The Conic Optimization Solver

The example in this section illustrates how you can use the OPTMODEL procedure to solve conic optimization problems. Suppose you want to solve the following second-order cone problem:

The fifth constraint is a rotated second-order cone that is defined on the variables , , , and . This cone constraint together with the fourth constraint transforms the following general nonlinear constraint into a second-order cone programming formulation:

You can use the following statements to call the OPTMODEL procedure to solve this SOCP problem:

/* getting started example for conic solver */
proc optmodel;
   /* declare variables */
   var x {1..6} >= 0;

   /* objective function */
   min Z = 5*x[1] + x[4] + x[5] + x[6];

   /* linear constraints */
   con c1: -3*x[3] + x[4] + x[5] >= 0;
   con c2: x[3] - 0.1*x[5] - 0.25*x[6] >= 0;
   con c3: x[4] + x[5] + x[6] >= 2;
   con c4: x[2] = 0.5;

   /* rotated second-order cone constraint */
   con cone: rsoc(x[1], x[2], x[3] x[4]);

   /* bounds on variables */
   x[3].lb = 0.2;
   x[3].ub = 1;
   for{i in 5..6} x[i].ub = 1;

   /* specify conic solver */
   solve with conic;

   /* print optimal primal solution and dual cone solution */
   print x x.dcone;
quit;

The output and optimal solution are displayed in Figure 1. The dual cone values are returned in each variable.dcone suffix.

The iteration log that displays problem statistics, the progress of the solution, and the optimal objective value is shown in Figure 2.