cvx_09_barrier_interior_point_method.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    },
    "gpuClass": "standard"
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "#### `Inequality constrained` optimization"
      ],
      "metadata": {
        "id": "D4P76UVhbYme"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Knowing how to use Newton's method for solving equality-constrained analytic centering problem, we now move on to solve optimization with both equality and inequality constraints"
      ],
      "metadata": {
        "id": "yCm6OV6BqjLL"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "For twice-differentiable convex functions $f_i$, we want to\n",
        "\n",
        "$$\\min f_0(x), \\text{s.t. } f_i(x)\\leq 0, i=1, \\cdots, m, \\,Ax=b$$\n",
        "\n",
        "where $A\\in \\mathbf{R}^{p \\times n}, \\, \\text{rank }A=p$, and $p^*$ is optimal value\n",
        "\n",
        "Assume the problem is `strictly feasible`, therefore strong duality holds (Slater's condition)\n",
        "\n",
        "Examples include LP, QP, QCQP, etc."
      ],
      "metadata": {
        "id": "19fx_-oNCIUd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Objective of interior point method"
      ],
      "metadata": {
        "id": "nE2R8RWuSI5L"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The idea of interior point methods is to `convert` solving an inequality constrained problem into solving a `sequence of small number` of unconstrained or equality constrained problems (which we know how to solve, e.g., Newton's method)\n",
        "\n",
        "The sequence of problems is `parameterized` by some parameter, say $t$"
      ],
      "metadata": {
        "id": "5SwbO921R3Gk"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Log-barrier function"
      ],
      "metadata": {
        "id": "34yiWBKuSNKM"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Define log-barrier function $\\phi(x)$ as\n",
        "\n",
        "$$\\phi(x)=-\\sum_{i=1}^m \\log (-f_i(x)), \\text{dom }\\phi=\\{x|f_1(x)<0,\\cdots,f_m(x)<0\\} $$\n",
        "\n",
        "The inequality constraints can be put into the objective\n",
        "\n",
        "$$\\min f_0(x)+\\frac{1}{t}\\phi(x), \\text{s.t. } Ax=b$$\n",
        "\n",
        "where $t>0$\n",
        "\n",
        "This is because as $f_i(x)$ approches 0 (from negative), the penalty increases drastically, making sure $x$ actually never reaches the boundary\n",
        "\n",
        "The approximation improves as $t\\rightarrow \\infty$"
      ],
      "metadata": {
        "id": "xPFLFk0DSUjh"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import cvxpy as cp\n",
        "import time\n",
        "np.set_printoptions(formatter={'float': '{: 0.4f}'.format})\n",
        "\n",
        "plt.style.use('dark_background')\n",
        "# color: https://matplotlib.org/stable/gallery/color/named_colors.htm"
      ],
      "metadata": {
        "id": "SJbo_862R0xc"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def log_barrier(fx, t):\n",
        "    return -np.log(-fx)/t\n",
        "\n",
        "x = np.linspace(-1.5, -1e-10, 400)\n",
        "\n",
        "plt.plot(x, log_barrier(x, 0.01), label=\"$t=0.01$\", linestyle='-', color='m')\n",
        "plt.plot(x, log_barrier(x, 0.02), label=\"$t=0.02$\", linestyle='-', color='c')\n",
        "plt.plot(x, log_barrier(x, 0.05), label=\"$t=0.05$\", linestyle='-', color='y')\n",
        "plt.plot(x, log_barrier(x, 0.2), label=\"$t=0.2$\", linestyle='-', color='r')\n",
        "\n",
        "plt.xlabel(\"$f(x)$\")\n",
        "plt.ylabel(\"$\\phi(x)/t$\")\n",
        "plt.ylim(-100, 600)\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 457
        },
        "id": "MdfDfS81Y64l",
        "outputId": "3078b3dc-40bc-49ac-ac2a-38fce956155e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We apply the chain rule to the log-barrier function\n",
        "\n",
        "$$\\phi(x)=-\\sum_{i=1}^m \\log (-f_i(x))$$\n",
        "\n",
        "where $g(y)=-\\log(y): \\mathbf{R}\\rightarrow \\mathbf{R}$ and $f(x)=-f_i(x): \\mathbf{R^n\\rightarrow\\mathbf{R}}$\n",
        "\n",
        "The chain rule for $h(x)=g(f(x))$ is expressed as\n",
        "\n",
        "$$\\begin{align*}\\nabla h(x) &= g'(f(x))\\nabla f(x)\\\\\n",
        "\\nabla^2 h(x)&=g'(f(x))\\nabla^2 f(x)+g''(f(x))\\nabla f(x) \\nabla f(x)^T\n",
        "\\end{align*}$$\n",
        "\n",
        "where $g'(y)=\\frac{1}{-y}, g''(y)=\\frac{1}{y^2}$\n",
        "\n",
        "Therefore, we have\n",
        "\n",
        "$$\\begin{align*}\n",
        "\\nabla \\phi(x)&=\\sum_{i=1}^m\\frac{1}{-f_i(x)}\\nabla f_i(x) \\\\\n",
        "\\nabla^2 \\phi(x)&=\\sum_{i=1}^m\\frac{1}{f_i(x)^2}\\nabla f_i(x)\\nabla f_i(x)^T+\\sum_{i=1}^m\\frac{1}{-f_i(x)}\\nabla^2 f_i(x)\n",
        "\\end{align*}$$\n"
      ],
      "metadata": {
        "id": "4cDAWrdyTMMo"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Centering problem and central path"
      ],
      "metadata": {
        "id": "Mdthc2mv217_"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "For $t>0$, let $x^*(t)$ as the solution of the `centering problem`\n",
        "\n",
        "$$\\boxed{\\min tf_0(x)+\\phi(x), \\text{s.t. } Ax=b}$$\n",
        "\n",
        "the `central path` is defined as the trajectory that traces the solution as $t$ changes\n",
        "\n",
        "$$\\{x^*(t)|t>0\\}$$\n",
        "\n",
        "(when $t=0$, the objective is to minimize the log-barrier function, which would keep $x$ as far away from each boundary as possible and the solution $x^*(0)$ is the analytic center)"
      ],
      "metadata": {
        "id": "G_sw-lPx26E1"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Optimality condition and Lagrangian"
      ],
      "metadata": {
