6.3 Conic Quadratic Optimization¶
The structure of a typical conic optimization problem is
(see Sec. 12 (Problem Formulation and Solutions) for detailed formulations). We recommend Sec. 6.2 (From Linear to Conic Optimization) for a tutorial on how problems of that form are represented in MOSEK and what data structures are relevant. Here we discuss how to set-up problems with the (rotated) quadratic cones.
MOSEK supports two types of quadratic cones, namely:
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, consider the following constraint:
which describes a convex cone in \(\real^3\) given by the inequality:
For other types of cones supported by MOSEK, see Sec. 15.11 (Supported domains) and the other tutorials in this chapter. Different cone types can appear together in one optimization problem.
6.3.1 Example CQO1¶
Consider the following conic quadratic problem which involves some linear constraints, a quadratic cone and a rotated quadratic cone.
The two conic constraints can be expressed in the ACC form as shown in (6.11)
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 Sec. 6.1 (Linear Optimization) 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
In order to append the conic constraints we first input the matrix \(\afef\) and vector \(\afeg\) appearing in (6.11). The matrix \(\afef\) is sparse and we input only its nonzeros using Task.putafefentrylist. Since \(\afeg\) is zero, nothing needs to be done about this vector.
Each of the conic constraints is appended using the function Task.appendacc. In the first case we append the quadratic cone determined by the first three rows of \(\afef\) and then the rotated quadratic cone depending on the remaining three rows of \(\afef\).
/* Create a matrix F such that F * x = [x(3),x(0),x(1),x(4),x(5),x(2)] */
task.appendafes(6);
task.putafefentrylist(new long[]{0, 1, 2, 3, 4, 5}, /* Rows */
new int[]{3, 0, 1, 4, 5, 2}, /* Columns */
new double[]{1.0, 1.0, 1.0, 1.0, 1.0, 1.0});
/* Quadratic cone (x(3),x(0),x(1)) \in QUAD_3 */
long quadcone = task.appendquadraticconedomain(3);
task.appendacc(quadcone, /* Domain */
new long[]{0, 1, 2}, /* Rows from F */
null); /* Unused */
/* Rotated quadratic cone (x(4),x(5),x(2)) \in RQUAD_3 */
long rquadcone = task.appendrquadraticconedomain(3);
task.appendacc(rquadcone, /* Domain */
new long[]{3, 4, 5}, /* Rows from F */
null); /* Unused */
The first argument selects the domain, which must be appended before being used, and must have the dimension matching the number of affine expressions appearing in the constraint. Variants of this method are available to append multiple ACCs at a time. It is also possible to define the matrix \(\afef\) using a variety of methods (row after row, column by column, individual entries, etc.) similarly as for the linear constraint matrix \(A\).
For a more thorough exposition of the affine expression storage (AFE) matrix \(\afef\) and vector \(\afeg\) see Sec. 6.2 (From Linear to Conic Optimization).
Source code
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];
// Create a task object.
using (mosek.Task task = new mosek.Task()) {
// 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]);
/* Create a matrix F such that F * x = [x(3),x(0),x(1),x(4),x(5),x(2)] */
task.appendafes(6);
task.putafefentrylist(new long[]{0, 1, 2, 3, 4, 5}, /* Rows */
new int[]{3, 0, 1, 4, 5, 2}, /* Columns */
new double[]{1.0, 1.0, 1.0, 1.0, 1.0, 1.0});
/* Quadratic cone (x(3),x(0),x(1)) \in QUAD_3 */
long quadcone = task.appendquadraticconedomain(3);
task.appendacc(quadcone, /* Domain */
new long[]{0, 1, 2}, /* Rows from F */
null); /* Unused */
/* Rotated quadratic cone (x(4),x(5),x(2)) \in RQUAD_3 */
long rquadcone = task.appendrquadraticconedomain(3);
task.appendacc(rquadcone, /* Domain */
new long[]{3, 4, 5}, /* Rows from F */
null); /* Unused */
task.putobjsense(mosek.objsense.minimize);
task.optimize();
// Print a summary containing information
// about the solution for debugging purposes
task.solutionsummary(mosek.streamtype.msg);
/* Get status information about the solution */
mosek.solsta solsta = task.getsolsta(mosek.soltype.itr);
double[] xx = task.getxx(mosek.soltype.itr); // Interior point solution
switch (solsta)
{
case mosek.solsta.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:
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;
}
}
}
}
}
