This article introduces an innovative mathematical framework designed to tackle non-linear convex variational problems in reflexive Banach spaces. Our approach employs a versatile technique that can handle a broad range of variational problems, including standard ones. To carry out the process effectively, we utilize specialized sets known as radial dictionaries, where these dictionaries encompass diverse data types, such as tensors in Tucker format with bounded rank and Neural Networks with fixed architecture and bounded parameters. The core of our method lies in employing a greedy algorithm through dictionary optimization defined by a multivalued map. Significantly, our analysis shows that the convergence rate achieved by our approach is comparable to the Method of Steepest Descend implemented in a reflexive Banach space, where the convergence rate follows the order of $O(m^{-1})$.

翻译：本文提出了一种创新的数学框架，用于解决自反巴拿赫空间中的非线性凸变分问题。我们的方法采用了一种通用技术，能够处理包括标准变分问题在内的广泛变分问题。为有效实施该过程，我们利用了称为径向字典的特殊集合，这些字典涵盖多种数据类型，例如具有有界秩的塔克格式张量以及具有固定架构和有界参数的神经网络。我们的方法核心在于通过由多值映射定义的字典优化来采用贪婪算法。重要的是，我们的分析表明，该方法实现的收敛速率与在自反巴拿赫空间中实现的梯度下降法相当，其收敛速率遵循 $O(m^{-1})$ 阶。