        "id": "YthKpx-G1y-o"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "For point on `central path` $x^*(t), t>0$, we can write the corresponding optimality condition of the `centering problem` (plug in the gradient of $\\phi(x)$)\n",
        "\n",
        "$$\\begin{align*}Ax&=b, \\,\\, t\\nabla f_0(x)+\\nabla \\phi(x)+A^Tw=0 \\\\ \\Longrightarrow Ax&=b, \\,\\, \\nabla f_0(x)+\\sum_{i=1}^m\\frac{1}{-tf_i(x)}\\nabla f_i(x)+\\frac{1}{t}A^Tw=0\n",
        "\\end{align*}$$\n",
        "\n",
        "We also write out the Lagrangian of the `original inequality constrained problem`\n",
        "\n",
        "$$L(x, \\lambda, \\nu)=f_0(x)+\\sum_{i=1}^m\\lambda_i f_i(x)+\\nu^T(Ax-b)$$\n",
        "\n",
        "Take gradient w.r.t. $x$ and set to zero\n",
        "\n",
        "$$\\nabla f_0(x)+\\sum_{i=1}^m\\lambda_i \\nabla f_i(x)+A^T\\nu=0$$\n",
        "\n",
        "We see that point on the central path $x^*(t)$ `minimizes the Lagrangian` for $\\lambda, \\nu$ given by\n",
        "\n",
        "$$\\lambda^*_i(t)=\\frac{1}{-tf_i(x^*(t))}>0, i=1, \\cdots, m, \\, \\nu^*(t)=\\frac{w}{t}$$"
      ],
      "metadata": {
        "id": "Pe4hvl6W12gp"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Duality gap on central path"
      ],
      "metadata": {
        "id": "TVuamhNV2Qoh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Therefore, by minimizing the centering problem to get $x^*(t)$ (e.g., through Newton's method), we also inadvertently have a point $\\lambda^*, \\nu^*$ that is `dual feasible`, which immediately gives a `lower bound` of optimal value to original problem $p^*$"
      ],
      "metadata": {
        "id": "DLG4RH0b-lLy"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "More specifically, through Lagrange dual function $g(\\lambda, \\nu)$, we have\n",
        "\n",
        "$$\\begin{align*}\n",
        "f_0(x^*(t))\\geq p^*\\geq g(\\lambda^*(t), \\nu^*(t))&=\\inf_{x\\in D} L(x, \\lambda^*(t), \\nu^*(t))\\\\\n",
        "&=L(x^*(t),\\lambda^*(t), \\nu^*(t))\\\\\n",
        "& \\text{plug these into Lagrangian expression}\\\\\n",
        "&=f_0(x^*(t))+\\sum_{i=1}^m\\lambda_i^*(t) f_i(x^*(t))+(\\nu^{*}(t))^T(Ax^*(t)-b) \\\\\n",
        "& x^*(t)\\text{ satisfies equality constraints in centering problem, so } Ax^*(t)=b\\\\\n",
        "&=f_0(x^*(t))+\\sum_{i=1}^m\\lambda_i^*(t) f_i(x^*(t))\\\\\n",
        "& \\text{plug in } \\lambda^*_i(t)=\\frac{1}{-tf_i(x^*(t))}\\\\\n",
        "&=f_0(x^*(t))-\\frac{m}{t}\n",
        "\\end{align*}$$\n",
        "\n",
        "$f_0(x^*(t))\\geq p^*$ holds because $p^*$ is optimal value of the original inequality constrained problem with objective function $f_0(x)$ and $x^*(t)$ is feasible for this problem\n",
        "\n",
        "This indicates that with $x^*(t)$, $f_0(x^*(t))$ would be `no more than` $\\frac{m}{t}$ suboptimal for the original problem"
      ],
      "metadata": {
        "id": "vKDerUHm5gjv"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### KKT conditions"
      ],
      "metadata": {
        "id": "vm6cQCWCF14Q"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "From above, we see that on central path, $x^*, \\lambda^*, \\nu^*$ almost satisfy the KKT conditions for the original problem\n",
        "\n",
        "* primal feasibility $f_i(x^*)\\leq 0, i=1, \\cdots, m,\\, Ax^*=b$\n",
        "* dual feasibility $\\lambda^*\\geq 0$\n",
        "* `approximate` complementary slackness $-\\lambda_i^*f_i(x^*)=1/t$\n",
        "* vanishing gradient\n",
        "$$\\nabla f_0(x^*)+\\sum_{i=1}^m\\frac{1}{-tf_i(x^*)}\\nabla f_i(x^*)+\\frac{1}{t}A^T\\nu^*=0$$\n"
      ],
      "metadata": {
        "id": "3r8tbcbTF4ZT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Barrier method"
      ],
      "metadata": {
        "id": "lhL0r5kFMzR-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Given strictly feasible $x, t=t^0>0, \\mu>1$ ($\\mu$ typically in 10-20), four steps in `outer loop`\n",
        "\n",
        "* compute $x^*(t)$ by solving centering problem (`inner loop` e.g., using Newton's method starting at current $x$)\n",
        "* $x \\leftarrow x^*(t)$\n",
        "* if $m/t< \\epsilon$, quit\n",
        "* $t \\leftarrow \\mu t$\n",
        "\n",
        "$\\mu$ determines `trade-off` between number of outer and inner iterations"
      ],
      "metadata": {
        "id": "jboKcD2VM00t"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Dual feasible points `near` central path"
      ],
      "metadata": {
        "id": "cXe1a7jgnciS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Due to iterative nature and computational accuracy, a point $x$ can be `close to` $x^*(t)$ but not quite centered. In such case, we cannot claim that $\\lambda_i=-\\frac{1}{tf_i(x)}$\n",
        "\n",
        "Instead, the dual feasible $\\lambda_i$ is represented as\n",
        "\n",
        "$$\\lambda_i = -\\frac{1}{tf_i(x)}\\left(1+\\frac{\\nabla f_i(x)^T \\Delta x_{nt}}{-f_i(x)}\\right)$$\n",
        "\n",
        "and we assume $\\Delta x_{nt}$ is small here (close to be centered)\n",
        "\n",
        "In addition, $x$ does `not` minimize $L(x, \\lambda)$ and therefore, we cannot just use the objective of the dual problem associated with $\\lambda$"
      ],
      "metadata": {
        "id": "zD26cGSvnf9W"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### LP example in inequality form"
      ],
      "metadata": {
        "id": "vVIho6WGUliz"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We write the primal problem\n",
        "\n",
        "$$\\min c^Tx, \\,\\, \\text{s.t. } Ax\\leq b, A\\in \\mathbf{R}^{m\\times n}$$\n",
        "\n",
        "and the dual\n",
        "\n",
        "$$\\max -b^T\\lambda, \\,\\, \\text{s.t. } c+A^T\\lambda=0, \\lambda\\geq 0$$"
      ],
      "metadata": {
