A tight degree 4 sum-of-squares lower bound for the Sherrington–Kirkpatrick Hamiltonian
Mathematical Programming(2020)
摘要
We show that, if W is an N × N matrix drawn from the gaussian orthogonal ensemble, then with high probability the degree 4 sum-of-squares relaxation cannot certify an upper bound on the objective N^-1·x^⊤Wx under the constraints x_i^2 - 1 = 0 (i.e. x∈{± 1 }^N ) that is asymptotically smaller than λ _max(W) ≈ 2 . We also conjecture a proof technique for lower bounds against sum-of-squares relaxations of any degree held constant as N →∞ , by proposing an approximate pseudomoment construction.
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关键词
Sum-of-squares, Convex optimization, Average-case computational complexity, Semidefinite programming, Spin glass, 60C05, 68Q17, 68Q87, 82D30, 90C22
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