sci.op-research

Using first order necessary condition of the inner problem as an
equality constraint for the outer problem is an interesting idea.
However, I do not understand how to handle the daul variables. Let's
go over a simple inner problem with only one variable (a):
Min (c_1*a^2+c_2*a) s.t. a>=3D0

where c_1 and c_2 are constant and c_1>0 to guarantee convexity.

Now writing the KKT conditions gives:
2*c_1*a+c_2-u*a=3D0
a>=3D0
u>=3D0
a*u=3D0

Now say the outer problem is Min (x^2+x) s.t. x>=3Da
So I get:
Min(x^2+x)
x>=3Da
2*c_1*a+c_2-u*a=3D0
a>=3D0
u>=3D0
a*u=3D0

The variables here are both a and x. However, what should I do with
the dual variable u? For example, if I want to feed this to a QP
solver, how should I represent the u? Do solvers have an option to
explicitly define a dual variable like u besides the main variables x
and a?

Your clarification would be appreciated.

G.D.

On Feb 25, 4:42=A0pm, Paul Rubin wrote:
> golabid...@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=3D0 or
> > a_k=3Dsomething_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)? =A0So 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 proble=
m.
>
> /Paul- Hide quoted text -
>
> - Show quoted text -