\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
qp_interior.cpp¶
View page sourceabs_normal qp_interior: 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
\[\begin{split}\begin{array}{rlr}
\R{minimize} & v & \R{w.r.t} \; (u,v) \in \B{R}^2 \\
\R{subject \; to} & u - 1 \leq v \\
& 1 - u \leq v
\end{array}\end{split}\]
This is equivalent to
\[\begin{split}\begin{array}{rlr}
\R{minimize}
& (0, 1) \cdot (u, v)^T & \R{w.r.t} \; (u,v) \in \B{R}^2 \\
\R{subject \; to}
&
\left( \begin{array}{cc} 1 & -1 \\ -1 & -1 \end{array} \right)
\left( \begin{array}{c} u \\ v \end{array} \right)
+
\left( \begin{array}{c} -1 \\ 1 \end{array} \right)
\leq
0
\end{array}\end{split}\]
which is in the form expected by qp_interior .
Source¶
# include <limits>
# include <cppad/utility/vector.hpp>
# include "qp_interior.hpp"
bool qp_interior(void)
{ bool ok = true;
typedef CppAD::vector<double> vector;
//
size_t n = 2;
size_t m = 2;
vector C(m*n), c(m), G(n*n), g(n), xin(n), xout(n), yout(m), sout(m);
C[ 0 * n + 0 ] = 1.0; // C(0,0)
C[ 0 * n + 1 ] = -1.0; // C(0,1)
C[ 1 * n + 0 ] = -1.0; // C(1,0)
C[ 1 * n + 1 ] = -1.0; // C(1,1)
//
c[0] = -1.0;
c[1] = 1.0;
//
g[0] = 0.0;
g[1] = 1.0;
//
// G = 0
for(size_t i = 0; i < n * n; i++)
G[i] = 0.0;
//
// If (u, v) = (0,2), C * (u, v) + c = (-2,-2)^T + (1,-1)^T < 0
// Hence (0, 2) is feasible.
xin[0] = 0.0;
xin[1] = 2.0;
//
double epsilon = 99.0 * std::numeric_limits<double>::epsilon();
size_t maxitr = 10;
size_t level = 0;
//
ok &= CppAD::qp_interior(
level, c, C, g, G, epsilon, maxitr, xin, xout, yout, sout
);
//
// check optimal value for u
ok &= std::fabs( xout[0] - 1.0 ) < epsilon;
// check optimal value for v
ok &= std::fabs( xout[1] ) < epsilon;
//
return ok;
}