Symmetry Reduction in AM/GM-Based Optimization
研究了在有限群线性对称作用下，如何利用AM/GM不等式约简优化问题的规模，特别分析了对称群情形下的复杂度增益和数值加速效果。
The arithmetic mean/geometric mean inequality (AM/GM inequality) facilitates classes of nonnegativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and develop techniques to reduce the size of the resulting relative entropy programs. We study in more detail the complexity gain in the case of the symmetric group. In this setup, we can show in particular certain stabilization results. We exhibit several sequences of examples in growing dimensions where the size of the reduced problem stabilizes. Finally, we provide some numerical results, emphasizing the computational speedup.