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degeneracyHunter3.m
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degeneracyHunter3.m
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function [nNonPivot, nDegenSets, degenEqnsNames] = degeneracyHunter3(varargin)
%% ***** Degeneracy Hunter *****
% This script helps identify rank deficiencies (degeneracies) in the
% Jacobian extracted from GAMS. This is particularly useful for model
% refinement of NLPs.
%
% Created by Alex Dowling at Carnegie Mellon University
% awdowlin@andrew.cmu.edu
% Last Updated: June 10th, 2014
%
% Number of inputs
% No inputs: act like a script
%
% Three inputs: Classic Degeneracy Hunter, MILP mode
% Input 1: Consider Weakly Active Constraints?
% Input 2: Consider Variable Bounds?
% Input 3: Filename to write output to ([ ] means display to console)
%% Toggle on/off core modules
% Run Degeneracy Hunter Module
module.degenHunt.tog = true;
% Run Second Order Condition Checker Module
module.SOCcheck.tog = false;
% Run Other Modules
module.other.tog = false;
% Run Setup Module
if(module.degenHunt.tog || module.SOCcheck.tog || module.other.tog)
module.setup.tog = true;
end
%% General Settings
% Should the code use sparse or dense linear algebra routines?
module.sparse = true;
% Tolerance for classifying multipliers
module.multTol = 1E-10;
% Tolerance for rank command
module.rankTol = 1E-10;
% Verbose output
module.verbose = false;
% Path to rank helper function
myrank = @(A) rankHelper(A,module.sparse,module.rankTol);
%% Degeneracy Hunter Module Specific Settings
% Should weakly active constraints be considered when checking for
% degeneracies?
module.degenHunt.weakAct = false;
% Should variables bounds be considered when checking for degeneracies?
module.degenHunt.varBounds = true;
% Use the heuristic search to identify degenerate equations
module.degenHunt.heurstic = false;
% Solve MILPs to identify degenerate equations
module.degenHunt.optimal = true;
% Tolerance for displaying equations in MILP degeneracy hunter
module.degenHunt.tol = 1E-6;
% Specify equations for Hueristic mode - only for debugging/special cases
% module.suspectEquations = [ ];
% module.suspectEqnGroup = [];
%% Second Order Condition Checker Settings
% Should weakly active constraints be considered when checking second
% order conditions?
module.SOCcheck.weakAct = true;
%% Other Module Specific Settings
% % Module 4 - This module identifies the largest and smallest elements of
% % the Jacobian.
module.other.four = true;
% % Order of magnitude for listing the largest (pos 1) and smallest (pos 2)
% % elements of the Jacobian
module.other.fourTol = [1, 6];
% % Module 5 - This module conducts analysis using the SVD of the Jacobian
% % to identify equations and variables contributing to ill-conditioning.
