$title Linear Multiplicative Programs - Type 3 (LMP3,SEQ=253)

$onText
Generate and solves random linear multiplicative models of
"Type 3." Problem instances are generated as proposed by
Benson and Boger. Model developed by N. Sahinidis.

H. P. Benson and G. M. Boger, "Multiplicative programming problems:
Analysis and efficient point search heuristic",
Journal of Optimization Theory and Applications, 94(487-510), 1997.

M. Tawarmalani and N. Sahinidis, Convexification and Global
Optimization in Continuous and Mixed-Integer Nonlinear
Programming: Theory, Algorithms, Software, and Applications,
Kluwer Academic Publishers, 2002.

Keywords: nonlinear programming, linear multiplicative programming,
          global optimization, concave minimization
$offText

option optCr = 0, optCa = 1.e-6, limRow = 0, limCol = 0, solPrint = off;

Set
   mm / m1*m220 /
   nn / n1*n200 /
   pp / p1*p4   /;

Set
   m(mm) 'constraints'
   n(nn) 'variables'
   p(pp) 'products'
   c     'cases'     / c1*c9 /
   i     'instances' / i1*i5 /;

* For each case to be solved, we use a different (m,n,p) triplet
Table cases(c,*)
         m    n  p
   c1   20   30  2
   c2  120  100  2
   c3  220  200  2
   c4   20   30  3
   c5  120  120  3
   c6  200  180  3
   c7   20   30  4
   c8  100  100  4
   c9  200  200  4;

Parameter
   cc(pp,nn) 'cost coefficients'
   A(mm,nn)  'constraint coefficients'
   b(mm)     'left-hand-side'
   rep(c,*)  'summary report'
   mactual
   nactual
   pactual
   ResMin
   Resmax
   NodMin
   Nodmax;

Variable
   y(pp)
   x(nn)
   obj;

Equation
   Objective
   Constraints(mm)
   Products(pp);

Objective..      obj  =e= prod(p, y(p));

Products(p)..    y(p) =e= sum(n, cc(p,n)*x(n));

Constraints(m).. b(m) =l= sum(n, A(m,n)*x(n));

x.lo(nn) = 1;

Model lmp3 / all /;

lmp3.workSpace = 32;

rep(c,'AvgResUsd') = 0;
rep(c,'AvgNodUsd') = 0;

loop(c,
   m(mm)   = ord(mm)<= cases(c,'m');
   n(nn)   = ord(nn)<= cases(c,'n');
   p(pp)   = ord(pp)<= cases(c,'p');
   mactual = cases(c,'m');
   nactual = cases(c,'n');
   pactual = cases(c,'p');
   ResMin  = inf;
   Resmax  = 0;
   NodMin  = inf;
   Nodmax  = 0;

   loop(i,
      cc(p,n) = round(uniform(1,10));
      A(m,n)  = round(uniform(1,10));
      b(m)    = sum(n, A(m,n)**2);
      x.up(n) = smax(m, b(m));

*     Set initial starting point for all models to 0
      x.l(n) = 0;
      y.l(p) = 0;

      Solve lmp3 minimizing obj using nlp;

      rep(c,'AvgResUsd') = rep(c,'AvgResUsd') + lmp3.resUsd;
      rep(c,'AvgNodUsd') = rep(c,'AvgNodUsd') + lmp3.nodUsd;
      ResMin = min(ResMin, lmp3.resUsd);
      NodMin = min(NodMin, lmp3.nodUsd);
      ResMax = max(ResMax, lmp3.resUsd);
      NodMax = max(NodMax, lmp3.nodUsd);
   );
   rep(c,'MinResUsd') = ResMin;
   rep(c,'MaxResUsd') = ResMax;
   rep(c,'MinNodUsd') = NodMin;
   rep(c,'MaxNodUsd') = NodMax;
);
rep(c,'AvgResUsd') = rep(c,'AvgResUsd')/card(i);
rep(c,'AvgNodUsd') = rep(c,'AvgnodUsd')/card(i);

display rep;