[Ipopt] non-convex model in IPOPT

[Christian]

Hello Hossein,

if a system is that general it can not be converted to any convex form.

If I understand you right, F(x,y) and mu*y both represent the "cost of power". 
This seems to be a doubleing to me, right? I guess there is still a 
missunderstanding in the problem formulation.

Why did you formulate it in the lagrangian form and some additional 
constraints? Either use the lagrangian form and the optimality constraints OR 
use pure bejective functions with constraints. Then ipopt will be able to 
solve it.

Christian

vom Wednesday 27 July 2011 06:17:56:
> Hello all,
> Many thanks for reply.
> it is a problem of minimum operating cost, in the context of electricity
> market.
> - F(x,y) is the cost of power, and "mu" is the nodal price in the market
> (Lagrange multiplier of the equality constraints ), and y is the power
> produced in the market.
> - "mu * y" represent the cost of power.
> 
> the point is that "mu" is only available post solution, but here it appears
> in the objective function.
> this problem seems to be non-convex because it entails the product of
> variables, and I am not sure if it can be converted into a convex form (if
> any).
> 
> Writing the optimality conditions is a good way, but we should note that
> they are stationary points (saddle-points + local optima)
> 
> Your comments are highly appreciated.
> 
> On Tue, Jul 26, 2011 at 9:36 PM, rony goldenthal <ronygold at gmail.com> wrote:
> > Hello Hossein,
> > 
> > The main challenge in implementing the function in IPOPT, as you
> > describe it, is that the Lagrange multipliers are not easily available
> > when evaluating the function to minimize.
> > Maybe you can try something like adding another set of variables, 'z',
> > and solve the following system instead:
> > 
> > Min L (x,y,z)= F(x)+ z * y      (a1)
> > s.t.
> > g(x,y) = 0                               (a2)
> > h(x,y)<= 0                              (a3)
> > z - mu = 0                              (a4)
> > Where mu are the Lagrange multipliers that correspond to g(x,y).
> > 
> > I cannot say much about the convexity of either of the approaches.
> > 
> > Good luck,
> > Rony

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