module.other.five = false;
%
% module.fiveVerbose = true;
%
% module.fiveCheckSmallSV = true;
%
% % Module 6 - Print all non-zero elements for a specified variable or
% % equation
module.other.six = false;
module.other.sixSearch = {'F(SV2)'};
module.other.sixType = {'v'};
tocM = @() toc;
disp = @(s) fprintf('%s\n',s);
%% Parse Function Input
if(nargin == 3)
module.degenHunt.tog = true;
module.SOCcheck.tog = false;
module.other.tog = false;
module.degenHunt.heurstic = false;
module.degenHunt.optimal = true;
module.degenHunt.weakAct = varargin{1};
module.degenHunt.varBounds = varargin{2};
if(~isempty(varargin{3}))
% Open file
fid = fopen(varargin{3},'w');
% Overload disp
disp = @(s) fprintf(fid,'%s\n',s);
% Redefine tocM
tocM = @() disp(['Elapsed time is ',num2str(toc),' seconds.']);
end
end
nNonPivot = -1;
nDegenSets = -1;
degenEqnsNames = [];
%% Setup Module
%%%%%%%%%%%%%%%
if(module.setup.tog)
% % % % % % % % % % % % % % % % %
% Classify equations and bounds %
% % % % % % % % % % % % % % % % %
% Extract equation names from GAMS
s.name = 'i';
eName = rgdx('hessian',s);
nEq = length(eName.val);
s.name = 'j';
vName = rgdx('hessian',s);
nVar = length(vName.val);
% Extract equation multipliers and bounds from GAMS
s.name = 'e';
s.form = 'full';
s.field = 'm';
eqnMult = rgdx('hessian',s);
s.field = 'lo';
eqnLow = rgdx('hessian',s);
s.field = 'up';
eqnUp = rgdx('hessian',s);
s.field = 'l';
eqnLvl = rgdx('hessian',s);
% Locate equality constraints
eqlCnst = find(eqnUp.val(1:nEq) - eqnLow.val(1:nEq) < module.multTol);
% Locate strongly active equality constraints
eqlStrg = intersect( eqlCnst, find(abs(eqnMult.val(1:nEq)) > module.multTol) );
% Locate weakly active equality constraints
eqlWeak = setdiff(eqlCnst, eqlStrg);
% Locate inequality constraints
ineqlCnst = setdiff(1:nEq, eqlCnst);
% Locate active inequality constraints
actv = intersect(ineqlCnst, ...
find(eqnLvl.val(1:nEq) - eqnLow.val(1:nEq) < module.multTol | eqnUp.val(1:nEq) - eqnLvl.val(1:nEq) < module.multTol ));
% Locate strongly active inequality constraints
actvS = intersect(actv, find(abs(eqnMult.val(1:nEq)) > module.multTol));
% Locate weakly active inequality constraints
actvW = setdiff(actv, actvS);
% Determine sign on inequality constraints. Want h(x) <= 0
ineqlSign = 1.0*(isfinite(eqnUp.val(1:nEq))) - 1.0*(isfinite(eqnLow.val(1:nEq)));
% Extract variable multiplers and bounds from GAMS
s.name = 'x';
s.form = 'full';
s.field = 'm';
varMult = rgdx('hessian',s);
s.field = 'lo';
varLow = rgdx('hessian',s);
s.field = 'up';
varUp = rgdx('hessian',s);
s.field = 'l';
varLvl = rgdx('hessian',s);
% Locate active variable bounds
actvB = find(varLvl.val(nEq+1:end) - varLow.val(nEq+1:end) < module.multTol | varUp.val(nEq+1:end) - varLvl.val(nEq+1:end) < module.multTol );
% Locate strongly active variable bounds
actvBS = intersect(actvB, find(abs(varMult.val(nEq+1:end)) > module.multTol));
% Locate weakly active variable bounds
actvBW = setdiff(actvB, actvBS);
% Determine which active variable bounds are lower
actvBLow = intersect(actvB, find(varLvl.val(nEq+1:end) - varLow.val(nEq+1:end) < module.multTol));
% Determine which active variables bounds are upper
actvBUp = intersect(actvB, find(varUp.val(nEq+1:end) - varLvl.val(nEq+1:end) < module.multTol));
% Note: With fixed variables there are two bounds and upper and lower.
% Only one may be strongly active. Both may be weakly active. For the
% purposes of null space calculations these will be treated as strongly
% active upper bounds. This is equivalent to treating them like
% equality constraints
actvBBoth = intersect(actvBLow,actvBUp);
actvBS = union(actvBS, actvBBoth);
actvBUp = union(actvBUp, actvBBoth);
actvBLow = setdiff(actvBLow, actvBBoth);
% % Determine which active variable bounds are upper
% actvBUp = setdiff(actvB, actvBLow);
% % % % % % % % % % % % %
% Assemble Jacobian(s) %
% % % % % % % % % % % % %
clear s;
% Extract the Jacobian from GAMS (GDX file)
s.name = 'A';
Jac = rgdx('hessian',s);
% Rows: Equations, Columns: Variables
i = Jac.val(:,1);
j = Jac.val(:,2) - nEq;
k = i > 0 & j > 0;
J = sparse(i(k), j(k), Jac.val(k,3), nEq, nVar);
% Assemble Jacobian matricies for bounds
Jbnd_actv = zeros(length(actvB),nVar);
Jbnd_Sactv = zeros(length(actvBS),nVar);
Jbnd_Wactv = zeros(length(actvBW),nVar);
if(~isempty(actvB) && (module.SOCcheck.tog || (module.degenHunt.tog && module.degenHunt.varBounds)))
% Note: For the SOC Checker Module needs two bound Jacobians: one with
% all active bounds. The other with only strongly active bounds.
