consistency of MOI cone definitions

[chriscoey]

to make it easier to remember the order of the variables in the various cone definitions, I think we should follow a convention of (epi/hypograph, perspective, argument(s)). note that a epi/hypograph variable and/or a perspective variable may not be present.

all the cones already follow this convention (though it's not immediately obvious from their names/definitions)

• SOC: x >= norm2(z)
• RSOC: x >= y*sumabs2(z/y)/2 (equivalent to current definition but maybe a little clearer)
• PSD: 0 <= eigmin(Z), Z symmetric
• Power: x >= y*abs(z/y)^q (equivalent to current definition where q = 1/exponent is positive)
• Exponential: x <= y*log(z/y), z >= 9
• RootDet: x <= (det(Z))^1/n, Z PSD
• LogDet: x <= y*log(det(Z/y)), Z PSD (see logdet cone definition is wrong (needs perspective variable) #546)
• Geomean: x <= prod(z)^1/n, z >= 0 (perhaps replace with generalized version below)
• generalized Geomean: x <= prod_i z_i^alpha_i, z >= 0 (for alpha on unit simplex; sometimes called power cone in literature, with x or abs(x) on LHS)

so to make things clear I would recommend

• noting the (epi/hypograph, perspective, argument(s)) ordering convention in the documentation
• changing docs to use the less arbitrary descriptions above
• potentially changing the names or adding name aliases for some cones, like MOI.EpiPerspSquare = MOI.RotatedSecondOrderCone (since this cone is the epigraph of the perspective of the 1/2 square function) and MOI.HypoPerspLog = MOI.ExponentialCone (since it is the (closure of the) hypograph of the perspective of the log function)

I am happy to make these non-breaking changes.

[chriscoey]

another advantage is that these definitions make it easier to derive the dual cones of whatever cones you define, via the construction on p19 of http://www.isye.umn.edu/courses/ie8534/pdf/Lecture-4.pdf

furthermore, these cone definitions are more intuitive from a modeling perspective.

[chriscoey]

Bit late for this to be feasible, so closing