Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

中文翻译：

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表征极小化子的存在性和非凸二次积分的最优性

二次函数在应用数学中扮演着重要的角色。在本文中，我们考虑在有限维空间上定义的非负可积函数的有界子集中最小化（不一定是凸的）二次函数的积分的问题，该空间对于任何（局部凸的）拓扑都不紧凑可积函数空间。我们根据被积函数所涉及的矩阵的（严格）共正性建立了关于解存在或不存在的完整描述。此外，我们通过哈密顿函数来表征最优性。