solution for nonlinear optimization problem

Hi,

I hope here is one matlab wizard who can help me with following non linear optimization problem. I have a vector of couple of thousands entries (probability vector) p~ and a vector x~ with real values (same number)

I would like to solve following optimization problem and solving for p

 min    sum log(p~i*x~i)*log((p~i*x~i)/(pi*x~i))
 p    1 o 3000

subject to (expectation and variance constraint)

 sum pi*x~i  >= some number

 sum pi(x~i-(sum (pi*x~i))^2) >=some number

I think you need to rewrite the problem with Lagrange duality because otherwise you have too many variables.

Thank you

Andreas

Answer 1

[Matt J]

Looks like FMINCON should be able to handle it. I don't see an immediate need to rewrite the problem in dual space. 3000 variables isn't absurdly large and your objective function has a highly separable form. My gut says it would converge pretty fast in the space of x(i).

[John D'Errico]

Don't forget a positivity constraint on the variables. Else those logs will cause all heck to break loose.

[Andreas Stach]

Wow thanks Matt and John, fast answer I am new to Matlab and apologize for stupid questions:

But any idea how I can formulate the constraints?

The first one i would use scalar multiplication p`x~ >=number

but how about the second one? The second one contains the first one?

What does separable means?

Appreciate your thoughts

Andy

[Matt J]

The first inequality is linear so you would use the A,b inputs of fmincon with

A=-pi(:).';
b= -somenumber

The nonlinear inequality would be implemented using the nonlcon input argument.