%
% The DH Module bound matrix (active vs strongly active) depends on the settings
% Note: This code won't work. Need to preserve variable order for DH module.
% countA = 0;
% countAS = 0;
% if(~isempty(actvBLow))
% for i = 1:length(actvBlow)
% Jbnd_actv(countA + i,actvBLow(i)) = -1;
% if(~isempty(intersect(actvBLow(i), actvBS)));
% Jbnd_Sactv(countAS + i, actvBLow(i)) = -1;
% end
% end
% countA = countA + length(actvBLow);
% countAS = countAS + length(intersect(actvBLow, actvBS));
% end
% if(~isempty(actvBUp))
% for i = 1:length(actvBlow)
% Jbnd_actv(countA + i,actvBLow(i)) = 1; % Check this. Should this be negative one?
% if(~isempty(intersect(actvBLow(i), actvBS)));
% Jbnd_Sactv(countAS + i, actvBLow(i)) = 1;
% end
% end
% countA = countA + length(actvBLow);
% countAS = countAS + length(intersect(actvBLow, actvBS));
% end
count = 1;
countW = 1;
chckS = false;
for i = 1:length(actvB)
if(isempty(intersect(actvBS,actvB(i)))) % Bound is weakly active
chckS = false;
else % Bound is strongly active
chckS = true;
end
if(~isempty(intersect(actvB(i), actvBLow))) % Lower bound
Jbnd_actv(i,actvB(i)) = -1;
% if(isempty(intersect(actvB(i), actvBBoth))) % Fixed variable. Both upper and lower bounds
% if(varMult.val(actvB(i)) > module.multTol)
% % Lower bound is strongly active, upper bound is
% % weakly active
% chckS = true;
% end
% end
if(chckS)
Jbnd_Sactv(count,actvB(i)) = -1;
count = count + 1;
else
Jbnd_Wactv(countW,actvB(i)) = -1;
countW = countW + 1;
end
end
if(~isempty(intersect(actvB(i), actvBUp))) % Upper bound
Jbnd_actv(i,actvB(i)) = 1;
% if(isempty(intersect(actvB(i), actvBBoth))) % Fixed variable. Both upper and lower bounds
% if(varMult.val(actvB(i)) > module.multTol)
% % Lower bound is strongly active, upper bound is
% % weakly active
% chckS = false;
% end
% end
if(chckS)
Jbnd_Sactv(count,actvB(i)) = 1;
count = count + 1;
else
Jbnd_Wactv(countW,actvB(i)) = 1;
countW = countW + 1;
end
end
end
end
% % % % % % % % % % % % %
% Parse dictionary file %
% % % % % % % % % % % % %
% dict.txt is created by GAMS. It containts variable and equation names.
