A new analytical envelope for multivariate global optimization

We propose a new global optimization method that combines an α–dense univariate reduction with explicitly constructed analytical envelopes: a piecewise concave underestimator (PCU) and a piecewise convex overestimator (PCO). By leveraging interval-based curvature bounds, the method provides rigorous global optimality certificates. An adaptive branch-and-bound strategy ensures rapid convergence by refining intervals based on theoretical envelope widths. Numerical experiments on challenging nonconvex and multimodal benchmarks demonstrate strong performance and efficiency.