非线性谱对偶
Nonlinear Spectral Duality
摘要：
齐次凸函数对的非线性本征值问题是一种特殊的非线性约束最优化问题，它出现在各种环境中，包括图挖掘、机器学习和网络科学。通过考虑不同的概念，从经典和最近的凸几何理论的对偶变换，在这项工作中，我们表明，一个可以从原始的对偶非线性本征值制定保持频谱，变分频谱以及相应的多重性不变。这些非线性谱对偶性质可用于将原始的最优化问题转化为各种可供选择的和可能更可处理的对偶问题。我们说明了使用非线性谱对偶在各种例子设置涉及图，非线性拉普拉斯算子，和凸体之间的距离的最优化问题。
英文摘要：
Nonlinear eigenvalue problems for pairs of homogeneous convex functions are particular nonlinear constrained optimization problems that arise in a variety of settings, including graph mining, machine learning, and network science. By considering different notions of duality transforms from both classical and recent convex geometry theory, in this work we show that one can move from the primal to the dual nonlinear eigenvalue formulation maintaining the spectrum, the variational spectrum as well as the corresponding multiplicities unchanged. These nonlinear spectral duality properties can be used to transform the original optimization problem into various alternative and possibly more treatable dual problems. We illustrate the use of nonlinear spectral duality in a variety of example settings involving optimization problems on graphs, nonlinear Laplacians, and distances between convex bodies.
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