Support QuadraticConstraint in MathematicalProgram

[hongkai-dai]

Gurobi natively supports the convex quadratic constraint x'*Q*x + q' * x <= b, we should add that into MathematicalProgram and GurobiSolver.

[RussTedrake]

Looks like Mosek supports them, too. It seems that we have QuadraticConstraint and the solvers declare if they support quadratic constraints, but we just don't have AddQuadraticConstraint nor AddConstraint<Binding<QuadraticConstraint>> on MathematicalProgram?

In the meantime, quadratic constraints can be written as Lorentz cone constraints.

[hongkai-dai]

Yes, and that is intentional. Mosek strongly suggests the users to reformulate the quadratic constraints as Lorentz cone constraints, mentioned here.

[hongkai-dai]

If it helps, I think we could provide this function

AddQuadraticConstraint(const Expression& quadratic_expr, double upper_bound, bool lorentz_cone=true);

which impose the constraint quadratic_expr <= upper_bound. And by default we enforce this constraint as a Lorentz cone constraint (with lorentz_cone=true).

[RussTedrake]

That could be nice, but it's certainly not urgent.

Should we perhaps delete QuadraticConstraint and kQuadraticConstraint if they are unused and if we have no intention of using them?

[hongkai-dai]

Currently QuadraticConstraint and kQuadraticConstraint are used in nonlinear optimization, for example by SnoptSolver. So I think it is useful to keep it around?

[RussTedrake]

How does a user create one of those constraints?

[hongkai-dai]

The user can do

prog.AddConstraint(std::make_shared<QuadraticConstraint>(...), vars)

as shown in snopt_solver_test.cc https://github.com/RobotLocomotion/drake/blob/6c5a106b1d3660a5e10c3667af4eddaf218a479c/solvers/test/snopt_solver_test.cc#L99-L102

Just curious, do you have a convex optimization problem with a convex quadratic constraint? I think that's a common use case, and I can add the function to parse the convex constraint to a lorentz cone constraint.

[RussTedrake]

I have one in my open PR: https://github.com/RobotLocomotion/drake/pull/15194/files#diff-eeb9eea73750e4da6b48cbda49815973217d01264b9ed577ea725ba477b2720bR95

[hongkai-dai]

As mentioned in https://github.com/RobotLocomotion/drake/issues/18940#issuecomment-1455154129.

I think I should add an API

MathematicalProgram::AddQuadraticConstraint(...);

which adds Binding<QuadraticConstraint> into MathematicalProgram. Then the solver will either treat it like a generic constraint (just compute the value and the gradient in SNOPT, IPOPT, etc), or a Lorentz cone constraint (by convex solvers like Mosek, Gurobi, SCS, etc).

This will involve quite some change in MathematicalProgram, as now we will add a new data member

std::vector<Binding<QuadraticConstraint>> quadratic_constraints_;

into MathematicalProgram. Also we need to mark each QuadraticConstraint whether it is convex or not.

[hongkai-dai]

Prioritizing this because we are solving QCQP.