6.4 Semidefinite Optimization

Semidefinite optimization is a generalization of conic quadratic optimization, allowing the use of matrix variables belonging to the convex cone of positive semidefinite matrices

\[\PSD^r = \left\lbrace X \in \Symm^r: z^T X z \geq 0, \quad \forall z \in \real^r \right\rbrace,\]

where \(\Symm^r\) is the set of \(r \times r\) real-valued symmetric matrices.

MOSEK can solve semidefinite optimization problems of the form

\[\begin{split}\begin{array}{lccccll} \mbox{minimize} & & & \sum_{j=0}^{n-1} c_j x_j + \sum_{j=0}^{p-1} \left\langle \barC_j, \barX_j \right\rangle + c^f & & &\\ \mbox{subject to} & l_i^c & \leq & \sum_{j=0}^{n-1} a_{ij} x_j + \sum_{j=0}^{p-1} \left\langle \barA_{ij}, \barX_j \right\rangle & \leq & u_i^c, & i = 0, \ldots, m-1,\\ & l_j^x & \leq & x_j & \leq & u_j^x, & j = 0, \ldots, n-1,\\ & & & x \in \K, \barX_j \in \PSD^{r_j}, & & & j = 0, \ldots, p-1 \end{array}\end{split}\]

where the problem has \(p\) symmetric positive semidefinite variables \(\barX_j\in \PSD^{r_j}\) of dimension \(r_j\) with symmetric coefficient matrices \(\barC_j\in \Symm^{r_j}\) and \(\barA_{i,j}\in \Symm^{r_j}\). We use standard notation for the matrix inner product, i.e., for \(A,B\in \real^{m\times n}\) we have

\[\left\langle A,B \right\rangle := \sum_{i=0}^{m-1} \sum_{j=0}^{n-1} A_{ij} B_{ij}.\]

6.4.1 Example SDO1

We consider the simple optimization problem with semidefinite and conic quadratic constraints:

(1)\[\begin{split}\begin{array} {llcc} \mbox{minimize} & \left\langle \left[ \begin{array} {ccc} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array} \right], \barX \right\rangle + x_0 & & \\ \mbox{subject to} & \left\langle \left[ \begin{array} {ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right], \barX \right\rangle + x_0 & = & 1, \\ & \left\langle \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right], \barX \right\rangle + x_1 + x_2 & = & 1/2, \\ & x_0 \geq \sqrt{{x_1}^2 + {x_2}^2}, & \barX \succeq 0, & \end{array}\end{split}\]

The problem description contains a 3-dimensional symmetric semidefinite variable which can be written explicitly as:

\[\begin{split}\barX = \left[ \begin{array} {ccc} \barX_{00} & \barX_{10} & \barX_{20} \\ \barX_{10} & \barX_{11} & \barX_{21} \\ \barX_{20} & \barX_{21} & \barX_{22} \end{array} \right] \in \PSD^3,\end{split}\]

and a conic quadratic variable \((x_0, x_1, x_2) \in \Q^3\). The objective is to minimize

\[2(\barX_{00} + \barX_{10} + \barX_{11} + \barX_{21} + \barX_{22}) + x_0,\]

subject to the two linear constraints

\[\begin{split}\begin{array}{ccc} \barX_{00} + \barX_{11} + \barX_{22} + x_0 & = & 1, \\ \barX_{00} + \barX_{11} + \barX_{22} + 2(\barX_{10} + \barX_{20} + \barX_{21}) + x_1 + x_2 & = & 1/2. \end{array}\end{split}\]

Setting up the linear and quadratic part

The linear and quadratic parts (constraints, variables, objective, cones) are set up using the methods described in the relevant tutorials; Sec. 6.1 (Linear Optimization) and Sec. 6.3 (Conic Quadratic Optimization). Here we only discuss the aspects directly involving semidefinite variables.

Appending semidefinite variables

First, we need to declare the number of semidefinite variables in the problem, similarly to the number of linear variables and constraints. This is done with the function Task.appendbarvars.

          task.appendbarvars(dimbarvar);

Appending coefficient matrices

Coefficient matrices \(\barC_j\) and \(\barA_{ij}\) are constructed as weighted combinations of sparse symmetric matrices previously appended with the function Task.appendsparsesymmat.

            idx[0] = task.appendsparsesymmat(dimbarvar[0],
                                             barc_i,
                                             barc_j,
                                             barc_v);

The arguments specify the dimension of the symmetric matrix, followed by its description in the sparse triplet format. Only lower-triangular entries should be included. The function produces a unique index of the matrix just entered in the collection of all coefficient matrices defined by the user.

After one or more symmetric matrices have been created using Task.appendsparsesymmat, we can combine them to set up the objective matrix coefficient \(\barC_j\) using Task.putbarcj, which forms a linear combination of one or more symmetric matrices. In this example we form the objective matrix directly, i.e. as a weighted combination of a single symmetric matrix.

            task.putbarcj(0, idx, falpha);

Similarly, a constraint matrix coefficient \(\barA_{ij}\) is set up by the function Task.putbaraij.

            task.putbaraij(0, 0, idx, falpha);

Retrieving the solution

After the problem is solved, we read the solution using Task.getbarxj:

              task.getbarxj(mosek.soltype.itr,    /* Request the interior solution. */
                            0,
                            barx);

The function returns the half-vectorization of \(\barX_j\) (the lower triangular part stacked as a column vector), where the semidefinite variable index \(j\) is passed as an argument.

