$title Standard QP Model with GDX Data Input (QP1X,SEQ=246)

$onText
The first in a series of variations on the standard
QP formulation. The subsequent models exploit data
and problem structures to arrive at formulations that
have sensational computational advantages. Additional
information can be found at:

/modlib/adddocs/qp1doc.htm

Kalvelagen, E, Model Building with GAMS. forthcoming
de Wetering, A V, private communication.

Keywords: nonlinear programming, quadratic programming, finance
$offText

* From StockMaster at MIT
Set
   days   '100 days from 95-11-27 to 96-04-29'
   stocks '170 selected stocks'
   upper(stocks,stocks)
   lower(stocks,stocks);

Alias (stocks,sstocks);

Parameter
   val(stocks,days)    'closing value'
   return(stocks,days) 'daily returns - derived';

* We use the (hopefully Posix-compliant) utility test to check if
* qpdata.gdx is more recent than qpdata.inc. If that's the case we can
* skip the processing of qpdata.inc. test can do more checks. Run test
* --help for a brief overview.
*
* WINDOWS ONLY: From within a GAMS program test and
* other Posix utilities are automatically available. From other
* environments, you need to add the gbin subdirectory of the GAMS
* system directory to your PATH (Windows only).

$call =test qpdata.gdx -nt qpdata.inc
$ifE errorLevel<>0 $call =gams qpdata.inc lo=0 a=c gdx=qpdata
$ifE errorLevel<>0 $abort problems creating qpdata.gdx
$gdxIn qpdata
$load days stocks

* Execution time load of closing value
execute_load 'qpdata', val;

return(stocks,days-1) = val(stocks,days) - val(stocks,days-1);
upper(stocks,sstocks) = ord(stocks) <= ord(sstocks);
lower(stocks,sstocks) = not upper(stocks,sstocks);

Set
   d(days)   'selected days'
   s(stocks) 'selected stocks';

Alias (s,t);

* setect subset of stocks and periods
d(days)   = ord(days)   >  1 and ord(days) < 31;
s(stocks) = ord(stocks) < 51;

Parameter
   mean(stocks)          'mean of daily return'
   dev(stocks,days)      'deviations'
   covar(stocks,sstocks) 'covariance matrix of returns (upper)'
   totmean               'total mean return';

mean(s)  = sum(d, return(s,d))/card(d);
dev(s,d) = return(s,d) - mean(s);

* calculate covariance
* to save memory and time we only compute the uppertriangular
* part as the covariance matrix is symmetric
covar(upper(s,t)) = sum(d, dev(s,d)*dev(t,d))/(card(d) - 1);

totmean = sum(s, mean(s))/(card(s));

Variable
   z         'objective variable'
   x(stocks) 'investments';

Positive Variable x;

Equation
   obj    'objective'
   budget
   retcon 'return constraint';

obj..    z =e= sum(upper(s,t), x(s)*covar(s,t)*x(t))
            +  sum(lower(s,t), x(s)*covar(t,s)*x(t));

budget.. sum(s, x(s))         =e= 1.0;

retcon.. sum(s, mean(s)*x(s)) =g= totmean*1.25;

Model qp1 / all /;

* Some solvers need more memory
qp1.workFactor = 10;
solve qp1 using nlp minimizing z;
display x.l;

* Dump solution into GDX file
execute_unload 'qp1xsol', x = xsol;

* Lets see if the solution was stored correctly
x.l(s) = 0;
execute_load 'qp1xsol', x = xsol;
qp1.iterLim = 20;
solve qp1 using nlp minimizing z;

* Dump reoptimized solution into GDX file
execute_unload 'qp1xso2', x = xsol;

* Compare both GDX files
execute 'gdxdiff qp1xsol qp1xso2 %system.redirlog%';

* Dump the diff GDX file
execute 'gdxdump diffile %system.redirlog%';