        "id": "W6UDhjQNUnlZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Since $Ax\\leq b, A^T\\lambda +c=0, \\lambda\\geq 0$, we have the following `duality gap` for inequality LP\n",
        "\n",
        "$$0\\leq \\lambda^T(b-Ax)=b^T\\lambda+c^Tx$$\n",
        "\n",
        "This lower bound immediately implies `weak` duality\n",
        "\n",
        "$$p^* \\geq d^*$$"
      ],
      "metadata": {
        "id": "g-_lwGKj3fYL"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "It can be shown that if primal and dual problems are feasible, then there exist $x^*$ and $\\lambda^*$ such that\n",
        "\n",
        "$$c^Tx^*=-b^T\\lambda^*, Ax^*\\leq b, A^T\\lambda^*+c=0, \\lambda^*\\geq 0$$\n",
        "\n",
        "That is, `strong` duality holds"
      ],
      "metadata": {
        "id": "oRPCAbVz50do"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.seed(42)\n",
        "\n",
        "m, n = 500, 200\n",
        "A = np.random.randn(m, n)\n",
        "x = np.random.rand(n)\n",
        "b0 = A @ x\n",
        "b = b0 + 1 # make x0 strictly feasible\n",
        "c = A.T @ np.random.rand(m)\n",
        "x0 = x\n",
        "print(\"Initial objective value\", c @ x0)"
      ],
      "metadata": {
        "id": "3zIkyjs-zydZ",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "7a00b259-be9a-4772-ff50-471382120eec"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Initial objective value 117.35316777345444\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Solve using CVXPY\n",
        "x_var = cp.Variable(n)\n",
        "constraints = [A @ x_var <= b]\n",
        "objective = cp.Minimize(c @ x_var)\n",
        "\n",
        "problem = cp.Problem(objective, constraints)\n",
        "\n",
        "start_time = time.time()\n",
        "result = problem.solve(solver=cp.SCS, eps=1e-5)\n",
        "print(\"Time taken:\", time.time() - start_time)\n",
        "\n",
        "print(\"Optimal objective value (CVXPY):\", result)\n",
        "# print(\"Optimal x (CVXPY):\", x_var.value)\n",
        "print(\"Feasibility (inequality):\", np.all(A @ x_var.value <= b))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "yYg5e3_PRc_D",
        "outputId": "aed58b68-2e95-4dfa-cfb2-5e0867e58cf3"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Time taken: 5.36395263671875\n",
            "Optimal objective value (CVXPY): -356.2157974048377\n",
            "Feasibility (inequality): False\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "With log-barrier function, the original problem becomes `unconstrained` and we can use Newton's method\n",
        "\n",
        "$$\\min tc^Tx-\\sum \\log(b-Ax)$$\n",
        "\n",
        "The gradient is\n",
        "\n",
        "$$\\nabla f(x)=tc+A^T\\left(\\frac{1}{b-Ax}\\right)$$\n",
        "\n",
        "The Hessian is\n",
        "\n",
        "$$\\nabla^2 f(x)=A^T\\text{diag}\\left(\\frac{1}{(b-Ax)^2}\\right)A$$"
      ],
      "metadata": {
        "id": "uswkZb9x7Sx1"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def cholesky_factorization(A):\n",
        "    m = A.shape[0]\n",
        "    l_mat = A.copy().astype(float)\n",
        "\n",
        "    for k in range(m):\n",
        "        if l_mat[k, k] <= 0:\n",
        "            raise ValueError('Matrix is not positive definite')\n",
        "\n",
        "        l_mat[k+1:, k+1:] -= np.outer(l_mat[k+1:, k], l_mat[k+1:, k]) / l_mat[k, k]\n",
        "        l_mat[k:, k] /= np.sqrt(l_mat[k, k])\n",
        "\n",
        "    return np.tril(l_mat)\n",
        "\n",
        "def forward_substitution(L, b):\n",
        "    m, n = L.shape\n",
        "    x = np.zeros(n)\n",
        "    for i in range(n):\n",
        "        x[i] = (b[i] - np.dot(L[i, :i], x[:i])) / L[i, i]\n",
        "    return x\n",
        "\n",
        "def back_substitution(R, b):\n",
        "    m, n = R.shape\n",
        "    x = np.zeros(n)\n",
        "    for i in range(n - 1, -1, -1):\n",
        "        x[i] = (b[i] - np.dot(R[i, i + 1:], x[i + 1:])) / R[i, i]\n",
        "    return x"
      ],
      "metadata": {
        "id": "n3T6P4M1-FZN"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def ipm_lp_ieq(A, b, c, x0, mu, tol, np_solver=False):\n",
        "    total_iter = 500\n",
        "    alpha = 0.01\n",
        "    beta = 0.5\n",
        "    tol = tol\n",
        "\n",
        "    m, n = A.shape\n",
        "    t = 1\n",
        "    x = x0.copy()\n",
        "\n",
        "    c = c.flatten() if c.ndim > 1 else c\n",
        "    x = x.flatten() if x.ndim > 1 else x\n",
        "\n",
        "    gaps = []\n",
        "    inner_iter = []\n",
        "\n",
        "    for k in range(total_iter):\n",
        "        d = b - A @ x # d = -f(x)\n",
        "        f = t * c @ x - np.sum(np.log(d))\n",
        "        grad = t * c + A.T @ (1 / d)\n",
        "        hess = A.T @ np.diag(1 / (d ** 2)) @ A\n",
        "\n",
        "        # dx = - hess^{-1} nabla f(x)\n",
        "        if np_solver:\n",
        "            dx = - np.linalg.solve(hess, grad)\n",
        "        else:\n",
        "            try:\n",
        "                L = cholesky_factorization(hess)\n",
        "                dx = forward_substitution(L, -grad)\n",
        "                dx = back_substitution(L.T, dx)\n",
        "            except Exception as e:\n",
        "                print(e)\n",
        "        nt_decre_sq = - grad @ dx\n",
        "\n",
        "        s = 1\n",
        "\n",
        "        # Ensure feasibility (domain of f(x))\n",
        "        while np.min(b - A @ (x + s * dx)) < 0:\n",
        "            s *= beta\n",
        "\n",
        "        # Line search\n",
        "        while t * c @ (x + s * dx) - np.sum(np.log(b - A @ (x + s * dx))) > f - alpha * s * nt_decre_sq:\n",
