带有耦合约束的概率测度上的二次优化
Quadratic Optimization over Probability Measures with Coupling Constraints
AI总结：
针对概率测度上二次优化问题，提出适配概率测度的矩平方和凸松弛层次，证明其收敛性并通过数值实验验证，为测度相关非线性问题提供方法路线。
AI中文摘要：
我们考虑求解一类优化问题，其目标为二次型，决策变量为概率测度。这类问题的应用背景来自最优传输（其中Gromov-Wasserstein问题是典型例子）以及能量景观最小化。由于目标对决策变量呈二次依赖，这类问题不属于广义矩问题的标准建模框架（该框架要求目标为线性）。为此，我们提出基于在基空间乘积上的概率测度中搜索的凸松弛层次，这类松弛可通过矩平方和（moment Sum-of-squares）层次得到自然解释——该框架是求解多项式优化问题的重要工具，我们对其进行调整以适配概率测度。一项关键的概念性贡献是引入了半正定性的概念，将矩阵上的常规概念进行扩展。在决策变量满足特定边际约束（如最优传输问题的Kantorovich形式中的约束）的假设下，我们证明该层次收敛到全局最优解。在目标为多项式的额外假设下，我们提出了一种矩-SOS类型的有限维半定规划层次，其最优解收敛到原测度上二次优化问题的解。我们通过数值实验验证了该框架。更广泛地说，目标和/或约束以多项式方式依赖决策的测度优化是一个基础问题，希望我们的工作能为平方和（SOS）的思想（通常针对多项式优化发展）如何应用到更广泛的涉及测度的非线性问题提供路线图。
英文摘要：
We consider solving an optimization instance in which the objective is quadratic and where the decision variable is a probability measure. Our class of problems are motivated by applications arising from optimal transport (with the Gromov-Wasserstein problem being a prominent example) as well as energy landscape minimization. Because the objective depends quadratically on the decision variable, our class of problems fall outside the standard modeling framework of the Generalized Moment Problems (which requires the objective to be linear). To this end, we propose a hierarchy of convex relaxations based on searching over probability measures over products of the base space. These have a natural interpretation with the moment Sum-of-squares hierarchy-a prominent framework for solving polynomial optimization instances, which we adapt to accommodate probability measures. A key conceptual contribution is to introduce a notion of positive-semidefiniteness that extends the usual notion over matrices. Under the assumption that the decision variables satisfy certain marginal constraints (as in the Kantorovich formulation of the optimal transport problem), we establish convergence of our hierarchy towards the globally optimal solution. Under the additional assumption that the objective is a polynomial, we propose a moment-SOS type hierarchy of finite dimensional semidefinite programs whose optimal solution converges to that of the original quadratic optimization over measures. We demonstrate our framework with numerical experiments. More generally, optimization over measures where the objective and/or constraint depends on the decision in a polynomial way is a fundamental problem. It is hoped that our work provides a road-map as to how the ideas of the SOS-ordinarily developed for polynomial optimization-may be applied to a broader class of non-linear problems involving measures.
