\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
simplex_method.cpp¶
View page sourceabs_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
\[\begin{split}\begin{array}{rlr}
\R{minimize} & v & \R{w.r.t} \; (u,v) \in \B{R}^2 \\
\R{subject \; to} & u - 1 \leq v \\
& 1 - u \leq v
\end{array}\end{split}\]
We know that the value of \(v\) at the solution is greater than or equal zero. Hence we can reformulate this problem as
\[\begin{split}\begin{array}{rlr}
\R{minimize} & v & \R{w.r.t} \; ( u_- , u_+ , v) \in \B{R}_+^3 \\
\R{subject \; to} & u_+ - u_- - 1 \leq v \\
& 1 - u_+ + u_- \leq v
\end{array}\end{split}\]
This is equivalent to
\[\begin{split}\begin{array}{rlr}
\R{minimize}
& (0, 0, 1) \cdot ( u_+, u_- , v)^T & \R{w.r.t} \; (u,v) \in \B{R}_+^3 \\
\R{subject \; to}
&
\left( \begin{array}{ccc}
+1 & -1 & -1 \\
-1 & +1 & +1
\end{array} \right)
\left( \begin{array}{c} u_+ \\ u_- \\ v \end{array} \right)
+
\left( \begin{array}{c} -1 \\ 1 \end{array} \right)
\leq
0
\end{array}\end{split}\]
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;
}