--amend

mp6/Project_6_Maggioni_Claudio/auxiliary.m

@@ -1,10 +1,8 @@
 function [B,D,c_B,c_D,x_B,x_D,index_B,index_D] = auxiliary (A_aug,c_aug,h,m,n)
 % The auxiliary problem is always a minimization problem
  
-% The output will be: B and D (basic and nonbasic matrices), c_B and c_D
+% The output will be: B and D (basic and nonbasic matrices), c_B and c_D (subdivision of the coefficient vector in basic and nonbasic parts), x_B
-% (subdivision of the coefficient vector in basic and nonbasic parts), x_B
+% and x_D (basic and nonbasic variables) and index_B and index_D (to keep track of the variables indices)
-% and x_D (basic and nonbasic variables) and index_B and index_D (to keep 
-% track of the variables indices)
 
 % Redefine the problem by introducing the artificial variables required by
 % the auxiliary problem (the objective function has to reach value 0)
@@ -39,8 +37,7 @@ r_D = c_Daux - c_Baux*BiD;
 while(z~=0)
     
     % Find nonnegative index
-    idxIN = find(r_D == min(r_D));
+    idxIN = find(r_D==min(r_D));
-    
     % Using Bland's rule to avoid cycling
     if(size(idxIN,2)>1)
         idxIN = min(idxIN);
@@ -85,21 +82,17 @@ while(z~=0)
     Bih = B\h;
     
     % Compute reduced cost coefficients
-    r_D = c_Daux - (c_Baux * BiD);
+    r_D = c_Daux - c_Baux*BiD;
- 
-    % Exercise 4: Uncomment this to experience "oscillation"
-    %fprintf("iteration %d: in %d out %d z %d\n", nIter+1, idxIN, idxOUT, z);
-    %B
-    %D
     
     % Detect inefficient loop if nIter > total number of basic solutions
     nIter = nIter + 1;
-    if nIter > itMax
+    if(nIter>itMax)
        error('The original LP problem does not admit a feasible solution.');
     end
 
-    x_B = Bih - BiD * x_D;
+    x_B = Bih - BiD*x_D;
-    z = c_Baux * x_B;
+    z = c_Baux*x_B;
+    
 end
 
 check = index_D<(n+m+1);

mp6/Project_6_Maggioni_Claudio/exercise4.m

@@ -1,9 +1,8 @@
 type = 'max';
-A =    [4 3; 4 1; 4 2; 1 0; 0 1];
+A =    [4 3; 4 1; 4 2];
-sign = [-1 ; -1 ; -1 ; 1  ; 1  ];
+sign = [-1 ; -1 ; -1 ];
-h =    [12 ; 8  ; 8  ; 0  ; 0  ];
+h =    [12 ; 8  ; 8  ];
 c = [3 4];
 
-
 [z,x_B,index_B] = simplex (type,A,h,c,sign);
 histogram([index_B x_B']')

mp6/Project_6_Maggioni_Claudio/simplexSolve.m

@@ -34,7 +34,7 @@ while optCheck ~= tol
     ratio = Bih ./ BiD(:,idxIN);
     
     % Find the smallest positive ratio
-    idxOUT = find(ratio == min(ratio(ratio > 0)));
+    idxOUT = find(ratio == min(ratio(ratio >= 0)));
     
     out = B(:,idxOUT);
     c_out = c_B(1,idxOUT);

mp6/project_6_Maggioni_Claudio.tex

@@ -361,16 +361,17 @@ The MATLAB implementation for this problem can be found under file
 \texttt{Project\_6\_Claudio\_Maggioni/}).
 
 The simplex method implementation for this problem fails with an error in the
-auxiliary problem initialization. The error message is ``The original LP problem does not admit a feasible
+\texttt{simpleSolve} iterative method. The error message is
-solution''. This error message appears when the simplex method does not
+``Incorrect loop, more iterations than the number of basic solutions''.
+This error message appears when the simplex method does not
 converge in the expected maximum number of iterations. The large number of
-iterations (793 before forced termination) is caused by an ``oscillation'' or
+iterations (11 before forced termination) is caused by an ``oscillation'' or
 vicious cycle of swap operations between matrix $B$ and matrix $D$. This
 oscillation is cause by the fact that entering and departing variable indexes do
-not change, and stay fixed at values 6 and 4 from the second iteration onwards.
+not change, and stay fixed at values 1 and 3 from the second iteration onwards.
 This causes an unnecessary continuous oscillation between two solutions that are
-equally not optimal ($z = 28$) according to the auxiliary problem definition,
+equally not optimal ($\text{optCheck} = 1$) according to the simplex algorithm definition,
-terminating the auxiliary algorithm abnormally after too many iterations.
+terminating the simplex algorithm algorithm abnormally after too many iterations.
 
 \subsection{Look at the number of constraints and at the number of unknowns:
 what can you notice about the underlying system of equations? Represent them
@@ -385,6 +386,14 @@ triangular shape, and it is made up only of one constraint other than the
 trivial positivity constraints at the axes (i.e. the constraints different from
 $4x +2y \leq 8$ do not make a difference in the feasability region).
 
+The system has 2 variables and 3 constraints, other than the positivity checks
+on the variables. But, for what said above, 2 out of the three constraints are
+useless and thus make the feasibility region defined like a 1-condition problem.
+
+This lack of ``useful'' condition may cause a ill-defined set of constraints
+that could produce two equally ``good'' not-optimal solutions, thus causing the
+infinite loop in the simplex algorithm.
+
 \begin{figure}[h]
   \centering
 \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45]