maggicl/NC
Archived
1
0
Fork 0
Code Issues Pull requests Projects Releases Wiki Activity

--amend

Browse source
This commit is contained in:
Claudio Maggioni 2020-12-19 23:50:07 +01:00
parent e5b5431e06
commit bf949c82e6
5 changed files with 27 additions and 26 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

17
mp6/Project_6_Maggioni_Claudio/auxiliary.m
View file

@ -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
% (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)
% 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
% 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)
@ -40,7 +38,6 @@ while(z~=0)
% Find nonnegative index
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);
% Exercise 4: Uncomment this to experience "oscillation"
%fprintf("iteration %d: in %d out %d z %d\n", nIter+1, idxIN, idxOUT, z);
%B
%D
r_D = c_Daux - c_Baux*BiD;
% 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;
z = c_Baux*x_B;
end
check = index_D<(n+m+1);

7
mp6/Project_6_Maggioni_Claudio/exercise4.m
View file

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

2
mp6/Project_6_Maggioni_Claudio/simplexSolve.m
View file

@ -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);

BIN
mp6/project_6_Maggioni_Claudio.pdf
View file

Binary file not shown.

21
mp6/project_6_Maggioni_Claudio.tex
View file

@ -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
solution''. This error message appears when the simplex method does not
\texttt{simpleSolve} iterative method. The error message is
``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,
terminating the auxiliary algorithm abnormally after too many iterations.
equally not optimal ($\text{optCheck} = 1$) according to the simplex algorithm definition,
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]