I need your help in getting an insightful information regarding the following problem:
Let $P \subset \mathbb{R}^d, Y \subset \left\lbrace 1,-1\right\rbrace$ a subset of points of cardinality $n$ and corresponding labels of cardinality $n$.
Let $\alpha_1, \alpha_2 \in \mathbb{R}$ such that $\alpha_1 > 0$, such that: $$ \begin{equation*} \begin{aligned} & \underset{w \in \mathbb{R}^d, b \in \mathbb{R}, \xi \in \mathbb{R}^n}{\text{minimize}} & & f_{\textit{SVM}}(P,w,b, \eta) = \frac{\left\| w\right\|_2^2}{2} + C \left\| \xi \right\|_1 \\ & \text{subject to} & & y_i(p_i^Tw + b) \geq 1 - \xi_i, \; i = 1, \ldots, n \\ &&& \left\| \xi_i \right\|_\infty \geq 0 \\ &&& \left\| w\right\|_2 \leq \alpha_1 \\\ &&& b \leq \alpha_2, \end{aligned} \end{equation*} $$ where $\forall_{i \in [n]} \xi_i$ is the $i^{\textit{th}}$ element of $\xi$.
Based on this, the duality conditions would change, resulting in different SVM solver which is usually weaker (higher error) however if the $\alpha_1, \alpha_2$ are actual upper bound to the optimal solution (as without these defined user constraints) then we would yield the same value as if we did not impose these constraints.
With that being said, is their a way to bound the approximation from above or at least approximate it w.r.t. the optimal value $\textit{OPT}$, i.e let $\phi \geq 1$, and let $\hat{w} \in \mathbb{R}^d, \hat{\beta} \in \mathbb{R}$ be the solutions of the defined optimization problem such that $$ \phi = \frac{f_{\textit{SVM}}(P,\hat{w},\hat{\beta})}{\textit{OPT}} $$
To clarify: we want to compare the optimal objective value of the traditional SVM problem with a variant that includes an additional constraint on the norm of $w$ and the value of bias $b$. We are interested in the optimality gap between the two problem variants. By inspection, it's clear that the SVM formulation with the additional constraints will be inferior to the one without the constraints, but we would like to quantify and bound the gap.
Please advise and thanks in advance.