$title Standard QP Model - intermediate Variables (QP3,SEQ=173)

$onText
Further speedup by simplifying the nonlinear terms.
Additional information can be found at:

/modlib/adddocs/qp3doc.htm

Kalvelagen, E, Model Building with GAMS. forthcoming

de Wetering, A V, private communication.

Keywords: nonlinear programming, quadratic programming, finance
$offText

$include qpdata.inc

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

Alias (s,t);

* select 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)'
   covarx(stocks,sstocks) 'covariance matrix - variation (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);
covarx(s,t)       = 2*covar(s,t);
covarx(s,s)       =   covar(s,s);
totmean           = sum(s, mean(s))/(card(s));

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

Positive Variable x;

Equation
   obj           'objective'
   budget
   retcon        'return constraint'
   ydefa(stocks) 'not exploiting symmetry'
   ydefb(stocks) 'exploiting symmetry';

obj..      z    =e= sum(s, y(s)*x(s));

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

ydefb(t).. y(t) =e= sum(s, x(s)*covarx(s,t));

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

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

Model
   qp3a / obj, ydefa, budget, retcon /
   qp3b / obj, ydefb, budget, retcon /;

solve qp3a using nlp minimizing z;

display x.l;

ydefb.m(t) = ydefa.m(t);

solve qp3b using nlp minimizing z;

display x.l;