sci.op-research

golabidoon@gmail.com wrote:
> Hello folks,
>
> I am dealing a convex quadratic problem in some variables, say x_i's
> and some constants a_k's. The constants a_k's are themselves the
> solution of another convex quadratic problem (which does not involve
> x_i's), so I call it nested programming.
>
> A naive attempt to solve this system is to find a closed form solution
> for the inner program (the one in a_k's) and the plug its solution in
> the outer program (the one in x_i's) and solve it again. Unfortunately
> closed form solution looks nasty in my case, due to constraints a_k>0,
> the stationary points of the lagrangian is of form a_k=0 or
> a_k=something_else, which is a disjunctive constraint and cannot be
> used in the other program.
>
> Is there any way other than closed form solution idea to solve a
> nested program like this? Is any trick such as introducing auxilary
> variables and additional terms possible here?
>
> Thank you in advance!
>
> G.D.

If the a_k's are right-hand sides (not coefficients) in the outer
optimization, what about making the a_k's variables in the outer
problem, throwing in the dual variables for the inner problem, and
replacing optimization of the inner objective with satisfaction of the
first-order necessary conditions (which should be sufficient if you're
minimizing and the inner objective is convex)? So the outer problem
becomes picking x, a and u (the new duals) such that x and a satisfy the
constraints of the outer problem, a satisfies the constraints of the
inner problem, and a and u satisfy the dual conditions of the inner problem.

/Paul