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arXiv 2110.03079math.OC
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积分变分问题的凸松弛：点态对偶松弛与平方和优化

Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization

Alexander Chernyavsky, Jason J. Bramburger, Giovanni Fantuzzi, David Goluskin

发表信息：
SIAM J. Optim. 33, 481-512 (2023)
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AI总结：

本文提出点态对偶松弛（PDR）方法，将非凸积分变分问题转化为有限维凸最大化问题以获取全局下界，并结合平方和优化实现可计算松弛，证明其在特征值与最优常数等问题上的精确性及收敛性。

摘要：

我们提出了一种求积分变分问题全局下确界下界的方法，其中在满足给定等式或不等式约束的函数 $u\colonΩ\subset\mathbb{R}^n\to\mathbb{R}^m$ 上最小化 $\int_Ωf(x,u(x),\nabla u(x)){\rm d}x$。每个约束可以施加在 $Ω$ 或其边界上，既可以逐点施加，也可以以积分意义施加。这些全局最小化问题通常是非凸的且难以求解。我们构造了一个特定的凸最大化问题，这里称为点态对偶松弛（PDR），其上确界是原问题下确界的一个下界。PDR 可以通过对原问题进行对偶化和松弛来导出；其约束是有限维集合上的逐点等式或不等式，而不是无限维函数空间上的约束。当原最小化问题可以由 $(x,u,\nabla u)$ 的多项式函数指定时，PDR 可以通过将逐点不等式替换为多项式平方和（SOS）条件来进一步松弛。当维数 $m,n$ 和约束数量不太大时，所得的 SOS 规划在计算上是可处理的。本文提出的框架推广了 Valmorbida、Ahmadi 和 Papachristodoulou（IEEE Trans. Automat. Contr., 61:1649--1654, 2016）的方法。我们证明了对于若干类问题，PDR 给出的最优下界是精确的，其特例包括 Sturm-Liouville 问题的主特征值和 Poincaré 不等式的最优常数。对于这些相同的问题类，我们证明了 PDR 的 SOS 松弛随着多项式次数的增加收敛到精确下界。通过若干例子说明了 SOS 计算在实际中的收敛性。

英文摘要：

We present a method for finding lower bounds on the global infima of integral variational problems, wherein $\int_Ωf(x,u(x),\nabla u(x)){\rm d}x$ is minimized over functions $u\colonΩ\subset\mathbb{R}^n\to\mathbb{R}^m$ satisfying given equality or inequality constraints. Each constraint may be imposed over $Ω$ or its boundary, either pointwise or in an integral sense. These global minimizations are generally non-convex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the original problem. The PDR can be derived by dualizing and relaxing the original problem; its constraints are pointwise equalities or inequalities over finite-dimensional sets, rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of $(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions $m,n$ and number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat. Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincaré inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several examples.

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