2
$\begingroup$

I believe, I have a parametric nonlinear optimization problem. The non-convex constraints depend on some parameters, and I seek a solution that satisfies these constraints for all parameters in a specified continuous range.

Let $ {\bf A}(\vec{x},\vec{\theta}) := {\bf A} \in\mathbb{R}^{n\times n}$ be a matrix-valued function of $\vec{x}\in \mathbb{R}^n$ and $\vec{\theta}\in \mathbb{R}^m$ (here $m\ll n$). Define $ {\bf L}(\vec{x}):= {\bf L} \in\mathbb{R}^{n\times n}$ to be a matrix which is a function of $\vec{x}$, and $ \vec{F}(\vec{x}):= \vec{F} $ is a vector-valued function of $\vec{x}$.

Main problem: The goal is to find any $\vec{x}$ such that $$ {\bf A}(\vec{x},\vec{\theta})\vec{y} = \vec{F}(\vec{x}), \\ {\bf L}(\vec{x})\vec{y}\ge \vec{0}, $$ holds for all $\vec{\theta}$ in some parameter space (ideally, $0<\vec{\theta}<\infty$ or $0<a\le \vec{\theta}\le b$ for scalars $a,b$).

For a fixed $\vec{\theta}$ I am able to solve the problem using nonlinear programming. I have tried discretizing the parameter space and then solving the resulting coupled nonlinear programs, but this only leads to a solution that is valid for the discrete parameter values.

Is there a better way to think about this problem? Or any standard ways to solve it? There is no objective function in the formulation I posed above, and I don't know if that is of any significance.

New contributor
vaiana is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
  • $\begingroup$ Looks like the parameter $\theta$ is a scalar - correct? $\endgroup$ – Maxim Umansky Jan 14 at 5:06
  • $\begingroup$ In general $\vec{\theta}$ is a vector, however I am open to the case when it is a scalar as well $\endgroup$ – vainia Jan 14 at 5:30
  • $\begingroup$ First: A problem without objective function is no optimization problem. Next, the following is important: Are you looking for any x that satisfies the constraint or for all x? In the first case, you could probably reformulate your problem as optimization problem to find an x that allows 'as many' $\theta$ as possible. The latter case would be harder, I'm not sure there is a way to solve it for general A, F and L $\endgroup$ – Yann 2 days ago
  • $\begingroup$ I am looking for any x that satisfies the constraint for as many $\theta$ as possible. How would I go about doing this reformulation? $\endgroup$ – vainia 2 days ago
  • $\begingroup$ To be honest I am not sure, it is a tricky question. However, one more thing baffels me: For fixed x, are you really expecting eq (1) to hold for multiple $\theta$? That would mean that $Ay$ is independent of $\theta$, at least in a certain range. Is that on purpose? $\endgroup$ – Yann yesterday

Your Answer

vaiana is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.