Source code

Listing 8 Source code solving problem (1). Click here to download.
using System;

namespace mosek.example
{
  public class sdo1
  {
    public static void Main(string[] args)
    {
      int    numcon    = 2;  /* Number of constraints.              */
      int    numvar    = 3;  /* Number of conic quadratic variables */
      int    numanz    = 3;  /* Number of non-zeros in A            */
      int    numbarvar = 1;  /* Number of semidefinite variables    */
      int[]  dimbarvar = { 3 };         /* Dimension of semidefinite cone */
      int[]  lenbarvar = { 3 * (3 + 1) / 2 }; /* Number of scalar SD variables  */

      mosek.boundkey[] bkc = { mosek.boundkey.fx, mosek.boundkey.fx };
      double[]     blc     = { 1.0, 0.5 };
      double[]     buc     = { 1.0, 0.5 };

      int[]        barc_i  = { 0,   1,   1,   2,   2 },
                   barc_j  = { 0,   0,   1,   1,   2 };
      double[]     barc_v  = { 2.0, 1.0, 2.0, 1.0, 2.0 };

      int[][]      asub    = { new int[] {0},   new int[] {1, 2}}; /* column subscripts of A */
      double[][]   aval    = { new double[] {1.0}, new double[] {1.0, 1.0}};

      int[][]      bara_i  = { new int[] {0,   1,   2},   new int[] {0,   1 ,  2,   1,   2,   2 } },
      bara_j  = { new int[] {0,   1,   2},   new int[] {0,   0 ,  0,   1,   1,   2 } };
      double[][]   bara_v  = { new double[] {1.0, 1.0, 1.0}, new double[] {1.0, 1.0, 1.0, 1.0, 1.0, 1.0}};
      int[]        conesub = { 0, 1, 2 };



      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);

          /* Append 'NUMBARVAR' semidefinite variables. */
          task.appendbarvars(dimbarvar);

          /* Optionally add a constant term to the objective. */
          task.putcfix(0.0);

          /* Set the linear term c_j in the objective.*/
          task.putcj(0, 1.0);

          for (int j = 0; j < numvar; ++j)
            task.putvarbound(j, mosek.boundkey.fr, -0.0, 0.0);

          /* Set the linear term barc_j in the objective.*/
          {
            long[] idx = new long[1];
            double[] falpha = { 1.0 };
            idx[0] = task.appendsparsesymmat(dimbarvar[0],
                                             barc_i,
                                             barc_j,
                                             barc_v);
            task.putbarcj(0, idx, falpha);
          }

          /* 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,           /* Index of constraint.*/
                             bkc[i],      /* Bound key.*/
                             blc[i],      /* Numerical value of lower bound.*/
                             buc[i]);     /* Numerical value of upper bound.*/

          /* Input A row by row */
          for (int i = 0; i < numcon; ++i)
            task.putarow(i,
                         asub[i],
                         aval[i]);

          /* Append the conic quadratic cone */
          task.appendcone(mosek.conetype.quad,
                          0.0,
                          conesub);

          /* Add the first row of barA */
          {
            long[] idx = new long[1];
            double[] falpha = {1.0};
            task.appendsparsesymmat(dimbarvar[0],
                                    bara_i[0],
                                    bara_j[0],
                                    bara_v[0],
                                    out idx[0]);

            task.putbaraij(0, 0, idx, falpha);
          }

          {
            long[] idx = new long[1];
            double[] falpha = {1.0};
            /* Add the second row of barA */
            task.appendsparsesymmat(dimbarvar[0],
                                    bara_i[1],
                                    bara_j[1],
                                    bara_v[1],
                                    out idx[0]);

            task.putbaraij(1, 0, idx, falpha);
          }

          /* Run optimizer */
          task.optimize();

          /* Print a summary containing information
             about the solution for debugging purposes*/
          task.solutionsummary (mosek.streamtype.msg);

          mosek.solsta solsta;
          task.getsolsta (mosek.soltype.itr, out solsta);

          switch (solsta)
          {
            case mosek.solsta.optimal:
            case mosek.solsta.near_optimal:
              double[] xx = new double[numvar];
              double[] barx = new double[lenbarvar[0]];

              task.getxx(mosek.soltype.itr, xx);
              task.getbarxj(mosek.soltype.itr,    /* Request the interior solution. */
                            0,
                            barx);
              Console.WriteLine("Optimal primal solution");
              for (int i = 0; i < numvar; ++i)
                Console.WriteLine("x[{0}]   : {1}", i, xx[i]);

              for (int i = 0; i < lenbarvar[0]; ++i)
                Console.WriteLine("barx[{0}]: {1}", i, barx[i]);
              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 certificate found.");
              break;
            case mosek.solsta.unknown:
              Console.WriteLine("The status of the solution could not be determined.");
              break;
            default:
              Console.WriteLine("Other solution status.");
              break;
          }
        }
      }
    }
  }



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

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