We present the Quadratic-Quasi-Newton (QQN) algorithm, a novel optimization method that combines gradient descent and quasi-Newton directions through quadratic interpolation. QQN constructs a parametric path d(t) = t(1 − t)(−∇f) + t 2 d L-BFGS and performs univariate optimization along this path, creating an adaptive interpolation that requires no additional hyperparameters beyond those of its constituent methods. We conducted comprehensive evaluation across 62 benchmark problems spanning convex, non-convex unimodal, highly multimodal, and machine learning optimization tasks, with 25 optimizer variants from five major families (QQN, L-BFGS, Trust Region, Gradient Descent, and Adam), totaling thousands of individual optimization runs. Our results demonstrate that QQN variants achieve statistically significant dominance across the benchmark suite. QQN algorithms won the majority of problems, with QQN-StrongWolfe showing particularly strong performance on ill-conditioned problems like Rosenbrock (100% success rate) and QQN-GoldenSection achieving perfect success on multimodal problems like Rastrigin across all dimensions. Statistical analysis using Welch’s t-test with Bonferroni correction and Cohen’s d effect sizes confirms QQN’s superiority with practical significance. While L-BFGS variants showed efficiency on well-conditioned convex problems and Adam-WeightDecay excelled on neural network tasks, QQN’s consistent performance across problem types establishes its practical utility as a robust general-purpose optimizer. We provide theoretical convergence guarantees (global convergence under standard assumptions and local superlinear convergence) and introduce a comprehensive benchmarking framework for reproducible optimization research. Code available at https://github.com/SimiaCryptus/qqn-optimizer/.