7.2 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:

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

In Fusion the coordinates of a cone are not restricted to single variables. They can be arbitrary linear expressions, and an auxiliary variable will be substituted by Fusion in a way transparent to the user.

7.2.1 Example CQO1

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

(1)\[\begin{split} \begin{array}{ll} \minimize & y_1 + y_2 + y_3 \\ \st & x_1 + x_2 + 2.0 x_3 = 1.0,\\ & x_1,x_2,x_3 \geq 0.0,\\ & (y_1,x_1,x_2) \in \Q^3,\\ & (y_2,y_3,x_3) \in \Qr^3. \end{array}\end{split}\]

We start by creating the optimization model:

    Model M = new Model("cqo1");

We then define variables x and y. Two logical variables (aliases) z1 and z2 are introduced to model the quadratic cones. These are not new variables, but map onto parts of x and y for the sake of convenience.

      Variable x = M.variable("x", 3, Domain.greaterThan(0.0));
      Variable y = M.variable("y", 3, Domain.unbounded());

      // Create the aliases
      //      z1 = [ y[0],x[0],x[1] ]
      //  and z2 = [ y[1],y[2],x[2] ]
      Variable z1 = Var.vstack(y.index(0),  x.slice(0, 2));
      Variable z2 = Var.vstack(y.slice(1, 3), x.index(2));

The linear constraint is defined using the dot product:

      // Create the constraint
      //      x[0] + x[1] + 2.0 x[2] = 1.0
      double[] aval = new double[] {1.0, 1.0, 2.0};
      M.constraint("lc", Expr.dot(aval, x), Domain.equalsTo(1.0));

The conic constraints are defined using the logical views z1 and z2 created previously. Note that this is a basic way of defining conic constraints, and that in practice they would have more complicated structure.

      // Create the constraints
      //      z1 belongs to C_3
      //      z2 belongs to K_3
      // where C_3 and K_3 are respectively the quadratic and
      // rotated quadratic cone of size 3, i.e.
      //                 z1[0] >= sqrt(z1[1]^2 + z1[2]^2)
      //  and  2.0 z2[0] z2[1] >= z2[2]^2
      Constraint qc1 = M.constraint("qc1", z1, Domain.inQCone());
      Constraint qc2 = M.constraint("qc2", z2, Domain.inRotatedQCone());

We only need the objective function:

      // Set the objective function to (y[0] + y[1] + y[2])
      M.objective("obj", ObjectiveSense.Minimize, Expr.sum(y));

Calling the Model.solve method invokes the solver:

      M.solve();

The primal and dual solution values can be retrieved using Variable.level, Constraint.level and Variable.dual, Constraint.dual, respectively:

      // Get the linear solution values
      double[] solx = x.level();
      double[] soly = y.level();
      // Get conic solution of qc1
      double[] qc1lvl = qc1.level();
      double[] qc1sn  = qc1.dual();
Listing 4 Fusion implementation of model (1). Click here to download.
package com.mosek.fusion.examples;
import mosek.fusion.*;

public class cqo1 {
  public static void main(String[] args)
  throws SolutionError {
    Model M = new Model("cqo1");
    try {
      Variable x = M.variable("x", 3, Domain.greaterThan(0.0));
      Variable y = M.variable("y", 3, Domain.unbounded());

      // Create the aliases
      //      z1 = [ y[0],x[0],x[1] ]
      //  and z2 = [ y[1],y[2],x[2] ]
      Variable z1 = Var.vstack(y.index(0),  x.slice(0, 2));
      Variable z2 = Var.vstack(y.slice(1, 3), x.index(2));

      // Create the constraint
      //      x[0] + x[1] + 2.0 x[2] = 1.0
      double[] aval = new double[] {1.0, 1.0, 2.0};
      M.constraint("lc", Expr.dot(aval, x), Domain.equalsTo(1.0));

      // Create the constraints
      //      z1 belongs to C_3
      //      z2 belongs to K_3
      // where C_3 and K_3 are respectively the quadratic and
      // rotated quadratic cone of size 3, i.e.
      //                 z1[0] >= sqrt(z1[1]^2 + z1[2]^2)
      //  and  2.0 z2[0] z2[1] >= z2[2]^2
      Constraint qc1 = M.constraint("qc1", z1, Domain.inQCone());
      Constraint qc2 = M.constraint("qc2", z2, Domain.inRotatedQCone());

      // Set the objective function to (y[0] + y[1] + y[2])
      M.objective("obj", ObjectiveSense.Minimize, Expr.sum(y));

      // Solve the problem
      M.solve();

      // Get the linear solution values
      double[] solx = x.level();
      double[] soly = y.level();
      System.out.printf("x1,x2,x3 = %e, %e, %e\n", solx[0], solx[1], solx[2]);
      System.out.printf("y1,y2,y3 = %e, %e, %e\n", soly[0], soly[1], soly[2]);

      // Get conic solution of qc1
      double[] qc1lvl = qc1.level();
      double[] qc1sn  = qc1.dual();
      
      System.out.printf("qc1 levels = %e", qc1lvl[0]);
      for (int i = 1; i < qc1lvl.length; ++i)
        System.out.printf(", %e", qc1lvl[i]);
      System.out.print("\n");

      System.out.printf("qc1 dual conic var levels = %e", qc1sn[0]);
      for (int i = 1; i < qc1sn.length; ++i)
        System.out.printf(", %e", qc1sn[i]);
      System.out.print("\n");

    } finally {
      M.dispose();
    }
  }
}