The first draft of simplex algorithm, simple tests.

What we have now corresponds to "formal simplex algorithm", described in
Cormen's "Intro to Algorithms". It will work *only* if the initial
problem has (0,0,0,...,0) as feasible solution (consequently, it will
work unpredictably if problem was unfeasible or did not have zero-vector as
feasible solution). Moreover, it might cycle.

TODO (first priority)
1. Implement initialize_simplex() procedure, that shall check for
feasibility and generate initial feasible solution. (in particular, code
should pass all 4 tests implemented at the moment)
2. Implement Bland's rule to avoid cycling.
3. Make the code more clear.
4. Implement several non-trivial tests (??) and check algorithm against
them. Debug if necessary.

TODO (second priority)
1. Concentrate on stability and speed (make difficult tests)

modules/optim/test/test_lpsolver.cpp

@@ -0,0 +1,61 @@
+#include "test_precomp.hpp"
+#include "opencv2/optim.hpp"
+
+TEST(Optim_LpSolver, regression)
+{
+    cv::Mat A,B,z,etalon_z;
+
+    if(true){
+    //cormen's example #1
+    A=(cv::Mat_<double>(1,3)<<3,1,2);
+    B=(cv::Mat_<double>(3,4)<<1,1,3,30,2,2,5,24,4,1,2,36);
+    std::cout<<"here A goes\n"<<A<<"\n";
+    cv::optim::solveLP(A,B,z);
+    std::cout<<"here z goes\n"<<z<<"\n";
+    etalon_z=(cv::Mat_<double>(1,3)<<8,4,0);
+    ASSERT_EQ(cv::countNonZero(z!=etalon_z),0);
+    }
+
+    if(true){
+    //cormen's example #2
+    A=(cv::Mat_<double>(1,2)<<18,12.5);
+    B=(cv::Mat_<double>(3,3)<<1,1,20,1,0,20,0,1,16);
+    std::cout<<"here A goes\n"<<A<<"\n";
+    cv::optim::solveLP(A,B,z);
+    std::cout<<"here z goes\n"<<z<<"\n";
+    etalon_z=(cv::Mat_<double>(1,2)<<20,0);
+    ASSERT_EQ(cv::countNonZero(z!=etalon_z),0);
+    }
+
+    if(true){
+    //cormen's example #3
+    A=(cv::Mat_<double>(1,2)<<5,-3);
+    B=(cv::Mat_<double>(2,3)<<1,-1,1,2,1,2);
+    std::cout<<"here A goes\n"<<A<<"\n";
+    cv::optim::solveLP(A,B,z);
+    std::cout<<"here z goes\n"<<z<<"\n";
+    etalon_z=(cv::Mat_<double>(1,2)<<1,0);
+    ASSERT_EQ(cv::countNonZero(z!=etalon_z),0);
+
+    }
+    if(false){
+    //cormen's example #4 - unfeasible
+    A=(cv::Mat_<double>(1,3)<<-1,-1,-1);
+    B=(cv::Mat_<double>(2,4)<<-2,-7.5,-3,-10000,-20,-5,-10,-30000);
+    std::cout<<"here A goes\n"<<A<<"\n";
+    cv::optim::solveLP(A,B,z);
+    std::cout<<"here z goes\n"<<z<<"\n";
+    etalon_z=(cv::Mat_<double>(1,2)<<1,0);
+    ASSERT_EQ(cv::countNonZero(z!=etalon_z),0);
+    }
+}
+
+//TODO
+// get optimal solution from initial (0,0,...,0) - DONE
+// milestone: pass first test (wo initial solution) - DONE
+    // learn how to get initial solution
+    // Blands_rule
+    // 1_more_test & make_more_clear
+    // -> **contact_Vadim**: min_l2_norm, init_optional_fsbl_check, error_codes, comment_style-too_many?, copyTo temp headers
+// ??how to get smallest l2 norm
+// FUTURE: compress&debug-> more_tests(Cormen) -> readNumRecipes-> fast&stable || hill_climbing