        "            s *= beta\n",
        "\n",
        "        x += s * dx\n",
        "\n",
        "        # Check Newton decrement\n",
        "        if abs(nt_decre_sq / 2) < tol:\n",
        "            inner_iter.append(k)\n",
        "\n",
        "            # # Check whether duality gap is within tolerance\n",
        "            # # Use expression of lambda where x is close to x^*\n",
        "            # # d_i = -f_i(x)\n",
        "            # # lambda_i = 1/(-t f_i)(1+{nabla f_i^T dx/-f_i})\n",
        "            # lamb = 1.0 / (t * d) * (1 + (A @ dx) / d)\n",
        "\n",
        "            # # -f(x)^T lambda\n",
        "            # gap_surrogate = d.T @ lamb\n",
        "            # gap = gap_surrogate\n",
        "\n",
        "            gap = m/t # upper bound\n",
        "\n",
        "            gaps.append(gap)\n",
        "            if gap < tol:\n",
        "                return x, inner_iter, gaps\n",
        "\n",
        "            # Increase t\n",
        "            t = t * mu\n",
        "\n",
        "    # If not converged to within tolerance\n",
        "    inner_iter.append(total_iter)\n",
        "    gaps.append(gap)\n",
        "    print(f\"Max # of iterations ({total_iter}) reached\")\n",
        "    return x, inner_iter, gaps"
      ],
      "metadata": {
        "id": "zMTBukgq8KVo"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "muvals = [2, 10, 50, 120]\n",
        "data = []\n",
        "for mu in muvals:\n",
        "    start_time = time.time()\n",
        "    x, iters, gaps = ipm_lp_ieq(A, b, c, x0, mu, 1e-5, np_solver=True)\n",
        "    print(\"Time taken:\", time.time() - start_time)\n",
        "    print(f\"Converged c^Tx (with mu={mu})\", c @ x)\n",
        "    print(\"Feasibility (inequality)\", np.all(A @ x < b))\n",
        "    print()\n",
        "    l = len(gaps)\n",
        "    iters_ext, gaps_ext = [], []\n",
        "    for i in range(l - 1):\n",
        "        iters_ext.extend([iters[i] - 1, iters[i + 1] - 1])\n",
        "        gaps_ext.extend([gaps[i], gaps[i]])\n",
        "    iters_ext.append(iters[-1] - 1)\n",
        "    gaps_ext.append(gaps[-1])\n",
        "    data.append((iters_ext, gaps_ext))\n",
        "\n",
        "plt.figure(1)\n",
        "for i, mu in enumerate(muvals):\n",
        "    plt.semilogy(data[i][0], data[i][1], label=f\"mu={mu}\")\n",
        "plt.xlabel(\"Newton iterations\")\n",
        "plt.ylabel(\"Duality gap\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 738
        },
        "id": "Fci2FFojKW0H",
        "outputId": "6c3a8288-a2a7-4f09-91e8-de3ba40e2d9d"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Time taken: 0.8245871067047119\n",
            "Converged c^Tx (with mu=2) -356.2155502252572\n",
            "Feasibility (inequality) True\n",
            "\n",
            "Time taken: 0.45833253860473633\n",
            "Converged c^Tx (with mu=10) -356.215551205489\n",
            "Feasibility (inequality) True\n",
            "\n",
            "Time taken: 0.4275951385498047\n",
            "Converged c^Tx (with mu=50) -356.215552565489\n",
            "Feasibility (inequality) True\n",
            "\n",
            "Time taken: 0.3841743469238281\n",
            "Converged c^Tx (with mu=120) -356.2155522409828\n",
            "Feasibility (inequality) True\n",
            "\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### LP with both inequality and equality constraints"
      ],
      "metadata": {
        "id": "Cs-ubodzicSp"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We write the primal problem\n",
        "\n",
        "$$\\min c^Tx, \\,\\, \\text{s.t. } Ax= b, -x\\leq 0, A\\in \\mathbf{R}^{m\\times n}$$\n",
        "\n",
        "and the dual\n",
        "\n",
        "$$\\max -b^T\\nu, \\,\\, \\text{s.t. } A^T\\nu-\\lambda+c=0, \\lambda\\geq 0$$\n",
        "\n",
        "or simply\n",
        "\n",
        "$$\\max -b^T\\nu, \\,\\, \\text{s.t. } A^T\\nu+c\\geq0$$"
      ],
      "metadata": {
        "id": "HKI5pZe4ijUc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "With log-barrier function, the original problem becomes\n",
        "\n",
        "$$\\min tc^Tx-\\sum \\log(x), \\,\\, \\text{s.t. } Ax= b$$\n",
        "\n",
        "The gradient is\n",
        "\n",
        "$$\\nabla f(x)=tc-\\text{diag}\\left(\\frac{1}{x}\\right)\\mathbf{1}$$\n",
        "\n",
        "The Hessian is\n",
        "\n",
        "$$\\nabla^2 f(x)=\\text{diag}\\left(\\frac{1}{x^2}\\right)$$"
      ],
      "metadata": {
        "id": "-kc81QQ-mDs8"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.seed(42)\n",
        "\n",
        "m, n = 50, 100\n",
        "A = np.random.randn(m, n)\n",
        "x0 = np.random.rand(n) # primal feasibility for inequality\n",
        "b = A @ x0 # primal feasibility for equality\n",
        "z = np.random.randn(m)\n",
        "c = A.T @ z + np.random.rand(n) # dual feasibility\n",
        "print(\"Initial objective value\", c @ x0)\n",
        "print(A.shape)\n",
        "print(b.shape)\n",
        "print(c.shape)\n",
        "print(x0.shape)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "THcXBx6dnKgN",
        "outputId": "573b5770-517b-4cd7-cf5c-910fe1ac776e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Initial objective value -0.9716215800840331\n",
            "(50, 100)\n",
            "(50,)\n",
            "(100,)\n",
            "(100,)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Solve using CVXPY\n",
        "x_var = cp.Variable(n)\n",
        "constraints = [A @ x_var == b, x_var >= 0]\n",
        "objective = cp.Minimize(c.T @ x_var)\n",
        "problem = cp.Problem(objective, constraints)\n",
        "\n",
        "start_time = time.time()\n",