% Read in file to string
text = fileread('dict.txt');
% Use regular expressions to parse string
str = regexp(text,' (x|e)(\d+) (\S+)','tokens');
% Manipulate to maintain similar structure as textscan results
dic = cell(1,3);
for i = 1:length(str)
for j = 1:3
dic{j}(i) = str{i}(j)';
end
end
np = length(dic{2});
% Create cell of equation and variable names
eqn = cell(np,1);
var = cell(np,1);
eqnCount = 0;
varCount = 0;
for i = 1:np
if(strcmpi(dic{1}(i), 'e'))
eqn(eqnCount+1) = dic{3}(i);
eqnCount = eqnCount + 1;
elseif(strcmpi(dic{1}(i), 'x'))
var(varCount+1) = dic{3}(i);
varCount = varCount + 1;
end
end
% Check eqnCount matches Hessian/Jacobian file(s)
if(eqnCount ~= nEq)
disp('Warning. Mismatch between number of equations is data');
disp('and dictionary files');
end
if(varCount ~= nVar)
disp('Warning. Mismatch between number of variables is data');
disp('and dictionary files');
end
eqn = eqn(1:eqnCount);
var = var(1:varCount);
disp('***************************************');
disp('********** Basic Information **********');
disp('***************************************');
disp(' ');
disp('********** Equations **********');
disp(['Total number: ', num2str(nEq)]);
disp(' ');
disp(['Number of equality constraints: ',num2str(length(eqlCnst))]);
disp(['Number of STRONGLY active equality constraints: ',num2str(length(eqlStrg))]);
disp(['Number of WEAKLY active equality constraints: ',num2str(length(eqlWeak))]);
disp(' ');
disp(['Number of inequality constraints: ',num2str(length(ineqlCnst))]);
disp(['Number of ACTIVE inequality constraints: ',num2str(length(actv))]);
disp(['Number of STRONGLY ACTIVE inequality constraints: ',num2str(length(actvS))]);
disp(['Number of WEAKLY ACTIVE inequality constraints: ',num2str(length(actvW))]);
disp(['Number of INACTIVE inequality constraints: ',num2str(length(ineqlCnst) - length(actv))]);
disp(' ');
disp('********** Variables **********');
disp(['Total number: ',num2str(varCount)]);
disp(['Number of ACTIVE variable bounds: ',num2str(length(actvB))]);
disp(['Number of STRONGLY ACTIVE variable bounds: ',num2str(length(actvBS))]);
disp(['Number of WEAKLY ACTIVE variable bounds: ',num2str(length(actvBW))]);
disp(' ');
disp('********** Degrees of Freedom **********');
disp('At analyzed point, considering...');
disp([' all ACTIVE inequalities and bounds: ',num2str(varCount - length(eqlCnst) - length(actv) - length(actvB))]);
disp([' only STRONGLY ACTIVE inequalities and bounds: ',num2str(varCount - length(eqlCnst) - length(actvS) - length(actvBS))]);
disp(' ');
end
%% Classic Degeneracy Hunter Module
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if(module.degenHunt.tog)
disp('***********************************************');
disp('********** Classic Degeneracy Hunter **********');
disp('***********************************************');
%% Part A: Rank Check
% (1) Assemble the "active Jacobian": Adh
% (1) Compute the rank of Adh
% (3) Identify linearly dependent equations
% Assemble the "active Jacobian" (Adh) and set of indicies (Adh_i) to map
% back to equations and bounds
if(module.degenHunt.weakAct)
if(module.degenHunt.varBounds)
Adh = [J(union(eqlCnst, actv),:); Jbnd_actv];
Adh_i = union(union(eqlCnst,actv),actvB + nEq);
disp('Analyzed Jacobian contains all active inequality constraints');
disp(' and all active variable bounds');
else
Adh = J(union(eqlCnst, actv),:);
Adh_i = union(eqlCnst, actv);
disp('Analyzed Jacobian contains all active inequality constraints');
disp(' and NO variable bounds');
end
else
if(module.degenHunt.varBounds)
Adh = [J(union(eqlCnst, actvS),:); Jbnd_Sactv];
Adh_i = union(union(eqlCnst, actvS), actvBS + nEq);
disp('Analyzed Jacobian contains only STRONGLY active inequality constraints');
disp(' and only STRONGLY active variable bounds');
else
Adh = J(union(eqlCnst, actvS),:);
Adh_i = union(eqlCnst, actvS);
disp('Analyzed Jacobian contains only STRONGLY active inequality constraints');
disp(' and NO variable bounds');
end
end
disp(' ');
if(sum(Adh_i > nEq) > 0)
Adh_names = [eqn(Adh_i(Adh_i <= nEq)) ; var(Adh_i(Adh_i > nEq) - nEq)];
else
Adh_names = eqn(Adh_i);
end
nRowAdh = size(Adh,1);
% Computer the rank of Adh
r = myrank(Adh);
disp(['Rank of analyzed Jacobian: ',num2str(r)]);
nDE = size(Adh,1) - r;
disp(['Estimated number of dependent equations (using rank): ',num2str(nDE)]);
if(module.sparse && max(size(Adh)) < 1000)
disp(['Estimated number of dependent equations (using dense rank calc.): ',num2str(size(Adh,1) - rank(full(Adh)))]);
end
disp(' ');
if(nDE > 0 || module.degenHunt.optimal)
disp('********** Suspect Equations (and Bounds) **********');
% Calculate the reduced row echelon form of the "small" Jacobian
% This also gives the pivot points, which are the linearly independent
% columns. Note: frref ignores the specified tolerance when operating
% on a sparse matrix.
