$title Linear Multiplicative Programs - Type 2 (LMP2,SEQ=252)

$onText
Generates and solves random linear multiplicative models of
"Type 2." Problem instances are generated as proposed by
Thoai. Model developed by N. Sahinidis.

N. V. Thoai, "A global optimization approach for solving
convex multiplicative programming problems",
Journal of Global Optimization, 1(341-357), 1991.

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
$offText

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

Set
   mm / m1*m200 /
   nn / n1*n200 /;

Set
   m(mm) 'constraints'
   n(nn) 'variables'
   p     'products'  / p1*p2 /
   c     'cases'     / c1*c5 /
   i     'instances' / i1*i5 /;

* For each case to be solved, we use a different (m,n) pair
Table cases(c,*)
         m    n
   c1   10   20
   c2   20   30
   c3   60  100
   c4  100  100
   c5  200  200;

Parameter
   cc(p,nn) 'cost coefficients'
   f(p)     'constants'
   A(mm,nn) 'constraint coefficients'
   b(mm)    'left-hand-side'
   rep(c,*) 'summary report'
   ResMin
   Resmax
   NodMin
   Nodmax;

Variable
   y(p)
   x(nn)
   obj;

Equation
   Objective
   Constraints(mm)
   Products(p);

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) = 0;

Model lmp2 / all /;

lmp2.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');
   ResMin  = inf;
   Resmax  = 0;
   NodMin  = inf;
   Nodmax  = 0;

   loop(i,
      f(p)    =  uniform(0,1);
      cc(p,n) =  uniform(0,1);
      A(m,n)  =  2*uniform(0,1) - 1;
      b(m)    = (sum(n, A(m,n)) + 2*uniform(0,1));

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

      solve lmp2 minimizing obj using nlp;

      rep(c,'AvgResUsd') = rep(c,'AvgResUsd') + lmp2.resUsd;
      rep(c,'AvgNodUsd') = rep(c,'AvgNodUsd') + lmp2.nodUsd;
      ResMin = min(ResMin, lmp2.resUsd);
      NodMin = min(NodMin, lmp2.nodUsd);
      ResMax = max(ResMax, lmp2.resUsd);
      NodMax = max(NodMax, lmp2.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;