        "result = problem.solve(solver=cp.SCS, eps=1e-6)\n",
        "print(\"Time taken:\", time.time() - start_time)\n",
        "\n",
        "print(\"Optimal objective value (CVXPY):\", result)\n",
        "# print(\"Optimal x (CVXPY):\", x_var.value)\n",
        "print(\"Feasibility (equality):\", np.allclose(A @ x_var.value, b))\n",
        "print(\"Feasibility (inequality):\", np.all(x_var.value >= 0))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HN9RysxP-hLQ",
        "outputId": "57f243df-5ff8-4928-a166-46959d16b670"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Time taken: 0.026960134506225586\n",
            "Optimal objective value (CVXPY): -14.901129108789444\n",
            "Feasibility (equality): True\n",
            "Feasibility (inequality): False\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We use standard Newton's method with equality constraints"
      ],
      "metadata": {
        "id": "u1e_UJmmn9Ft"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def ipm_lp_eq_ieq(A, b, c, x_0, mu, n_iter=100, np_solver=False):\n",
        "    alpha = 0.01\n",
        "    beta = 0.5\n",
        "    tol = 1e-3  # Tolerance for duality gap\n",
        "    tol_nt = 1e-5  # Tolerance for Newton decrement\n",
        "    n_iter = n_iter\n",
        "\n",
        "    f_list = []\n",
        "    gaps = []\n",
        "    inner_iter = []\n",
        "    x_list = []\n",
        "    t = 1.0\n",
        "\n",
        "    x=x_0.copy()\n",
        "\n",
        "    c = c.flatten() if c.ndim > 1 else c\n",
        "    x = x.flatten() if x.ndim > 1 else x\n",
        "\n",
        "    m, n = A.shape\n",
        "\n",
        "    for k in range(n_iter):\n",
        "        f = t * c @ x - np.sum(np.log(x))\n",
        "        g = t * c - 1.0 / x\n",
        "        f_list.append(f)\n",
        "        hess_f_inv = np.diag(x**2)\n",
        "\n",
        "        # Solve KKT equations use block elimination\n",
        "\n",
        "        # A hess^-1 A^T nu = - A hess^-1 g\n",
        "        if np_solver:\n",
        "            nu = np.linalg.solve(A @ hess_f_inv @ A.T, - A @ hess_f_inv @ g)\n",
        "        else:\n",
        "            try:\n",
        "                L = cholesky_factorization(A @ hess_f_inv @ A.T)\n",
        "                nu = forward_substitution(L, - A @ hess_f_inv @ g)\n",
        "                nu = back_substitution(L.T, nu)\n",
        "            except Exception as e:\n",
        "                print(e)\n",
        "                break\n",
        "\n",
        "        # hess dx = - (g + A^T nu)\n",
        "        dx = - hess_f_inv @ (g + A.T @ nu)\n",
        "\n",
        "        # Newton decrement\n",
        "        nt_decre_sq = - g @ dx\n",
        "\n",
        "        s = 1.0\n",
        "\n",
        "        # Ensure feasibility\n",
        "        while np.min(x + s * dx) <= 0:\n",
        "            s *= beta\n",
        "\n",
        "        # Line search on f (merge f from rhs with lhs)\n",
        "        while s * t * c @ dx - np.sum(np.log(1 + s * dx / x)) > alpha * s * (g @ dx):\n",
        "            s *= beta\n",
        "\n",
        "        if abs(nt_decre_sq / 2) < tol_nt:\n",
        "            # Evaluate duality gap in original problem\n",
        "            gap = m / t # upper bound\n",
        "\n",
        "            inner_iter.append(k)\n",
        "            gaps.append(gap)\n",
        "            if gap < tol_nt:\n",
        "                return x, inner_iter, gaps, x_list\n",
        "\n",
        "            t *= mu\n",
        "\n",
        "        x += s * dx\n",
        "        x_list.append(x)\n",
        "\n",
        "    return x, inner_iter, gaps, x_list"
      ],
      "metadata": {
        "id": "Go23WoWvoAvW"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "muvals = [2, 10, 50, 120]\n",
        "data = []\n",
        "for mu in muvals:\n",
        "    start_time = time.time()\n",
        "    x, iters, gaps, x_list = ipm_lp_eq_ieq(A, b, c, x0, mu, np_solver=False)\n",
        "    print(\"Time taken:\", time.time() - start_time)\n",
        "    print(f\"Converged c^Tx (with mu={mu})\", c @ x)\n",
        "    print(\"Feasibility (inequality)\", np.all(x >= 0))\n",
        "    print(\"Feasibility (equality)\", np.allclose(A @ x, b))\n",
        "    print()\n",
        "    l = len(gaps)\n",
        "    iters_ext, gaps_ext = [], []\n",
        "    for i in range(l - 1):\n",
        "        iters_ext.extend([iters[i] - 1, iters[i + 1] - 1])\n",
        "        gaps_ext.extend([gaps[i], gaps[i]])\n",
        "    iters_ext.append(iters[-1] - 1)\n",
        "    gaps_ext.append(gaps[-1])\n",
        "    data.append((iters_ext, gaps_ext))\n",
        "\n",
        "plt.figure(1)\n",
        "for i, mu in enumerate(muvals):\n",
        "    plt.semilogy(data[i][0], data[i][1], label=f\"mu={mu}\")\n",
        "plt.xlabel(\"Newton iterations\")\n",
        "plt.ylabel(\"Duality gap\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 806
        },
        "id": "lV2DHF5K-I_D",
        "outputId": "cea92330-7b95-4213-9da2-4643bdfcaa6a"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Time taken: 0.32435035705566406\n",
            "Converged c^Tx (with mu=2) -14.899600106192244\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Time taken: 1.3618998527526855\n",
            "Converged c^Tx (with mu=10) -14.900625682618752\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Time taken: 1.8965120315551758\n",
            "Converged c^Tx (with mu=50) -14.900725576855768\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Time taken: 2.0053184032440186\n",
