A Measure-Theoretic Formulation of Monte Carlo Stochastic Optimization: Unifying Continuous and Discrete Domains

This study presents a unified measure-theoretic formulation of the Monte Carlo Stochastic Optimization Technique (MOST), establishing a rigorous framework that encompasses both continuous and discrete optimization. Unlike conventional optimization methods that operate on pointwise evaluations, MOST is based on regional evaluation through normalized integrals, enabling robust and global exploration of the search space. We first reformulate MOST within a finite measure space, where the evaluation of a region is defined as the measure-weighted average of the objective function. This formulation naturally connects regional optimization with expectation under an induced probability measure and provides a theoretical foundation for Monte Carlo approximation. Building upon this framework, we construct a discrete version of MOST by introducing the counting measure and extend it further using weighted measures to rigorously handle odd-cardinality partitions via midpoint sharing. A central contribution of this work is the demonstration that continuous and discrete MOST are structurally identical algorithms arising from a single measure-based principle, differing only in the choice of underlying measure. This result eliminates the traditional separation between continuous and discrete optimization within the MOST framework. Theoretical analysis reveals that MOST is particularly effective when near-optimal regions possess non-negligible measure, while its performance may degrade in the presence of isolated global minima. These properties are validated through numerical experiments using benchmark functions, including the Ackley and Sphere functions, under uniform discretization. The results confirm that discrete MOST achieves accurate approximations of global optima, with errors controlled by discretization resolution and strong robustness in multimodal landscapes. Overall, this work establishes MOST as a measure-based optimization paradigm, offering a unified, theoretically grounded, and practically robust approach to global optimization across continuous and discrete domains.