# 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:

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

      using (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.AsExpr(), Domain.InQCone());
Constraint qc2 = M.Constraint("qc2", z2.AsExpr(), 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.
using System;
using mosek.fusion;

namespace mosek.fusion.example
{
public class cqo1
{
public static void Main(string[] args)
{
using (Model M = new Model("cqo1"))
{

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.AsExpr(), Domain.InQCone());
Constraint qc2 = M.Constraint("qc2", z2.AsExpr(), 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();
Console.WriteLine("x1,x2,x3 = {0}, {1}, {2}", solx[0], solx[1], solx[2]);
Console.WriteLine("y1,y2,y3 = {0}, {1}, {2}", soly[0], soly[1], soly[2]);

// Get conic solution of qc1
double[] qc1lvl = qc1.Level();
double[] qc1sn  = qc1.Dual();

Console.Write("qc1 levels = {0}", qc1lvl[0]);
for (int i = 1; i < qc1lvl.Length; ++i)
Console.Write(", {0}", qc1lvl[i]);
Console.WriteLine();

Console.Write("qc1 dual conic var levels = {0}", qc1sn[0]);
for (int i = 1; i < qc1sn.Length; ++i)
Console.Write(", {0}", qc1sn[i]);
Console.WriteLine();
}
}
}
}