Improve the scaling

[zaikunzhang]

This has been requested a few times. See

libprima/PRIMA.jl#4
#95
#181

Currently, the MATLAB and Julia interfaces do scale the problem if requested and if possible:

prima/matlab/interfaces/private/preprima.m

Line 297 in 67b6084

% Scale the problem if necessary and if intended.

https://github.com/libprima/PRIMA.jl/blob/2714339cfecf6b895e7b1ef5f278f1c105c4be1a/src/PRIMA.jl#L120

Maybe we can add two optional arguments scaling_factor and shift, to perform the scaling

x_before_scaling = scaling_factor*x_after_scaling + shift

or, equivalently,

x_after_scaling = (x_before_scaling - shift) / scaling_factor

In the high-level interfaces, there should be an option scale taking three possible values

• "none" (default)
• "auto"
• a vector of positive numbers

where "auto" means that the scaling factor will be decided automatically according to the given problem. Not that it is nontrivial to decide a suitable scaling factor.

Similarly, there should be an option shift taking three possible values

• "none" (default)
• "auto"
• a vector of real numbers

where "auto" means that the shift will be decided automatically according to the given problem (e.g., shift = x0).