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.

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  • $\begingroup$ 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

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