The soft-margin SVM primal attains its minimum, including the unpenalized-bias boundary

The tuple $w=0$, $b=0$, and ${\xi }_i=1$ is feasible and has objective $Cn$. Every feasible objective is nonnegative, so the primal infimum lies in $[0,Cn]$.

Let $M=\mathrm{span}\{\varphi (x_i):1\le i\le n\}$. For any feasible tuple, project $w$ onto $M$ using [F1]. The residual is orthogonal to every $\varphi (x_i)$, so all sample scores and constraints are unchanged, while the norm does not increase. Thus minimizing over $M\times \mathbb{R}\times {\mathbb{R}}^n$ has the same infimum as minimizing over the full feature space.

If all labels are $+1$, then $w=0$, $b=1$, and $\xi =0$ is feasible with objective $0$, which is globally minimal by step 1.1. If all labels are $-1$, then $w=0$, $b=-1$, and $\xi =0$ gives the same conclusion.

Suppose both labels occur. Set $R={\max }_i\mathrm{\parallel }\varphi (x_i)\mathrm{\parallel }$ and let $S$ be the subset of $M\times \mathbb{R}\times {\mathbb{R}}^n$ consisting of feasible tuples with objective at most $Cn$. By step 1.1, $S$ is nonempty. It is closed because the constraints are continuous affine inequalities and the objective is continuous; continuity of each score follows from [F2]. Every tuple in $S$ satisfies $\mathrm{\parallel }w\mathrm{\parallel }\le \sqrt{2Cn}$ and $0\le {\xi }_i\le n$. Choose indices $i_{+}$ and $i_{-}$ with labels $+1$ and $-1$. Their constraints and [F2] give $b\ge 1-n-\sqrt{2Cn}R$ and $b\le n-1+\sqrt{2Cn}R$, so $b$ is bounded. Equip the product space with the inner product $⟨(w,b,\xi ),(w^{\mathrm{\prime }},b^{\mathrm{\prime }},{\xi }^{\mathrm{\prime }})⟩=⟨w,w^{\mathrm{\prime }}⟩+b{b}^{\mathrm{\prime }}+{\xi }^T{\xi }^{\mathrm{\prime }}$. It has finite dimension $d=\dim M+1+n\ge 1$, so [F3] gives an orthonormal basis identifying it with ${\mathbb{R}}^d$. Hence $S$ is a nonempty closed bounded subset of that Euclidean space. By [F4], the continuous objective attains a minimum on $S$. Any feasible tuple outside $S$ has objective greater than $Cn$, whereas $S$ contains a feasible tuple of objective $Cn$; therefore this minimum is also the global primal minimum. By step 1.2 it is a full-space minimizer with $w\in M$.