We study hyperparameter optimization (HPO) from a numerical-optimization perspective and propose a multi-objective, damped Gauss--Newton search method. Rather than treating model evaluations as independent trials, the method estimates a finite-difference Jacobian that captures the local sensitivity of multiple validation metrics to hyperparameter perturbations. A Tikhonov-regularized Gauss--Newton system then produces a directed joint update, addressing the underdetermined setting in which the number of hyperparameters exceeds the number of performance objectives. We evaluate the method on three public classification datasets by tuning four XGBoost hyperparameters and compare it with exhaustive grid search, random search, and tree-structured Parzen estimator (TPE) optimization. On a controlled Breast Cancer split, the proposed method matches the best validation accuracy of a 320-configuration grid search while obtaining slightly better log loss and ROC--AUC. Across three datasets and three seeds, performance is competitive but does not consistently exceed random search or TPE. Ablations further show sensitivity to damping and initialization. These results position damped Gauss--Newton search as a directed local alternative to exhaustive HPO, while also clarifying the additional model-evaluation cost required for finite-difference Jacobian estimation.

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