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This question pertains to the Conic Benchmark Format (CBF) for specifying a convex optimization problem. Here's a link to the specification.

In the CBF specification, there are separate areas for positive semidefinite (psd) constraints and psd variables. But why should there be two separate areas? This seems completely redundant. No other convex cone is treated separately like this--variables are declared and then later constrained to be in a specific cone.

Are there any benefits (other than a shortcut for declaring variables subject to a psd constraint) that justify why the PSDVAR section should exist? I'm thinking specifically about benefits to the solver. Could any solvers be expected to take advantage of the fact that a variable was declared psd instead of a symmetric matrix later being constrained psd?

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  • $\begingroup$ Maybe @ErlingMOSEK can help? The CBF format seems to have been created by Mosek. $\endgroup$
    – Robert Bassett
    Commented Mar 25 at 18:01
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    $\begingroup$ I would like to note that CBF is a byproduct of the conic benchmark library (cblib.zib.de), attempting to simplify benchmarks across conic optimization software. You'll find many additional details on that website, including the fact that all cones can be used both as variable domains (like PSDVAR) or affine map domains (like PSDCON). $\endgroup$
    – Henrik Alsing Friberg
    Commented Apr 2 at 7:07

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We have included PSDCON and PSDVAR because

  • in some cases, it is more natural/convenient to use one over the other.
  • for instance, Mosek can only handle PSDVAR and must convert PSDCON to the PSDVAR form which often results in a significant loss in efficiency. (LMIs should be dualized.)

PS: There is a lot of redundancy in the CBF format. A scalar linear variable is a matrix variable of dimension 1.

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