Hi I am considering an optimization problem that requires a condition to hold over a continuous range, for example: \begin{equation} \begin{aligned} \max_{\boldsymbol{x}, \gamma} & \ \gamma \\ s.t. & \ f(\boldsymbol{x}, y) \ge \gamma, \ \forall y\in[\alpha, \beta] \end{aligned} \end{equation} where $\boldsymbol{x}\in\mathbb{R}^{N}$. Since $f(\boldsymbol{x}, y)$ is not convex in $y$ so it cannot be directly packed into an optimization problem. So I sampled the range with a step size $\delta = (\beta - \alpha)/K$ as $y_k = \alpha + (k - 1)\delta, k = 1, \cdots, K + 1$ where $K$ is a positive integer. Then I used the following conditions to approximate the condition $f(\boldsymbol{x}, y)\ge \gamma$: \begin{equation} f(\boldsymbol{x}, y_k) \ge \gamma, \ k = 1, \cdots, K + 1 \end{equation} Then I have $K + 1$ conditions which can be handled by optimization toolbox.
Clearly when $K\to\infty$ an equivalence can be expected between the sampled conditions and the original one, but too large $K$ brings higher complexity and is impractical in reality. Now what is bothering me is how to specify the trade-off between complexity and performance, i.e. how to give a bound of the performance given a step size $\delta$ (or $K$). For example, given $\delta$ and $\hat{\boldsymbol{x}}, \hat{\gamma}$ is the optimal solution to the sampled problem, we can guarantee $f(\hat{\boldsymbol{x}}, y_k)\ge\hat{\gamma}, \forall k$, but can we give a bound $\xi\in[0, 1]$ such that $f(\hat{\boldsymbol{x}}, y)\ge \xi\hat{\gamma}, \forall y\in[\alpha, \beta]$? Obviously we should have $\xi\to 1$ monotonically as $\delta\to 0$, so that a proper $\delta$ could be chosen to minimize the complexity given any performance requirement.
I have tried bounding the derivatives $|\partial f/\partial y|\le \Theta$, and then bound $f(\hat{\boldsymbol{x}}, y)\ge \hat{\gamma} - \delta\Theta/2$ but it turned out that $\Theta\sim O(\sqrt{N})$ which is far from what simulation results suggest. I am wondering whether there is some efficient way to find a bound $\xi$ for the performance and how to approach it. Also I am not clear about how I can take full advantage of $f(\hat{\boldsymbol{x}}, y_k)\ge\hat{\gamma}$ to get a tighter bound. Any suggestions will be appreciated. Thank you!