Progressive Hedging (1)

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  Olafur P. Pallson, Hans F. Ravn
  Scenario Analysis and the Progressive Hedging Algorithm
    - Simple Numerical Examples
  The Technical University of Denmark, 1993


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set
   iter 'iteration number'/it0*it9/
   s    'scenarios'/s1*s2/
;

table ds(s,*) 'single stochastic parameter'
      value   prob
  s1    5     0.6
  s2    2     0.4
;

parameter
   Xs(s)
   Ws(s)  'price'
   report(iter,*,*) 'table 1: results from iterations'
;
Ws(s) = 0;

scalars
   iterno
   continue /1/
   d   /0/
   xhat 'conditional expectation'/0/
   r  'penalty parameter'/0/
   w  /0/
;


variables
   x  'for submodel'
   z  'for submodel'
;

equations
   obj
;


obj.. z =e= sqr(x-d) + w*x + 0.5*r*sqr(x-xhat);
x.lo=3;
x.up=6;
model m /obj/;

m.solprint = 2;
m.solvelink = %Solvelink.LoadLibrary%;

loop(iter$continue,

  iterno = ord(iter);
  display'***** iteration ******',iterno;

* r = 0 in first iteration (it0)
  r = 1$(iterno>1);

* step 1
* solve each scenario
  loop(s,
     d = ds(s,'value');
     w = ws(s);
     solve m minimizing z using nlp;
     Xs(s) = x.l;
  );
  display Xs;

* step 2
* Calculate policy Xhat
  Xhat = sum(s, Xs(s)*ds(s,'prob'));
  display Xhat;

* step 3
* Update W
  Ws(s) = Ws(s) + r*(xs(s) - xhat);
  display Xhat,Ws;


  report(iter,'x',s) = Xs(s);
  report(iter,'xhat','-') = xhat;
  report(iter,'w',s) = Ws(s);
  report(iter,'z','-') = sum(s,ds(s,'prob')*sqr(Xs(s)-ds(s,'value')));
  report(iter,'zhat','-') = sum(s,ds(s,'prob')*sqr(xhat-ds(s,'value')));

);

option report:2:1:2;
display report;



*
* comparison
* this should give the same solution
*

equation obj2;
obj2.. z =e= sum(s,ds(s,'prob')*sqr(x-ds(s,'value')));
model m2 /obj2/;
solve m2 minimizing z using nlp;
display x.l;