tic
[R, licol] = frref(Adh',1E-6,'s');
tocM;
% Determine the linearly dependent columns in the "small" Jacobian
dcol = setdiff(1:size(Adh,1), licol);
nNonPivot = length(dcol);
if(isempty(dcol))
disp('No degenerate equations detected during factorization.');
else
% Display the linearly dependent equation number and names
st1 = [];
st2 = [];
disp(sprintf('Index\tActive\tType\tName\n'));
for i = 1:length(dcol)
if(sum(i == eqlCnst))
st2 = 'Eqlty.';
if(sum(i == eqlStrg))
str1 = 'Strong';
else
str1 = 'Weak';
end
elseif sum(i == ineqlCnst)
st2 = 'Ineql.';
else
str2 = 'V Bnd.';
end
if(sum(i == actvS) || sum(i == actvBS + nEq) || sum(i == eqlStrg))
st1 = 'Strong';
else
st1 = 'Weak';
end
disp(sprintf('%i\t%s\t%s\t%s\n',Adh_i(dcol(i)),st1,st2,Adh_names{dcol(i)}));
end
end
else
dcol = [];
end
disp(' ');
%% Part B: Heuristic Degeneracy Hunter
% Parts 1 - 3
% Overview - This module expands the set of suspect equations in search of
% degeneracy. This is done iteratively.
%
% (1) Identify variables in suspect equations with non-zero entries in the
% Jacobian.
% (2) Identify other equations with a non-zero entry in the Jacobian for
% the variables found in step 1. Add these equations to the suspect
% equation set if they are part of the active Jacobian (not inactive
% equality constraints.)
% (3) If the expanded set of suspect equations is rank deficiency, stop.
% Other go to step 1.
%
% Part (4)
% This module attempts to identify the smallest subset of the initial
% suspect equations required to maintain rank decifiency.
module.degenHunt.suspectEquations = [];
module.degenHunt.suspectEqnGroup = [ ];
if(module.degenHunt.heurstic)
% Subparts 1 - 3
% Process supplied suspect equations or get them from previous results
if(isempty(module.degenHunt.suspectEquations))
if(~isempty(dcol))
% problemEqns = Adh_i(dcol);
problemEqns = dcol;
else
problemEqns = [];
end
else
problemEqns = module.degenHunt.suspectEquations;
end
% How is problemEqns indexed?
