6.3 Conic Quadratic Optimization

Conic optimization is a generalization of linear optimization, allowing constraints of the type

\[x^t \in \K_t,\]

where \(x^t\) is a subset of the problem variables and \(\K_t\) is a convex cone. Since the set \(\real^n\) of real numbers is also a convex cone, we can simply write a compound conic constraint \(x\in \K\) where \(\K=\K_1\times\cdots\times\K_l\) is a product of smaller cones and \(x\) is the full problem variable.

MOSEK can solve conic quadratic optimization problems of the form

\[\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x + c^f & & \\ \mbox{subject to} & l^c & \leq & A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x, \\ & & & x \in \K, & & \end{array}\end{split}\]

where the domain restriction, \(x \in \K\), implies that all variables are partitioned into convex cones

\[x = (x^0, x^1, \ldots , x^{p-1}),\quad \mbox{with } x^t \in \K_t \subseteq \real^{n_t}.\]

For convenience, a user defining a conic quadratic problem only needs to specify subsets of variables \(x^t\) belonging to quadratic cones. These are:

  • Quadratic cone:

    \[\Q^n = \left\lbrace x \in \real^n: x_0 \geq \sqrt{\sum_{j=1}^{n-1} x_j^2} \right\rbrace.\]
  • Rotated quadratic cone:

    \[\Qr^n = \left\lbrace x \in \real^n: 2 x_0 x_1 \geq \sum_{j=2}^{n-1} x_j^2,\quad x_0\geq 0,\quad x_1 \geq 0 \right\rbrace.\]

For example, the following constraint:

\[(x_4, x_0, x_2) \in \Q^3\]

describes a convex cone in \(\real^3\) given by the inequality:

\[x_4 \geq \sqrt{x_0^2 + x_2^2}.\]

Furthermore, each variable may belong to one cone at most. The constraint \(x_i - x_j = 0\) would however allow \(x_i\) and \(x_j\) to belong to different cones with same effect.

6.3.1 Example CQO1

Consider the following conic quadratic problem which involves some linear constraints, a quadratic cone and a rotated quadratic cone.

(1)\[\begin{split}\begin{array} {lccc} \mbox{minimize} & x_4 + x_5 + x_6 & & \\ \mbox{subject to} & x_1+x_2+ 2 x_3 & = & 1, \\ & x_1,x_2,x_3 & \geq & 0, \\ & x_4 \geq \sqrt{x_1^2 + x_2^2}, & & \\ & 2 x_5 x_6 \geq x_3^2 & & \end{array}\end{split}\]

Setting up the linear part

The linear parts (constraints, variables, objective) are set up using exactly the same methods as for linear problems, and we refer to Section 6.1 for all the details. The same applies to technical aspects such as defining an optimization task, retrieving the solution and so on.

Setting up the conic constraints

A cone is defined using the function Task.appendcone:

          csub[0] = 3;
          csub[1] = 0;
          csub[2] = 1;
          task.appendcone(mosek.conetype.quad,
                          0.0, /* For future use only, can be set to 0.0 */
                          csub);

The first argument selects the type of quadratic cone, in this case either conetype.quad for a quadratic cone or conetype.rquad for a rotated quadratic cone. The second parameter is currently ignored and passing 0.0 will work.

The last argument is a list of indexes of the variables appearing in the cone.

Variants of this method are available to append multiple cones at a time.

Source code

Listing 6.4 Source code solving problem (1). Click here to download.
using System;

namespace mosek.example
{
  class msgclass : mosek.Stream
  {
    string prefix;
    public msgclass (string prfx)
    {
      prefix = prfx;
    }

    public override void streamCB (string msg)
    {
      Console.Write ("{0}{1}", prefix, msg);
    }
  }

  public class cqo1
  {
    public static void Main ()
    {
      const int numcon = 1;
      const int numvar = 6;

