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FEAS_TOL = 1e-9\n",
    "\n",
    "function SDPA_GMPFactory()\n",
    "    return SDPAFamily.Optimizer{Float64}(\n",
    "        presolve = true,\n",
    "        params = ( epsilonDash = FEAS_TOL, epsilonStar = FEAS_TOL)\n",
    "        )\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It remains to calculate lower and upper bounds on the log-Sobolev constant $\\alpha(p)$ for a range of values of $p$. These are displayed graphically and saved in the vectors `α_lower` and `α_upper`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = vcat([0.001], 0.01:0.02:0.49)\n",
    "α_upper = Float64[]\n",
    "α_lower = Float64[]\n",
    "\n",
    "for i in 1:length(p)\n",
    "    \n",
    "    kernel = [0 1 0; p[i] 0 1-p[i]; 0 1 0]\n",
    "    relaxation = default_relaxation(kernel)\n",
    "    \n",
    "    push!(α_lower, solve_relaxation!(relaxation, SDPA_GMPFactory; tol=0.0))\n",
    "    push!(α_upper, get_upper_bound(relaxation, repeats=10)[2])\n",
    "    \n",
    "    IJulia.clear_output(true)\n",
    "    plt = scatter(p[1:i], α_upper, markeralpha = 0.5, markersize = 5, \n",
    "                    markershape=:circle, label=raw\"$\\alpha_{upper}$\",\n",
    "                    ylim=(0,0.55), xlim=(-0.01,0.5), xlabel=raw\"$p$\", legend=:outertopright)\n",
    "    plt = scatter!(plt, p[1:i], α_lower, markeralpha = 1, markersize = 5, markercolor=\"black\",\n",
    "                    markershape=:cross, label=raw\"$\\alpha_{lower}$\") \n",
    "    IJulia.display(plt)\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.5.3",
   "language": "julia",
   "name": "julia-1.5"
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  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.5.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}