            "Converged c^Tx (with mu=120) -14.90109655543085\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Feasibility problem (phase I)"
      ],
      "metadata": {
        "id": "M5BZvuefNrY9"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We need to compute a `strictly feasible starting point` for barrier method\n",
        "\n",
        "Find $x$, such that $f_i(x)< 0, i=1, \\cdots, m, \\, Ax=b$"
      ],
      "metadata": {
        "id": "-whonWlxNt6-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##### Basic method"
      ],
      "metadata": {
        "id": "ozwJ9AiZN8zo"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$$\\min_{x, s} s\\,\\,\\, \\text{s.t. } f_i(x)\\leq s, i=1, \\cdots, m,\\,\\, Ax=b$$\n",
        "\n",
        "* if $x, s$ feasible, with $s<0$, $x$ is strictly feasible for feasibility problem, and we can start barrier iterations\n",
        "* if optimal $s$ is positive, then no $x$ is feasible\n",
        "* if zero and attained, then $x$ is feasible (but not strictly)\n",
        "* if zero, not attained, then no $x$ is feasible"
      ],
      "metadata": {
        "id": "bNmUwhsmN_Rd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##### LP example"
      ],
      "metadata": {
        "id": "PoMZ7ivUjmGN"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "For our LP (assume `strictly` feasible, otherwise boundary situation can be tricky)\n",
        "\n",
        "$$\\min c^Tx, \\,\\, \\text{s.t. } Ax= b, -x\\leq 0, A\\in \\mathbf{R}^{m\\times n}$$\n",
        "\n",
        "we can form the feasibility problem as\n",
        "\n",
        "$$\\min_{t, x} t, \\,\\, \\text{s.t. } Ax= b, x\\geq(1-t)\\mathbf{1}, t\\geq 0 $$\n",
        "\n",
        "The original LP is strictly feasible if and only if $t^*<1$"
      ],
      "metadata": {
        "id": "zW2EfxIUiGi5"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Rewrite using $z=\\begin{bmatrix}x+(t-1)\\mathbf{1} \\\\t\\end{bmatrix}$ as input, we have\n",
        "\n",
        "$$\\min_z c_1^Tz, \\,\\, \\text{s.t. } A_1z=b_1, -z \\leq 0$$\n",
        "\n",
        "where\n",
        "\n",
        "$$c_1=\\begin{bmatrix}0\\\\0\\\\\\vdots\n",
        "\\\\0\\\\1\\end{bmatrix},\\,\\,A_1=\\begin{bmatrix}A & -A\\mathbf{1}\\end{bmatrix},\\,\\,b_1=b-A\\mathbf{1}$$\n",
        "\n",
        "We solve this feasibility problem (which is an LP itself), and we are done if the last entry in $z<1$"
      ],
      "metadata": {
        "id": "aTiagMFoihyR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "x_feas = np.linalg.lstsq(A, b, rcond=None)[0].reshape(-1, 1)  # Get feasible x\n",
        "print(\"x_feas >= 0?\", np.all(x_feas >= 0)) # If this is True, then we are done\n",
        "\n",
        "if not np.all(x_feas >= 0):\n",
        "    t0 = -np.min(x_feas) + 2\n",
        "    print(t0)\n",
        "    print(\"t0 >= 0?\", t0 >= 0)\n",
        "\n",
        "    A1 = np.hstack((A, -A @ np.ones((n, 1))))\n",
        "    b1 = b - A @ np.ones((n, 1))\n",
        "\n",
        "    x0 = x_feas + t0 * np.ones((n, 1)) - np.ones((n, 1))\n",
        "    print(\"x0 >= 0?\", np.all(x0 >= 0))\n",
        "\n",
        "    c1 = np.vstack((np.zeros((n, 1)), np.array([[1]])))\n",
        "    z0 = np.vstack((x0, np.array([[t0]])))\n",
        "\n",
        "    print(\"A1.shape:\", A1.shape)\n",
        "    print(\"b1.shape:\", b1.shape)\n",
        "    print(\"c1.shape:\", c1.shape)\n",
        "    print(\"z0.shape:\", z0.shape)\n",
        "    print(z0[-1])"
      ],
      "metadata": {
        "id": "XpZ-R1WfcaJu",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "f3a3138c-ed91-43c2-dcc7-7e95a834a294"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "x_feas >= 0? False\n",
            "2.460745600994705\n",
            "t0 >= 0? True\n",
            "x0 >= 0? True\n",
            "A1.shape: (50, 101)\n",
            "b1.shape: (50, 50)\n",
            "c1.shape: (101, 1)\n",
            "z0.shape: (101, 1)\n",
            "[ 2.4607]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "_, _, _, z_list = ipm_lp_eq_ieq(A1, b1, c1, z0, mu=50, n_iter=50, np_solver=False)"
      ],
      "metadata": {
        "id": "2hGba11Fz7Xm"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "for z in z_list:\n",
        "    # Check t < 1, if so, original problem is strictly feasible\n",
        "    if z[-1] < 1:\n",
        "        # Verify x+(t-1)1 >= 0\n",
        "        x_strictly_feas = np.all(z[:-1] >= 0)\n",
        "        if x_strictly_feas:\n",
        "            x0 = z[:-1] - z[-1]  + 1\n",
        "            print(\"Original LP strictly feasible?\", x_strictly_feas)\n",
        "            break"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "65sajDM_qIn2",
        "outputId": "cf79becd-144e-4f21-ecf7-f9d33eb1168c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Original LP strictly feasible? True\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We will solve the original LP using this initial point x0 from the solution of the feasibility problem"
      ],
      "metadata": {
        "id": "FhcbqIxYrpRg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"Initial objective value\", c @ x0)\n",
        "print()\n",
        "muvals = [2, 10, 50, 120]\n",
        "data = []\n",
        "for mu in muvals:\n",
        "    start_time = time.time()\n",
        "    x, iters, gaps, x_list = ipm_lp_eq_ieq(A, b, c, x0, mu, np_solver=False)\n",
        "    print(\"Time taken:\", time.time() - start_time)\n",
        "    print(f\"Converged c^Tx (with mu={mu})\", c @ x)\n",
        "    print(\"Feasibility (inequality)\", np.all(x >= 0))\n",
        "    print(\"Feasibility (equality)\", np.allclose(A @ x, b))\n",
        "    print()\n",
        "    l = len(gaps)\n",
        "    iters_ext, gaps_ext = [], []\n",
        "    for i in range(l - 1):\n",