% Option 1: These are the indicies of Adh
% Option 2: These are the indicies relating to all equations
% Let's try option 1...
if(isempty(problemEqns))
if(module.verbose)
disp('Warning: No linearly dependent equations. Skipping expansion step');
end
else
flag = true;
iter = 1;
if(nDE ~= length(problemEqns) && isempty(module.degenHunt.suspectEquations))
disp(['Due to numerical reasons the "rank" command predicted ',num2str(nDE),' linearly dependent equations']);
disp(['whereas sparse QR factorization identified ',num2str(length(problemEqns)),]);
disp(['For speed this module uses the "rank" command and searches for a rank deficiency of ',num2str(nDE)]);
goalRankDef = nDE;
else
goalRankDef = length(problemEqns);
end
disp('Beginning to Expand Set of Suspect Equations (and Bounds).');
while(flag)
disp(['Iteration ',num2str(iter),' ...']);
% Determine variables with non-zero entries for problem equations
problemVar = zeros(varCount,1);
for i = 1:length(problemEqns)
j = (Adh(problemEqns(i),:) ~= 0);
problemVar = logical(problemVar + j');
if(module.verbose)
disp('Problem equation/bound...');
disp(eqn(problemEqns(i)));
disp('Variables with nonzero entries...');
disp(var(j));
end
end
disp(['Number of variables identified: ',num2str(sum(problemVar))]);
% Need to expand this section to include logic for variable
% bounds
% Determine equations with non-zero entries for the identified variables
pV = find(problemVar);
suspectEqn = zeros(nRowAdh,1);
for i = 1:length(pV)
k = (Adh(:,pV(i)) ~= 0);
suspectEqn = logical(suspectEqn + k);
if(module.verbose)
disp('Identified variable...');
disp(var(pV(i)));
disp('Equations/bounds with nonzero entries for this variable');
disp(Adh_names(k));
end
end
nSE = sum(suspectEqn);
disp(['Number of suspect equations (and bounds): ', num2str(nSE)]);
% Check "reduced" Jacobian of only suspect equatons for rank deficiency
redJac = Adh(suspectEqn,:);
r = myrank(redJac');
disp(['Rank: ',num2str(r)]);
problemEqns = find(suspectEqn);
iter = iter + 1;
disp(' ');
% Check if rank deficiency identified
if(r + goalRankDef <= nSE)
flag = false;
end
end
if(module.verbose)
disp('Final Suspect Equations (and Bounds)');
disp(eqn(problemEqns));
end
end
% Subpart 4
% Process supplied suspect equations or get them from Module 2 results
if(~isempty(module.degenHunt.suspectEqnGroup))
problemEqns = module.suspectEqnGroup;
end
if(isempty(problemEqns))
if(module.verbose)
disp('Warning: No linearly dependent equations. Skipping shrinking step');
end
else
incldEqns = zeros(length(eqn),1);
incldEqns(problemEqns) = 1;
incldEqns = logical(incldEqns);
nPE = sum(incldEqns);
r = myrank(Adh(problemEqns,:)');
flag = true;
iter = 1;
disp('Beginning to Shrink Set of Degenerate Equations (and Bounds)...');
disp(['Number of equations (and bounds): ',num2str(nPE)]);
disp(['Rank: ',num2str(r)]);
disp(' ');
while(flag)
flag = false;
nPE = sum(incldEqns);
for i = 1:length(problemEqns)
incldEqns(problemEqns(i)) = false;
tr = myrank(Adh(incldEqns,:));
if(tr == r - 1)
% Remove equation
r = tr;
flag = true;
else
% Keep equation... add it back in
incldEqns(problemEqns(i)) = true;
end
end
nnPE = sum(incldEqns);
disp(['Iteration ',num2str(iter),' ...']);
disp(['New number of equation: ',num2str(nnPE)]);
