We propose a numerical algorithm for computing approximately optimal solutions of the matching for teams problem. Our algorithm is efficient for problems involving large number of agent categories and allows for non-discrete agent type measures. Specifically, we parametrize the so-called transfer functions and develop a parametric version of the dual formulation, which we tackle to produce feasible and approximately optimal solutions for the primal and dual formulations. These solutions yield upper and lower bounds for the optimal value, and the difference between these bounds provides a direct sub-optimality estimate of the computed solutions. Moreover, we are able to control the sub-optimality to be arbitrarily close to 0. 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. In the numerical experiments, we study three matching for teams problems: a problem of business location distribution, the well-known 2-Wasserstein barycenter problem, and a high-dimensional problem involving 100 agent categories. Through the numerical results, we showcase that the proposed algorithm can produce high-quality approximate matching equilibria in these settings, provide quantitative insights about the optimal city structure in the business location distribution problem, and that the sub-optimality estimates computed by our algorithm are much less conservative than theoretical estimates.

翻译：我们提出了一种数值算法，用于计算团队匹配问题的近似最优解。该算法适用于涉及大量智能体类别的问题，并允许非离散的智能体类型测度。具体而言，我们对所谓的传递函数进行参数化，并构建了对偶公式的参数化版本，进而求解得到原问题和对偶问题的可行近似最优解。这些解为最优值提供了上界和下界，而上下界之差则直接给出了所求解的次优性估计。此外，我们能够将次优性控制到任意接近0的程度。随后，我们证明了当次优性趋于0时，近似原问题解和对偶问题解收敛，且它们的极限构成真实的匹配均衡。因此，我们算法的输出被视为近似匹配均衡。我们还分析了参数化公式的理论计算复杂度以及由此产生的近似匹配均衡的稀疏性。在数值实验中，我们研究了三个团队匹配问题：商业区位分布问题、著名的2-Wasserstein重心问题以及一个涉及100个智能体类别的高维问题。数值结果表明，所提出的算法能够在这些场景下生成高质量的近似匹配均衡，为商业区位分布问题中的最优城市结构提供了定量见解，并且我们算法计算的次优性估计远不如理论估计保守。