        "        iters_ext.extend([iters[i] - 1, iters[i + 1] - 1])\n",
        "        gaps_ext.extend([gaps[i], gaps[i]])\n",
        "    iters_ext.append(iters[-1] - 1)\n",
        "    gaps_ext.append(gaps[-1])\n",
        "    data.append((iters_ext, gaps_ext))\n",
        "\n",
        "plt.figure(1)\n",
        "for i, mu in enumerate(muvals):\n",
        "    plt.semilogy(data[i][0], data[i][1], label=f\"mu={mu}\")\n",
        "plt.xlabel(\"Newton iterations\")\n",
        "plt.ylabel(\"Duality gap\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 842
        },
        "id": "YqymhC7EqLMV",
        "outputId": "5507e898-67cc-44ac-8dd8-cfc03cb51eb5"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Initial objective value 41.35722626454216\n",
            "\n",
            "Time taken: 0.34185290336608887\n",
            "Converged c^Tx (with mu=2) -14.89960005095006\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Time taken: 1.2361891269683838\n",
            "Converged c^Tx (with mu=10) -14.9006257608807\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Time taken: 0.3501744270324707\n",
            "Converged c^Tx (with mu=50) -14.900725562605299\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Time taken: 2.470705032348633\n",
            "Converged c^Tx (with mu=120) -14.90109695114552\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Infeasible start Newton's method for LP"
      ],
      "metadata": {
        "id": "ArqZg41YcV7q"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We can also use infeasible start Newton's method without having to relying on phase I\n",
        "\n",
        "We write the problem objective again\n",
        "\n",
        "$$\\min tc^Tx-\\sum \\log(x), \\,\\, \\text{s.t. } Ax= b$$\n",
        "\n",
        "The gradient is\n",
        "\n",
        "$$\\nabla f(x)=tc-\\text{diag}\\left(\\frac{1}{x}\\right)\\mathbf{1}$$\n",
        "\n",
        "The Hessian is\n",
        "\n",
        "$$\\nabla^2 f(x)=\\text{diag}\\left(\\frac{1}{x^2}\\right)$$\n",
        "\n",
        "Recall for infeasible start Newton's method, the KKT equations are\n",
        "\n",
        "$$\\boxed{\\begin{bmatrix}\\nabla^2 f(x) & A^T\\\\ A & 0 \\end{bmatrix}\\begin{bmatrix} \\Delta x_{nt} \\\\ \\Delta \\nu_{nt} \\end{bmatrix}=-\\begin{bmatrix}\\nabla f(x)+A^T\\nu \\\\ Ax-b \\end{bmatrix}}$$\n",
        "\n",
        "and we will solve through block elimination\n",
        "\n",
        "$$\\begin{bmatrix}\n",
        "H & A^T \\\\ A & 0\n",
        "\\end{bmatrix}\\begin{bmatrix}\n",
        "v\\\\ w\n",
        "\\end{bmatrix}=-\\begin{bmatrix}\n",
        "g\\\\h\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "* if $H$ is nonsingular\n",
        "$$AH^{-1}A^Tw=h-AH^{-1}g,\\,\\, Hv=-(g+A^Tw)$$"
      ],
      "metadata": {
        "id": "NSpn0AN1ckHv"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The following steps are followed for each $t$\n",
        "\n",
        "Start with a point $x\\in \\text{dom }f$ and $\\nu$\n",
        "* compute primal and dual Newton `step` $\\Delta x_{nt}, \\Delta \\nu_{nt}$\n",
        "* line search for `step size`, starting at $s=1$, backtrack $s\\leftarrow \\beta s$ until\n",
        "$$\\|r(x+s\\Delta x_{nt}, \\nu+s\\Delta \\nu_{nt})\\|_2\\leq (1-\\alpha s)\\|r(x, \\nu)\\|_2$$\n",
        "* update\n",
        "$$x\\leftarrow x+s\\Delta x_{nt}, \\nu\\leftarrow \\nu+s\\Delta \\nu_{nt}$$\n",
        "* terminate if $Ax=b$ and $\\|r(x,\\nu)\\|_2\\leq \\epsilon$\n",
        "\n",
        "Update $t\\leftarrow \\mu t$, repeat these steps"
      ],
      "metadata": {
        "id": "YxTTsd4zdDGz"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def ipm_lp_eq_ieq_infeasible_start(A, b, c, mu, n_iter=500, np_solver=False):\n",
        "    alpha = 0.01\n",
        "    beta = 0.3\n",
        "    tol_norm = 1e-6\n",
        "    tol_gap = 1e-5\n",
        "    m, n = A.shape\n",
        "\n",
        "    rp_list = []\n",
        "    rd_list = []\n",
        "    r_list = []\n",
        "    gaps = []\n",
        "    inner_iter = []\n",
        "\n",
        "    # infeasible start\n",
        "    np.random.seed(42)\n",
        "    t = 1.0\n",
        "\n",
        "    # x must still be in the domain of f(x), that is, x > 0\n",
        "    x = np.random.rand(n) + 0.1\n",
        "    print(\"Feasible initial x?\", np.allclose(A @ x, b))\n",
        "    print(\"Initial objective value\", c @ x)\n",
        "    nu = np.zeros(m)\n",
        "\n",
        "    # Outer loop: increase t until gap is small enough or max iterations reached\n",
        "    total_iters = 0\n",
        "    while total_iters < n_iter:\n",
        "        # Inner loop: solve centering problem for fixed t\n",
        "        for k in range(n_iter):\n",
        "            total_iters += 1\n",
        "            rp = A @ x - b\n",
        "            rd = t * c - 1.0 / x + A.T @ nu\n",
        "            rp_list.append(np.linalg.norm(rp))\n",
        "            rd_list.append(np.linalg.norm(rd))\n",
        "            norm_r = np.linalg.norm(np.hstack([rd, rp]))\n",
        "            r_list.append(norm_r)\n",
        "\n",
        "            # Check convergence for the current t\n",
        "            if norm_r < tol_norm:\n",
        "                gap = m / t # upper bound\n",
        "                inner_iter.append(k + total_iters) # k is local, so need to add total_iters\n",
        "                gaps.append(gap)\n",
        "                if gap < tol_gap:\n",
        "                    # Sufficiently close to optimal solution\n",
        "                    return x, inner_iter, gaps\n",
        "                else:\n",
        "                    t *= mu\n",
        "                    break\n",
