Given a binary-labeled linearly separable dataset, and the objective is to compute the maximum-margin separating hyperplane, also known as the hard-margin Support Vector Machine (SVM) classifier. This paper investigates whether, if given an initial separating hyperplane, can it be exploited to reach this unique optimum more efficiently. We present a geometric approach that gradually improves the alignment of the hyperplane, starting from an initial separating hyperplane, while preserving separation and continuously increasing its margin until convergence to the global optimum. At each iteration, the method considers only local information, namely the current active set, and aims to re-align the hyperplane according to the optimal separating hyperplane of this reduced subset. Consequently, the original convex quadratic optimization problem is addressed through a sequence of smaller subproblems. The paper presents the algorithm in detail, together with a preliminary experimental evaluation and several theoretical findings. The results suggest that, when an initial separating hyperplane is available, the proposed method can be competitive on larger datasets and, in some cases, can outperform state-of-the-art approaches that solve the optimization problem directly.

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