MIP model 2: imposing a minimum investment in each share

To formulate the second MIP model, we start again with the LP model from Chapters 2 and 10. The new constraint we wish to formulate is `if a share is bought, at least a minimum amount 10% of the budget is spent on the share.' Instead of simply constraining every variable frac_s to take a value between 0 and 0.3, it now must either lie in the interval between 0.1 and 0.3 or take the value 0. This type of variable is known as semi-continuous variable. In the new model, we replace the bounds on the variables frac_s by the following constraint:

Implementation with BCL

The following program implements the MIP model 2. The semi-continuous variables are defined by the type XPRB_SC. By default, BCL assumes a continuous limit of 1, se we need to set this value to 0.1 with the method setLim.

A similar type is available for integer variables that take either the value 0 or an integer value between a given limit and their upper bound (so-called semi-continuous integers): XPRB_SI. A third composite type is a partial integer which takes integer values from its lower bound to a given limit value and is continuous beyond this value (marked by XPRB_PI).

#include <iostream>
#include "xprb_cpp.h"

using namespace std;
using namespace ::dashoptimization;

#define NSHARES 10                   // Number of shares
#define NRISK 5                      // Number of high-risk shares
#define NNA 4                        // Number of North-American shares

double RET[] = {5,17,26,12,8,9,7,6,31,21};  // Estimated return in investment
int RISK[] = {1,2,3,8,9};            // High-risk values among shares
int NA[] = {0,1,2,3};                // Shares issued in N.-America

int main(int argc, char **argv)
{
 int s;
 XPRBprob p("FolioSC");              // Initialize a new problem in BCL
 XPRBexpr Risk,Na,Return,Cap;
 XPRBvar frac[NSHARES];              // Fraction of capital used per share

// Create the decision variables
 for(s=0;s<NSHARES;s++)
 {
  frac[s] = p.newVar("frac", XPRB_SC, 0, 0.3);
  frac[s].setLim(0.1);
 }

// Objective: total return
 for(s=0;s<NSHARES;s++) Return += RET[s]*frac[s];
 p.setObj(Return);                   // Set the objective function

// Limit the percentage of high-risk values
 for(s=0;s<NRISK;s++) Risk += frac[RISK[s]];
 p.newCtr(Risk <= 1.0/3);

// Minimum amount of North-American values
 for(s=0;s<NNA;s++) Na += frac[NA[s]];
 p.newCtr(Na >= 0.5);

// Spend all the capital
 for(s=0;s<NSHARES;s++) Cap += frac[s];
 p.newCtr(Cap == 1);

// Solve the problem
 p.setSense(XPRB_MAXIM);
 p.mipOptimize("");

// Solution printing
 cout << "Total return: " << p.getObjVal() << endl;
 for(s=0;s<NSHARES;s++)
  cout << s << ": " << frac[s].getSol()*100 << "%" << endl;

 return 0;
}

When executing this program we obtain the following output (leaving out the part printed by the Optimizer):

Total return: 14.0333
0: 30%
1: 0%
2: 20%
3: 0%
4: 10%
5: 26.6667%
6: 0%
7: 0%
8: 13.3333%
9: 0%

Now five securities are chosen for the portfolio, each forming at least 10% and at most 30% of the total investment. Due to the additional constraint, the optimal MIP solution value is again lower than the initial LP solution value.