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:

M = 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.

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

% Create the aliases 
%      z1 = [ y[1],x[1],x[2] ]
%  and z2 = [ y[2],y[3],x[3] ]
z1 = Var.vstack(y.index(1),   x.slice(1,3));
z2 = Var.vstack(y.slice(2,4), x.index(3));

The linear constraint is defined using the dot product:

% Create the constraint
%      x[0] + x[1] + 2.0 x[2] = 1.0
M.constraint('lc', Expr.dot([1.0, 1.0, 2.0], 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
qc1 = M.constraint('qc1', z1, Domain.inQCone())
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 linearsolution values
solx = x.level();
soly = y.level();
% Get conic solution of qc1
qc1lvl = qc1.level();
qc1sn  = qc1.dual();
Listing 7.2 Fusion implementation of model (1). Click here to download.
function cqo1()
import mosek.fusion.*;

M = Model('cqo1');
        
x = M.variable('x', 3, Domain.greaterThan(0.0));
y = M.variable('y', 3, Domain.unbounded());

% Create the aliases 
%      z1 = [ y[1],x[1],x[2] ]
%  and z2 = [ y[2],y[3],x[3] ]
z1 = Var.vstack(y.index(1),   x.slice(1,3));
z2 = Var.vstack(y.slice(2,4), x.index(3));
        
% Create the constraint
%      x[0] + x[1] + 2.0 x[2] = 1.0
M.constraint('lc', Expr.dot([1.0, 1.0, 2.0], 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
qc1 = M.constraint('qc1', z1, Domain.inQCone())
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 linearsolution values
solx = x.level();
soly = y.level();
disp(['[x1 x2 x3] = ', mat2str(solx',7)]);
disp(['[y1 y2 y3] = ', mat2str(soly',7)]);

% Get conic solution of qc1
qc1lvl = qc1.level();
qc1sn  = qc1.dual();
disp(['qc1 levels = ', mat2str(qc1lvl',7)]);
disp(['qc1 dual conic var levels = ', mat2str(qc1sn',7)]);