Mosek uses imposes linear or rotated quadratic constraint for small-size LMI constraints.

[hongkai-dai]

For LMI on a 1x1 matrix, we impose linear inequality constraint. For LMI on a 2x2 matrix, we impose a rotated quadratic constraint (rotated Lorentz cone constraint)

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This change is 

[hongkai-dai]

+@AlexandreAmice for feature review please, thanks!

Reviewable status: LGTM missing from assignee AlexandreAmice, needs platform reviewer assigned, needs at least two assigned reviewers

[AlexandreAmice]

Reviewed 2 of 2 files at r1, all commit messages.
Reviewable status: 3 unresolved discussions, LGTM missing from assignee AlexandreAmice, needs platform reviewer assigned, needs at least two assigned reviewers

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solvers/mosek_solver_internal.h line 651 at r1 (raw file):

        const double tmp = dual_sol(1);
        dual_sol(1) = dual_sol(2) / 2;
        dual_sol(2) = tmp;

I find this inconsistent reshuffling a bit confusing. I would introduce the mapping

[y0, y1] -> [y0, y2, y1*sqrt(2)]
[y1, y2]

this being in mosek's rotated lorentz cone is the condition

2*y0*y2 >=  2 * y1**2 

which is the determinant condition and this is just a reshuffling of the standard PSD matrix vectorization.

Then denote the dual solution of Mosek's rotated second order cone as

[z0, z2, z1]

which makes it clear that under the inverse mapping, to preserve the inner product we must have

[z0, z2, z1] -> [z0, z1/sqrt(2)]
                       [z1/sqrt(2), z2]

This makes everything symmetric and much more similar to the normal case.

Suggestion:

        // We impose the LMI constraint as a rotated Quadratic cone constraint
        // in Mosek. For a constraint that
        // [y0 y1]
        // [y1 y2]
        // is in psd, this is equivalent to
        // [y0/2, y2, y1] in Mosek's rotated quadratic cone.
        // If we denote the dual solution of the Mosek's rotated quadratic cone
        // as [z0, z1, z2], then by the duality theory, that the complementary
        // slackness (namely the inner product between the primal and the dual)
        // should be the same for both the PSD cone and the rotated quadratic
        // cone, hence the dual solution for the PSD constraint is
        // [z0/2 z2/2]
        // [z2/2   z1]
        // We store the lower triangular part of the dual PSD matrix as the dual
        // solution, namely [z0/2, z2/2, z1]
        dual_sol(0) *= 0.5;
        const double tmp = dual_sol(1);
        dual_sol(1) = dual_sol(2) / 2;
        dual_sol(2) = tmp;  

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solvers/mosek_solver_internal.cc line 969 at r1 (raw file):

      A.setFromTriplets(A_triplets.begin(), A_triplets.end());
      MSKdomaintypee domain_type =
          matrix_rows == 1 ? MSK_DOMAIN_RPLUS : MSK_DOMAIN_SVEC_PSD_CONE;

It might be nice to hard-code the 1x1 case similar to the 2x2 case. It would help the reader separate the three cases more easily.

Code quote:

      MSKdomaintypee domain_type =
          matrix_rows == 1 ? MSK_DOMAIN_RPLUS : MSK_DOMAIN_SVEC_PSD_CONE;

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solvers/mosek_solver_internal.cc line 981 at r1 (raw file):

      // [y1, y2]
      // We impose it as a rotated Lorentz cone constraint, namely
      // [y0/2, y2, y1] is in Mosek's rotated Quadratic cone.

see discussion on the dual solution.

Code quote:

      // For PSD constraint on a 2x2 matrix
      // [y0, y1] = F0 + F1*x0 + ... + Fk*xk
      // [y1, y2]
      // We impose it as a rotated Lorentz cone constraint, namely
      // [y0/2, y2, y1] is in Mosek's rotated Quadratic cone.

[hongkai-dai]

Reviewable status: 3 unresolved discussions, LGTM missing from assignee AlexandreAmice, needs platform reviewer assigned, needs at least two assigned reviewers

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solvers/mosek_solver_internal.h line 651 at r1 (raw file):

Previously, AlexandreAmice (Alexandre Amice) wrote…

I find this inconsistent reshuffling a bit confusing. I would introduce the mapping

[y0, y1] -> [y0, y2, y1*sqrt(2)]
[y1, y2]

this being in mosek's rotated lorentz cone is the condition

2*y0*y2 >=  2 * y1**2 

which is the determinant condition and this is just a reshuffling of the standard PSD matrix vectorization.

Then denote the dual solution of Mosek's rotated second order cone as

[z0, z2, z1]

which makes it clear that under the inverse mapping, to preserve the inner product we must have

[z0, z2, z1] -> [z0, z1/sqrt(2)]
                       [z1/sqrt(2), z2]

This makes everything symmetric and much more similar to the normal case.

I agree scaling y2 by sqrt(2) is aesthetically better (it is symmetric).

On the other hand, all the rotated Lorentz cone constraint in MosekSolver is parsed by scaling y0 with a factor of 0.5. To stay consistent with all the other code in side this same file, I think we should do scaling of 0.5 on y0 here as well.

We could have a separate PR to change the scaling for all rotated Lorentz cone constraint.

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solvers/mosek_solver_internal.cc line 981 at r1 (raw file):

Previously, AlexandreAmice (Alexandre Amice) wrote…

see discussion on the dual solution.

Done. Ditto

[AlexandreAmice]

Reviewable status: 2 unresolved discussions, needs platform reviewer assigned, needs at least two assigned reviewers

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solvers/mosek_solver_internal.h line 651 at r1 (raw file):

Previously, hongkai-dai (Hongkai Dai) wrote…

I agree scaling y2 by sqrt(2) is aesthetically better (it is symmetric).

On the other hand, all the rotated Lorentz cone constraint in MosekSolver is parsed by scaling y0 with a factor of 0.5. To stay consistent with all the other code in side this same file, I think we should do scaling of 0.5 on y0 here as well.

We could have a separate PR to change the scaling for all rotated Lorentz cone constraint.

Very well. I would still adjust the labeling in the comment so that it appears as [z0 z2 z1]

[hongkai-dai]

Reviewable status: 2 unresolved discussions, needs platform reviewer assigned, needs at least two assigned reviewers

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solvers/mosek_solver_internal.h line 651 at r1 (raw file):

Previously, AlexandreAmice (Alexandre Amice) wrote…

Very well. I would still adjust the labeling in the comment so that it appears as [z0 z2 z1]

Done.

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solvers/mosek_solver_internal.cc line 969 at r1 (raw file):

Previously, AlexandreAmice (Alexandre Amice) wrote…

It might be nice to hard-code the 1x1 case similar to the 2x2 case. It would help the reader separate the three cases more easily.

Done.