Add scaled diagonally dominant matrix constraint.

[hongkai-dai]

@shensquared

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This change is 

[hongkai-dai]

+@frankpermenter for feature review please. Thanks!

Reviewable status: all discussions resolved, platform LGTM missing

[frankpermenter]

Reviewed 3 of 4 files at r1.
Reviewable status: 1 unresolved discussion, LGTM missing from assignee frankpermenter, platform LGTM missing

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solvers/test/scaled_diagonally_dominant_matrix_test.cc, line 58 at r1 (raw file):

he product DXD is diagonally dominant with non-negative diagonal entries

This looks like a linear program. If so, then there is a bug in the formulation. (You cannot check membership in the SDD cone by solving a linear program since it is nonpolyhedral.) I think the constraint you mean to check is existence of D for which A=DXD. This is quadratic, not linear, in D.

[hongkai-dai]

Reviewable status: 1 unresolved discussion, LGTM missing from assignee frankpermenter, platform LGTM missing

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solvers/test/scaled_diagonally_dominant_matrix_test.cc, line 58 at r1 (raw file):

Previously, frankpermenter wrote…

he product DXD is diagonally dominant with non-negative diagonal entries

This looks like a linear program. If so, then there is a bug in the formulation. (You cannot check membership in the SDD cone by solving a linear program since it is nonpolyhedral.) I think the constraint you mean to check is existence of D for which A=DXD. This is quadratic, not linear, in D.

If we are searching for a matrix X being SDD, then the program is not LP, but SOCP. On the other hand, if X is given, but only need to determine that if the given X is SDD, then we can do this through an LP.

The idea is that if X is SDD, then there exist a diagonal matrix D, such that A = DXD is diagonally dominant. Namely A(i, j) = d(i) * d(j) * X(i, j). The constraint that A being DD is that A(i, i) >= sum_j A(i, j), which is equivalent to d(i)X(i, i) >= sum_j d(j) X(i, j).

This idea is mentioned as remark 6 of Amir Ali's paper. I added it in the documentation.

[frankpermenter]

Reviewable status: 1 unresolved discussion, platform LGTM missing

[jwnimmer-tri]

+@soonho-tri for platform review per schedule please.

[soonho-tri]

@frankpermenter, could you check the pending issue?

Reviewed 3 of 4 files at r1, 1 of 1 files at r2.
Reviewable status: 4 unresolved discussions, platform LGTM from [soonho-tri], commit history needs curation

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solvers/mathematical_program.h, line 2053 at r2 (raw file):

   * represents the absolute value |X(i, j)| ∀ j ≠ i. The diagonal entries
   * Y(i, i) = X(i, i)
   * The users can refer to DSOS and SDSOS Optimization: More Tractable

nit: Please quote the cited article such as

"More Tractable..."

Also, how about having the arxiv link of it: https://arxiv.org/abs/1706.02586 ?

(FYI: the article appears again below)

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solvers/mathematical_program.h, line 2082 at r2 (raw file):

   * X.
   * @pre X(i, j) should be a linear expression of decision variables.
   * @return M for i < j, M[i][j] is the 2 x 2 symmetric matrix

I think the @return part is not clear enough. How about the following?

   * @return A vector of vectors of 2 x 2 symmetric matrices M.
   * For i < j, M[i][j] is
   * <pre>
   * [Mⁱʲ(i, i), Mⁱʲ(i, j)]
   * [Mⁱʲ(i, j), Mⁱʲ(j, j)].
   * </pre>
   * Note that M[i][j](0, 1) = Mⁱʲ(i, j) = (X(i, j) + X(j, i)) / 2.
   * For i >= j, M[i][j] is the zero matrix.

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solvers/test/scaled_diagonally_dominant_matrix_test.cc, line 62 at r2 (raw file):

  // d, such that the matrix A defined as A(i, j) = d(j) * X(i, j) is diagonally
  // dominant with positive diagonals.
  // This is explained as Remark 6 of DSOS and SDSOS optimization: more

nit: please consider quoting "DSOS and SDSOS ..."

[hongkai-dai]

Reviewable status: 2 unresolved discussions, platform LGTM from [soonho-tri], commit history needs curation

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solvers/mathematical_program.h, line 2053 at r2 (raw file):

Previously, soonho-tri (Soonho Kong) wrote…

nit: Please quote the cited article such as

"More Tractable..."

Also, how about having the arxiv link of it: https://arxiv.org/abs/1706.02586 ?

(FYI: the article appears again below)

Done

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solvers/mathematical_program.h, line 2082 at r2 (raw file):

Previously, soonho-tri (Soonho Kong) wrote…

I think the @return part is not clear enough. How about the following?

   * @return A vector of vectors of 2 x 2 symmetric matrices M.
   * For i < j, M[i][j] is
   * <pre>
   * [Mⁱʲ(i, i), Mⁱʲ(i, j)]
   * [Mⁱʲ(i, j), Mⁱʲ(j, j)].
   * </pre>
   * Note that M[i][j](0, 1) = Mⁱʲ(i, j) = (X(i, j) + X(j, i)) / 2.
   * For i >= j, M[i][j] is the zero matrix.

Done. Thanks!

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solvers/test/scaled_diagonally_dominant_matrix_test.cc, line 62 at r2 (raw file):

Previously, soonho-tri (Soonho Kong) wrote…

nit: please consider quoting "DSOS and SDSOS ..."

Done.

[soonho-tri]

Reviewed 2 of 2 files at r3.
Reviewable status: 1 unresolved discussion, platform LGTM from [soonho-tri], commit history needs curation

[hongkai-dai]

Dismissed @frankpermenter from a discussion.
Reviewable status: all discussions resolved, platform LGTM from [soonho-tri], commit history needs curation