Nonlinear MIPs

Modelers may wish to impose integer restrictions on nonlinear formulations.  Today GAMS contains the DICOPT and CBC solvers that permit this.  They tie together other solvers.  For example both can use CONOPT to solve the nonlinear sub-problems.  DICOPT also uses MIP solvers on the integer part of the problem while CBC contains an internal integer solution algorithm.

For example suppose we impose restrictions in a portfolio problem that a minimum of 10 shares be bought if any and that we buy integer numbers of shares (intev.gms)

 Integer VARIABLES  INVEST(STOCKS)       MONEY INVESTED IN EACH STOCK

 binary variables   mininv(stocks)       at least 10 shares bought

 VARIABLE              OBJ               NUMBER TO BE MAXIMIZED ;

 EQUATIONS             OBJJ              OBJECTIVE FUNCTION

                       INVESTAV          INVESTMENT FUNDS AVAILABLE

                       minstock(stocks)  at least 10 units to be bought

                       maxstock(stocks)  Set up indicator variable ;

 OBJJ.. OBJ =E=   SUM(STOCKS, MEAN(STOCKS) * INVEST(STOCKS))

                - RAP*(SUM(STOCK, SUM(STOCKS,

                       INVEST(STOCK)* COVAR(STOCK,STOCKS) * INVEST(STOCKS))));

 INVESTAV..     SUM(STOCKS, PRICES(STOCKS) * INVEST(STOCKS)) =L= FUNDS;

 minstock(stocks)..    invest(stocks) =g= 10*mininv(stocks);

 maxstock(stocks)..    invest(stocks)=l=1000*mininv(stocks);

 MODEL EVPORTFOL /ALL/ ;

 SOLVE EVPORTFOL USING MINLP MAXIMIZING OBJ ;

When using DICOPT and CBC it is very important to have the constraints represent to the full extent possible the link between continuous and integer variables.