[R] Optimization inconsistencies

[peter dalgaard]

On May 18, 2012, at 09:10 , Hans W Borchers wrote:

> peter dalgaard <pdalgd <at> gmail.com> writes:
>> 
>> On May 18, 2012, at 00:14 , Nathan Stephens wrote:
>> 
>>> I have a very simple maximization problem where I'm solving for the vector
>>> x:
>>> 
>>> objective function:
>>> w'x = value to maximize
>>> 
>>> box constraints (for all elements of w):
>>> low < x < high
>>> 
>>> equality constraint:
>>> sum(x) = 1
>>> 
>>> But I get inconsistent results depending on what starting values I. I've
>>> tried various packages but none seem to bee the very solver in Excel. Any
>>> recommendations on what packages or functions I should try?
> 
> Use the equality constraint to reduce the dimension of the problem by one.
> Then formulate the inequality constraints and apply, e.g., constrOptim().
> You can immediately write down and use the gradient, so method "L-BFGS-B" is
> appropriate.

I considered making a similar remark, then realized that lpSolve actually allows equality constraints, so why not just use the tool that is designed for the job?


> Because the problem is linear, there is only one maximum and no dependency
> on starting values. 

However, with a linear objective function, the Hessian is 0 and the maximum is attained at a corner point, which is likely to confuse algorithms that expect a locally quadratic function. 

> If you had supplied some data and code (which packages did you try, and how?),
> a more concrete answer would have been possible. My own test example worked
> out of the box.
> 

Yes, also from the development perspective. We need to see more of these hard examples. 


> Hans Werner
> 
> 
>> Sounds like a linear programming problem, so perhaps one of the packages 
>> that are specialized for that? lpSolve looks like it should do it.
>> 
>> (As a general matter: There's nothing simple about constrained maximization 
>> problems, and generic optimizers aren't really geared towards dealing with 
>> large sets of constraints.)
>> 
>>> 
>>> --Nathan
> 
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-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
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