modeling a power cone

[Jack1]

Given a vector decision variable x and vectors a, b of real scalars, I am trying to model the following quantity to be used inside an optimization objective:

x = sdpvar(5,1); a = rand(5,1); b = rand(5,1);
b'*(abs(x - a).^(3/2))

and I get the following output:

Nonlinear scalar (real, homogeneous, 5 variables)
Coeffiecient range: 0.09754 to 0.96489

I am wondering if Yalmip would represent the above quantity (i.e., in absolute value raised to (3/2)) as a power cone once Mosek is used as the solver? [mosek documentation: linked]

And if it would be ok for me to proceed with a cvx. optimization? Thank you.

[Jack1]

I get the following error when I optimize:

sol = 
  struct with fields:
    solvertime: 0
          info: 'Solver not applicable (mosek)'
       problem: -4
    yalmiptime: 8.418

 maybe I can express the objective I have in another format? thank you. 

[Jack1]

Not sure if cpower might come in handy here? But, do I need to do a cpower for each variable difference (with a) and product (with b) and then sum all of these together? 

[Johan Löfberg]

power via ^ or .^ leads to general nonlinear program (except for quadratics). To get conic programming modelling, you have to use cpower. The case you show complies with convexity propagation rules and will thus work

YALMIP does not support power cone in mosek. Instead, cpower is implemented using an socp model for rational powers

[Jack1]

Thank you for your reply. So, should I just use cpower on my expression separated by my rational power of 3/2? But, how do I go about tossing this into an optimization objective and solving the problem with additional linear, quadratic objective terms and constraints using the mosek solver? If possible, I would appreciate a small working example for my expression above?

[Jack1]

Something like
cpower(b'*abs(x - a), 3/2)
? But, not sure if this does the operation element by element which is what I am after. Thanks again.

[Johan Löfberg]

you simply use cpower and abs as you've written it and yalmip will automatically convert the model to an socp, and call mosek if that is the solver you are using

optimize(x^2 <= 1,cpower(abs(x-4),3/2) + abs(x))

[Johan Löfberg]

cpower acts elementwise

[Jack1]

Also, is there any reason as to why yalmip's cpower takes much longer to convert the model to socp than cvx's power_pos(x,p) fcn [http://cvxr.com/cvx/doc/funcref.html]? I need this conversion many times and so would have helped me immensely if cpower was faster. Thank you. 

[Johan Löfberg]

you would have to supply an example to see where the bottle-neck is for your specific case

[Jack1]

Here is a simple example for yalmip and cvx:

tic
x = sdpvar(3000,1);
optimize(norm(x,2) <= 1,ones(1,3000)*cpower(abs(x-4),3/2))
toc

tic
cvx_begin
    variable y(3000);
    minimize( ones(1,3000)*pow_pos(abs(y-4),3/2) );    
    subject to
        norm(y,2) <= 1;
cvx_end
toc

The first one takes quite some time. Is it because my decision variable is very large? Thank you. 

[Johan Löfberg]

yes that is nasty. I have no quick fix, will have to dig deep into that one

[Jack1]

Would you be able to suggest some ways that I can try to minimize element-wise deviation? 

For instance, given decision variable x and a known vector b, I would like to minimize the element-wise deviation between them --> so, I tried adding 

sum(x-b) == 0

as a constraint. The solution was infeasible so I tried to relax the lower+upper bounds, but, the same issue persisted. 

Any advice on this would be much appreciated.

[Johan Löfberg]

norm(x-b,p)<=tolerance for standard norms such as 2, inf, 1 would be the vanilla way

[Jack1]

Thank you for getting back! Ideally, I would like my optimized variable x to respect the same sign and relative magnitudes (element-wise) of the quantity b that I am optimizing. Any advice on how I could do so would be greatly appreciated, thank you

[Johan Löfberg]

same sign simply means you want to add x(find(b>0))>=0 and x(find(b<0))<=0, and the relative size probably means you want to use weights when you measure the deviation using the norm

[Jack1]

Thank you -- tried using the find fcn. but running into infeasbility issues. Any other suggestion would be very appreciative

[Johan Löfberg]

well then you have to add slacks or any hack you find reasonable