We propose a numerical algorithm for computing approximately optimal solutions of the matching for teams problem. Our algorithm is efficient for problems involving a large number of agent categories and allows for the measures describing the agent types to be non-discrete. Specifically, we parametrize the so-called transfer functions and develop a parametric version of the dual formulation. Our algorithm tackles this parametric formulation and produces feasible and approximately optimal solutions for the primal and dual formulations of the matching for teams problem. These solutions also yield upper and lower bounds for the optimal value, and the difference between the upper and lower bounds provides a direct sub-optimality estimate of the computed solutions. Moreover, we are able to control a theoretical upper bound on the sub-optimality to be arbitrarily close to 0 under mild conditions. We subsequently prove that the approximate primal and dual solutions converge when the sub-optimality goes to 0 and their limits constitute a true matching equilibrium. Thus, the outputs of our algorithm are regarded as an approximate matching equilibrium. We also analyze the theoretical computational complexity of our parametric formulation as well as the sparsity of the resulting approximate matching equilibrium. Through numerical experiments, we showcase that the proposed algorithm can produce high-quality approximate matching equilibria and is applicable to versatile settings, including a high-dimensional setting involving 100 agent categories.

翻译：我们提出一种数值算法，用于计算团队匹配问题的近似最优解。该算法在处理包含大量智能体类别的场景时具有高效性，且允许描述智能体类型的度量指标为非离散形式。具体而言，我们将所谓的传递函数进行参数化，并发展出对偶公式的参数化版本。该算法针对参数化公式进行求解，为团队匹配问题的原始-对偶公式生成可行且近似最优的解。这些解同时给出了最优值的上界与下界，上下界之差可直接评估所求解的次优性程度。进一步，在温和条件下，我们能够将次优性的理论上限控制到任意接近0。随后证明当次优性趋近于0时，近似原始解与对偶解将收敛，其极限构成真实的匹配均衡。因此，算法输出可视为近似匹配均衡。我们还分析了参数化公式的理论计算复杂度以及所得近似匹配均衡的稀疏性。通过数值实验证明，该算法能够生成高质量的近似匹配均衡，适用于多种场景设置，包括包含100个智能体类别的高维场景。