disp(['Number of equations dropped this iteration: ',num2str(nPE - nnPE)]);
disp(['Final rank: ',num2str(r)]);
disp(' ');
problemEqns = find(incldEqns);
iter = iter + 1;
end
disp('Smallest set of rank deficient equations...');
disp(Adh_names(problemEqns));
% Find multipliers for each equation
miniJac = Adh(problemEqns,:);
[mR, mLiCol] = frref(miniJac',1E-6,'s');
% Determine the linearly dependent columns in the "mini" Jacobian
mDCol = setdiff(1:size(miniJac,1), mLiCol);
% Identify multipliers for each equation
disp('***** Smallest Sets of Linearly Dependent Equations *****');
for i = 1:length(mDCol)
j = find(abs(mR(:,mDCol(i))) > 1E-6);
disp(Adh_names(problemEqns(mDCol(i))));
count = 1;
for k = 1:length(j)
% disp(eqn(problemEqns(j)));
ii = problemEqns(j(k));
if(sum(ii == eqlCnst))
st2 = 'Eqlty.';
if(sum(ii == eqlStrg))
str1 = 'Strong';
else
str1 = 'Weak';
end
elseif sum(ii == ineqlCnst)
st2 = 'Ineql.';
else
str2 = 'V Bnd.';
end
if(sum(ii == actvS) || sum(ii == actvBS + nEq) || sum(ii == eqlStrg))
st1 = 'Strong';
else
st1 = 'Weak';
end
fprintf('%f\t%s\t%s\t%s\n',full(mR(j(k), mDCol(i))),st1,st2, eqn{problemEqns(j(k))});
count = count + 1;
end
disp(['Set Size: ',num2str(count),' Eqns (and Bounds)']);
disp('***');
end
end
end
%% Part C: MILP Degeneracy Hunter
if(module.degenHunt.optimal)
% Setup function output
nDegenSets = 0;
degenEqnsNames = cell(1,nNonPivot);
% Write active equations to GDX file
actE.name = 'actE';
actE.type = 'Set';
actE.uels = cell(1,length(Adh_i));
for i = 1:length(Adh_i)
actE.uels{1,i} = ['e',num2str(Adh_i(i))];
end
for j = 1:length(dcol)
SE.name = 'SE';
SE.type = 'Set';
SE.uels = {['e',num2str(Adh_i(dcol(j)))]};
wgdx('degenData',actE,SE);
disp(['Consider degeneracy with ',Adh_names{dcol(j)}]);
tic
gamso.form = 'full';
[x, y, Z, MIPstat] = gams('../GenerateMinDegenerateSet_MIP2.gms');
tocM;
switch MIPstat.val
case 1
disp(['Optimal: This equation is part of a degenerate set with ',num2str(Z.val),' equations (total)']);
% n = find(y.val(Adh_i) > 0);
n = find(abs(x.val(Adh_i)) > module.degenHunt.tol);
for k = 1:length(n)
disp(sprintf('%f \t\t %s',x.val(Adh_i(n(k))),Adh_names{n(k)}));
end
if(length(n) < round(Z.val))
disp(['Only ',num2str(length(n)),' equation displayed. The remaining ', num2str(Z.val - length(n)),' equations`']);
disp(['singular vector components are smaller than ',num2str(module.degenHunt.tol),' in magnitude.']);
end
nDegenSets = nDegenSets + 1;
degenEqnsNames{nDegenSets} = Adh_names{dcol(j)};
otherwise
disp('Infeasible: This equation is not part of a degenerate set');
end
disp(' ');
end
if(nDegenSets < nNonPivot)
degenEqnsNames = degenEqnsNames(1:nDegenSets);
end
end
end
%% Second Order Condition Checker
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if(module.SOCcheck.tog)
disp('*****************************************************');
disp('********** Second Order Conditions Checker **********');
disp('*****************************************************');
% Calculate null space
if(module.SOCcheck.weakAct)
A2 = [J(union(eqlCnst, actv),:); Jbnd_actv];
else
A2 = [J(union(eqlCnst, actvS),:); Jbnd_Sactv];
end
Z = nulls(A2);
% Z = null(full(A2));
% Extract objective coefficient (min or max) from GAMS
s.name = 'objcoef';