        "\n",
        "            hess_f_inv = np.diag(x**2)\n",
        "\n",
        "            # Solve KKT equations use block elimination\n",
        "            # A hess^-1 A^T dnu = rp - A hess^-1 rd\n",
        "            if np_solver:\n",
        "                dnu = np.linalg.solve(A @ hess_f_inv @ A.T, rp - A @ hess_f_inv @ rd)\n",
        "            else:\n",
        "                try:\n",
        "                    L = cholesky_factorization(A @ hess_f_inv @ A.T)\n",
        "                    dnu = forward_substitution(L, rp - A @ hess_f_inv @ rd)\n",
        "                    dnu = back_substitution(L.T, dnu)\n",
        "                except Exception as e:\n",
        "                    print(e)\n",
        "                    break\n",
        "\n",
        "            # hess dx = - (rd + A^T dnu)\n",
        "            dx = - hess_f_inv @ (rd + A.T @ dnu)\n",
        "\n",
        "            s = 1.0\n",
        "            # Ensure feasibility\n",
        "            while np.min(x + s * dx) <= 0:\n",
        "                s *= beta\n",
        "            # Line search on norm of residual\n",
        "            while np.linalg.norm(\n",
        "                np.hstack([t * c - 1.0 / (x + s * dx) + A.T @ (nu + s * dnu), A @ (x + s * dx) - b])\n",
        "            ) > (1 - alpha * s) * norm_r:\n",
        "                s *= beta\n",
        "\n",
        "            x += s * dx\n",
        "            nu += s * dnu\n",
        "        else:\n",
        "            # If we complete the for-loop without a break, it means we didn't\n",
        "            # meet the norm_r < tol condition and didn't return.\n",
        "            # We've hit max iterations without convergence.\n",
        "            break\n",
        "\n",
        "    return x, inner_iter, gaps"
      ],
      "metadata": {
        "id": "nRNIKELPcX-M"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "muvals = [2, 10, 50, 120]\n",
        "data = []\n",
        "for mu in muvals:\n",
        "    start_time = time.time()\n",
        "    x, iters, gaps = ipm_lp_eq_ieq_infeasible_start(A, b, c, mu=mu, np_solver=False)\n",
        "    print(\"Time taken:\", time.time() - start_time)\n",
        "    print(f\"Converged c^Tx (with mu={mu})\", c @ x)\n",
        "    print(\"Feasibility (inequality)\", np.all(x >= 0))\n",
        "    print(\"Feasibility (equality)\", np.allclose(A @ x, b))\n",
        "    print()\n",
        "    l = len(gaps)\n",
        "    iters_ext, gaps_ext = [], []\n",
        "    for i in range(l - 1):\n",
        "        iters_ext.extend([iters[i] - 1, iters[i + 1] - 1])\n",
        "        gaps_ext.extend([gaps[i], gaps[i]])\n",
        "    iters_ext.append(iters[-1] - 1)\n",
        "    gaps_ext.append(gaps[-1])\n",
        "    data.append((iters_ext, gaps_ext))\n",
        "\n",
        "plt.figure(1)\n",
        "for i, mu in enumerate(muvals):\n",
        "    plt.semilogy(data[i][0], data[i][1], label=f\"mu={mu}\")\n",
        "plt.xlabel(\"Newton iterations\")\n",
        "plt.ylabel(\"Duality gap\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 955
        },
        "id": "3QVpTY3TgnNE",
        "outputId": "3d33ad8c-ac1d-4947-fff9-5209a1c50e69"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Feasible initial x? False\n",
            "Initial objective value 29.846488899047426\n",
            "Time taken: 0.9432690143585205\n",
            "Converged c^Tx (with mu=2) -14.90111962565468\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Feasible initial x? False\n",
            "Initial objective value 29.846488899047426\n",
            "Time taken: 0.5318305492401123\n",
            "Converged c^Tx (with mu=10) -14.901120586118418\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Feasible initial x? False\n",
            "Initial objective value 29.846488899047426\n",
            "Time taken: 0.33930444717407227\n",
            "Converged c^Tx (with mu=50) -14.90111758612111\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n",
            "Feasible initial x? False\n",
            "Initial objective value 29.846488899047426\n",
            "Time taken: 2.450629472732544\n",
            "Converged c^Tx (with mu=120) -14.901125344990183\n",
            "Feasibility (inequality) True\n",
            "Feasibility (equality) True\n",
            "\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Note on standard LP form"
      ],
      "metadata": {
        "id": "H7VGoYIJ6mt1"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Commonly, a `general LP` has the form\n",
        "\n",
        "$$\\min c^Tx,\\,\\, \\text{s.t. }Gx\\leq h, Ax=b$$\n",
        "\n",
        "We can show that this general form LP can be transformed into the `standard form` that we know how to solve\n",
        "\n",
        "\n",
        "$$\\min c^Tx, \\,\\, \\text{s.t. } Ax= b, x\\geq 0$$"
      ],
      "metadata": {
        "id": "8wIjBeyB6omw"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "To do this, we introduce slack variable $s_i$ for the inequalities\n",
        "\n",
        "$$\\min c^Tx,\\,\\, \\text{s.t. }Gx+s = h, Ax=b, s\\geq 0$$\n",
        "\n",
        "and we express $x$ using its positive and negative part\n",
        "\n",
        "$$x=x^+-x^-, x^+, x^-\\geq 0$$\n",
        "\n",
        "and we can rewrite the general form LP as `standard form` (of variables $x^+, x^-, s$)\n",
        "\n",
        "$$\\min c^Tx^+-c^Tx^-, \\,\\, \\text{s.t.  } Gx^+-Gx^-+s=h,\\, Ax^+-Ax^-=b, \\,x^+, x^-, s \\geq 0$$"
      ],
      "metadata": {
        "id": "8B7cy4YH7-Oz"
      }
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "by3wH6op_L3n"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}