Optimization of the SherringtonKirkpatrick Hamiltonian
Abstract
Let ${\boldsymbol A}\in{\mathbb R}^{n\times n}$ be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing $\langle{\boldsymbol \sigma},{\boldsymbol A}{\boldsymbol \sigma}\rangle$ over binary vectors ${\boldsymbol \sigma}\in\{+1,1\}^n$. In the language of statistical physics, this amounts to finding the ground state of the SherringtonKirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any $\varepsilon>0$, outputs ${\boldsymbol \sigma}_*\in\{1,+1\}^n$ such that $\langle{\boldsymbol \sigma}_*,{\boldsymbol A}{\boldsymbol \sigma}_*\rangle$ is at least $(1\varepsilon)$ of the optimum value, with probability converging to one as $n\to\infty$. The algorithm's time complexity is $C(\varepsilon)\, n^2$. It is a messagepassing algorithm, but the specific structure of its update rules is new. As a side result, we prove that, at (low) nonzero temperature, the algorithm constructs approximate solutions of the ThoulessAndersonPalmer equations.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.10897
 Bibcode:
 2018arXiv181210897M
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Optimization and Control
 EPrint:
 27 pages