objcoef = rgdx('hessian',s);
% Extract the Hessian from GAMS
s.name = 'W';
% s.form = 'sparse';
Hes = rgdx('hessian',s);
i = Hes.val(:,1) - nEq;
j = Hes.val(:,2) - nEq;
k = i > 0 & j > 0;
W = sparse(i(k), j(k), Hes.val(k,3), nVar, nVar);
% W = sparse(Hes.val(:,1),Hes.val(:,2),Hes.val(:,3));
% W = W(nEq+1:end,nEq+1:end);
% This is needed b/c only an upper triangle is extracted from GAMS
W = objcoef.val*(W + W' - diag(diag(W)));
if(isempty(Z))
disp('The null space is empty, and thus');
disp('second order conditions are vacuously true.');
else
Wr = Z'*W*Z;
[V, D] = eigs(Wr);
% Z = nulls(A2);
d = diag(D);
i = find(d < 0);
disp('Smallest Eigenvalue of Reduced Hessian');
disp(min(real(d)));
disp('Largest Eigenvalue of Reduced Hessian');
disp(max(real(d)));
if(~isempty(i))
if(isempty(actvW) && isempty(actvBW))
disp('This point violates second order conditions.');
disp('There are no weakly active constraints or bounds');
disp('and the reduced Hessian has a negative eigenvalue.');
else
disp('This point may violate second order conditions.');
disp(' ');
disp('Beginning to search for an allowable direction with');
disp('negative curvature...');
disp(' ');
if(isempty(actvW) && ~isempty(actvBW))
A2w = Jbnd_Wactv;
elseif(~isempty(actvW) && isempty(actvBW))
A2w = diag(ineqlSign(actvW))*J(actvW,:);
else
A2w = [diag(ineqlSign(actvW))*J(actvW,:); Jbnd_Wactv];
end
end
f = zeros(length(Wr),1);
b = zeros(size(A2w,1),1);
x0 = V(:,real(d) == min(real(d)));
bnd = ones(size(x0));
optns = optimset('algorithm','active-set');
[x, fval] = quadprog(full(Wr),f,full(A2w*Z),b,[],[],-bnd,bnd,x0,optns);
xFull = Z*x;
if(fval < 0)
disp('Second Order Conditions (SOC) are violated. Located');
disp('a feasible direction with negative curvature:');
disp(' ');
for k = 1:nVar
if(xFull(k) ~= 0)
disp(sprintf('%s \t%f',var{k},full(xFull(k))));
end
end
else
disp('Inconclusive. Failed to find feasible direction with');
disp('negative curvature. Need to solve non-convex QP to');
disp('global optimality to prove second order conditions are');
disp('satified.');
end
else
disp('There are no negative eigenvalues of the reduced Hessian');
disp('at this point, thus second order conditions are satisfied.');
% The LP version does not work!!!!
% % Time to solve LPs
% np = find(real(d) <= 0); % Locate non-positive eigenvalues
% f = zeros(length(np)+1,1);
% f(end) = 1;
% Alp = [A2w(np,:)*Z*V(np,:), -1*ones(length(np),1)];
% blp = zeros(1,size(A2w,1));
%
% beq = 1;
%
% % Verify this tomorrow
% flag = true;
% i = 1;
% optn = optimset('Display','off');
% while(flag)
%
% Aeq = zeros(1,length(np)+1);
% Aeq(ceil(i/2)) = 1;
% beq = 2*mod(i,2) - 1;
%
% v_lo = [-Inf(length(np),1); -1];
% v_up = [ ];
%
% vWght = linprog(f,Alp,blp,Aeq,beq,v_lo,v_up,f,optn);
%
% deltaX = Z*V(np,:)*vWght(1:end-1);
%
% if(vWght(end) < 0)
% disp('Allowable negative curvature direction found.')
% disp('Second order conditions violated.');
% disp('Direction stored in "deltaX"');
% flag = false;
% else
% i = i + 1;
% if(i > 2*length(np))
% flag = false;
% disp('Failed to find an allowable direction with negative curvature')
% disp('Second order conditions are satisfied.')
% end
% end
%
% end
end
end
end
%% Other Module
%%%%%%%%%%%%%%%
if(module.other.tog)
if(module.other.four)