论文标题
变分问题与其职业度量放松之间的差距
The gap between a variational problem and its occupation measure relaxation
论文作者
论文摘要
最近的工作提出了基于职业量度形式主义的非线性PDE约束的变异优化问题的线性编程松弛。这些方法的主要吸引力是它们依赖于凸优化（通常是半决赛编程）。在这项工作中，我们结束了与这种方法有关的一个公开问题。我们证明，当未知函数的代码剂的维度等于一个，既是变化的计算和最佳控制问题，因此在域的维度等于一个等于一个时，因此，经典和放松的最小值同时重合一个。为此，我们证明了适用于我们环境中的正常电流的硬质壁板分解的概括。我们还通过反例显示，如果域和代码域的尺寸大于一个，则可能存在一个正差距。我们构建的示例表明后者也表明，有时放松的职业措施可能代表着比其经典对应物更令人满意的“解决方案”，因此，即使它们可能并不等效，算法在较大的放松占用度量的较大空间中可访问的算法仍然非常有价值。最后，我们表明，在存在整体约束的情况下，在域和代码域的任何维度上都可能发生正差距。
Recent works have proposed linear programming relaxations of variational optimization problems subject to nonlinear PDE constraints based on the occupation measure formalism. The main appeal of these methods is the fact that they rely on convex optimization, typically semidefinite programming. In this work we close an open question related to this approach. We prove that the classical and relaxed minima coincide when the dimension of the codomain of the unknown function equals one, both for calculus of variations and for optimal control problems, thereby complementing analogous results that existed for the case when the dimension of the domain equals one. In order to do so, we prove a generalization of the Hardt-Pitts decomposition of normal currents applicable in our setting. We also show by means of a counterexample that, if both the dimensions of the domain and of the codomain are greater than one, there may be a positive gap. The example we construct to show the latter serves also to show that sometimes relaxed occupation measures may represent a more conceptually-satisfactory "solution" than their classical counterparts, so that -- even though they may not be equivalent -- algorithms rendering accessible the minimum in the larger space of relaxed occupation measures remain extremely valuable. Finally, we show that in the presence of integral constraints, a positive gap may occur at any dimension of the domain and of the codomain.