      // Since the value infinity is never used, we define
      // 'infinity' symbolic purposes only
      double infinity = 0;

      mosek.boundkey[] bkc    = { mosek.boundkey.fx };
      double[] blc = { 1.0 };
      double[] buc = { 1.0 };

      mosek.boundkey[] bkx = {mosek.boundkey.lo,
                              mosek.boundkey.lo,
                              mosek.boundkey.lo,
                              mosek.boundkey.fr,
                              mosek.boundkey.fr,
                              mosek.boundkey.fr
                             };
      double[] blx = { 0.0,
                       0.0,
                       0.0,
                       -infinity,
                       -infinity,
                       -infinity
                     };
      double[] bux = { +infinity,
                       +infinity,
                       +infinity,
                       +infinity,
                       +infinity,
                       +infinity
                     };

      double[] c   = { 0.0,
                       0.0,
                       0.0,
                       1.0,
                       1.0,
                       1.0
                     };

      double[][] aval = {new double[] {1.0},
               new double[] {1.0},
               new double[] {2.0}
      };

      int[][] asub = {new int[] {0},
            new int[] {0},
            new int[] {0}
      };

      int[] csub = new int[3];

      // Make mosek environment.
      using (mosek.Env env = new mosek.Env())
      {
        // Create a task object.
        using (mosek.Task task = new mosek.Task(env, 0, 0))
        {
          // Directs the log task stream to the user specified
          // method msgclass.streamCB
          task.set_Stream (mosek.streamtype.log, new msgclass (""));

          /* Append 'numcon' empty constraints.
             The constraints will initially have no bounds. */
          task.appendcons(numcon);

          /* Append 'numvar' variables.
             The variables will initially be fixed at zero (x=0). */
          task.appendvars(numvar);

          for (int j = 0; j < numvar; ++j)
          {
            /* Set the linear term c_j in the objective.*/
            task.putcj(j, c[j]);
            /* Set the bounds on variable j.
                   blx[j] <= x_j <= bux[j] */
            task.putvarbound(j, bkx[j], blx[j], bux[j]);
          }

          for (int j = 0; j < aval.Length; ++j)
            /* Input column j of A */
            task.putacol(j,          /* Variable (column) index.*/
                         asub[j],     /* Row index of non-zeros in column j.*/
                         aval[j]);    /* Non-zero Values of column j. */

          /* Set the bounds on constraints.
               for i=1, ...,numcon : blc[i] <= constraint i <= buc[i] */
          for (int i = 0; i < numcon; ++i)
            task.putconbound(i, bkc[i], blc[i], buc[i]);

          csub[0] = 3;
          csub[1] = 0;
          csub[2] = 1;
          task.appendcone(mosek.conetype.quad,
                          0.0, /* For future use only, can be set to 0.0 */
                          csub);
          
          csub[0] = 4;
          csub[1] = 5;
          csub[2] = 2;
          task.appendcone(mosek.conetype.rquad, 0.0, csub);
          
          task.putobjsense(mosek.objsense.minimize);
          task.optimize();
          // Print a summary containing information
          //   about the solution for debugging purposes
          task.solutionsummary(mosek.streamtype.msg);

          mosek.solsta solsta;
          /* Get status information about the solution */
          task.getsolsta(mosek.soltype.itr, out solsta);

          double[] xx  = new double[numvar];

          task.getxx(mosek.soltype.itr, // Basic solution.
                     xx);

          switch (solsta)
          {
            case mosek.solsta.optimal:
            case mosek.solsta.near_optimal:
              Console.WriteLine ("Optimal primal solution\n");
              for (int j = 0; j < numvar; ++j)
                Console.WriteLine ("x[{0}]: {1}", j, xx[j]);
              break;
            case mosek.solsta.dual_infeas_cer:
            case mosek.solsta.prim_infeas_cer:
            case mosek.solsta.near_dual_infeas_cer:
            case mosek.solsta.near_prim_infeas_cer:
              Console.WriteLine("Primal or dual infeasibility.\n");
              break;
            case mosek.solsta.unknown:
              Console.WriteLine("Unknown solution status.\n");
              break;
            default:
              Console.WriteLine("Other solution status");
              break;
          }
        }
      }
    }
  }
}