Introduction. The significance of methods and algorithms for solving discrete optimization problems in mathematical supporting computer technologies of diverse levels and objectives is increasing. Consequently, the efficacy of discrete optimization methods deserves particular attention, as it drives the advancement of techniques capable of solving complex real-world problems. This paper introduces the Global Equilibrium Search (GES) method as a highly effective approach for solving Boolean programming problems, thus contributing to the field's progress and applicability. Purpose. We describe the successful application of the approximate probabilistic GES method for effectively solving various Boolean programming problems. Results. This paper explores the application of sequential GES algorithms for solving Boolean linear, Boolean quadratic programming, and other related problems with their specific characteristics. In our study, we conducted a comparative analysis to assess the effectiveness of GES algorithms by evaluating them against state-of-the-art approaches. Additionally, to parallelize the optimization process for discrete programming problems, we introduced algorithm unions, specifically portfolios, and teams. The efficiency of GES algorithm portfolios and teams is investigated by solving the maximum weighted graph cut problem, with subsequent comparisons to identify distinctions between them. Conclusions. Based on the accumulated experience of applying GES algorithms and their modifications to solve discrete optimization problems, this study establishes the GES method as the leading approximate approach for Boolean programming. The results demonstrate the GES algorithm unions experience a significant boost in the optimization process speed, whereas algorithm teams demonstrate higher efficiency. Keywords: global equilibrium search method, Boolean programming problems, experimental studies, algorithm efficiency, algorithm unions (portfolios and teams).