$title Standard QCP Model (QCP1,SEQ=283)

$onText
This is the gamslib model QP1 expressed as a QCP. Also
note that the full sized data set is used and the
handling of the Q matrix is simplified.

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: quadratic constraint programming, finance, portfolio optimization,
          investment planning
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$eolCom //
$include qpdata.inc

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

Alias (s,t);

* note that we have to drop the first day because of the definition of
* return(stocks,days-1) = val(stocks,days) - val(stocks,days-1);

d(days+1) = yes;   // this will drop the first day
s(stocks) = yes;

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);
covar(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((s,t), x(s)*covar(s,t)*x(t));

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

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

Model qcp1 / all /;

option limCol = 0, limRow = 0;
qcp1.workFactor = 20;

solve qcp1 using qcp minimizing z;