abs_normal simplex_method: Example and Test

simplex_method.cpp

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ abs_normal simplex_method: Example and Test
Problem
Our original problem is @[@ \R{minimize} \; | u - 1| \; \R{w.r.t} \; u \in \B{R} @]@ We reformulate this as the following problem @[@ \begin{array}{rlr} \R{minimize} & v & \R{w.r.t} \; (u,v) \in \B{R}^2 \\ \R{subject \; to} & u - 1 \leq v \\ & 1 - u \leq v \end{array} @]@ We know that the value of @(@ v @)@ at the solution is greater than or equal zero. Hence we can reformulate this problem as @[@ \begin{array}{rlr} \R{minimize} & v & \R{w.r.t} \; ( u_- , u_+ , v) \in \B{R}_+^3 \\ \R{subject \; to} & u_+ - u_- - 1 \leq v \\ & 1 - u_+ + u_- \leq v \end{array} @]@ This is equivalent to @[@ \begin{array}{rlr} \R{minimize} & (0, 0, 1) \cdot ( u_+, u_- , v)^T & \R{w.r.t} \; (u,v) \in \B{R}_+^3 \\ \R{subject \; to} & \left( \begin{array}{ccc} +1 & -1 & -1 \\ -1 & +1 & +1 \end{array} \right) \left( \begin{array}{c} u_+ \\ u_- \\ v \end{array} \right) + \left( \begin{array}{c} -1 \\ 1 \end{array} \right) \leq 0 \end{array} @]@ which is in the form expected by simplex_method .

Source


# include <limits>
# include <cppad/utility/vector.hpp>
# include "simplex_method.hpp"

bool simplex_method(void)
{     bool ok = true;
     typedef CppAD::vector<double> vector;
     double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
     //
     size_t n = 3;
     size_t m = 2;
     vector A(m * n), b(m), c(n), xout(n);
     A[ 0 * n + 0 ] =  1.0; // A(0,0)
     A[ 0 * n + 1 ] = -1.0; // A(0,1)
     A[ 0 * n + 2 ] = -1.0; // A(0,2)
     //
     A[ 1 * n + 0 ] = -1.0; // A(1,0)
     A[ 1 * n + 1 ] = +1.0; // A(1,1)
     A[ 1 * n + 2 ] = -1.0; // A(1,2)
     //
     b[0]           = -1.0;
     b[1]           =  1.0;
     //
     c[0]           =  0.0;
     c[1]           =  0.0;
     c[2]           =  1.0;
     //
     size_t maxitr  = 10;
     size_t level   = 0;
     //
     ok &= CppAD::simplex_method(level, A, b, c,  maxitr, xout);
     //
     // check optimal value for u
     ok &= std::fabs( xout[0] - 1.0 ) < eps99;
     //
     // check optimal value for v
     ok &= std::fabs( xout[1] ) < eps99;
     //
     return ok;
}

Input File: example/abs_normal/simplex_method.cpp