We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an $$\varepsilon $$ε-suboptimal global solution within finite run-time for any given termination tolerance $$\varepsilon > 0$$ε>0. Finally, we illustrate these results for a problem of calculus of variations.

中文翻译：

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希尔伯特空间中的全局优化

我们提出了一种完整搜索算法，用于解决一类非凸、可能是无限维的优化问题，以获得全局最优性。我们假设优化变量位于希尔伯特空间的有界子集中，并且我们在成本函数和约束集的某些规律性条件下确定算法的最坏情况运行时界限。因为这些运行时界限与优化变量的数量无关，特别是对于具有无限多个优化变量的优化问题有效，所以我们证明该算法收敛到 $$\varepsilon $$ε 次优全局解对于任何给定的终止容差 $$\varepsilon > 0$$ε>0，有限运行时间。最后，我们说明了变分法问题的这些结果。