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Convergence of ADMM for multi-block nonconvex separable optimization models
Ke GUO, Deren HAN, David Z. W. WANG, Tingting WU
Front. Math. China. 2017, 12 (5): 1139-1162.
https://doi.org/10.1007/s11464-017-0631-6
For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers (ADMM) has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function, ADMM or its direct extended version may not converge. In this paper, we consider the multi-block separable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.
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Contact process on regular tree with random vertex weights
Yu PAN, Dayue CHEN, Xiaofeng XUE
Front. Math. China. 2017, 12 (5): 1163-1181.
https://doi.org/10.1007/s11464-017-0633-4
This paper is concerned with the contact process with random vertex weights on regular trees, and studies the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices xand yat rate λρ(x)ρ(y) for some λ>0,where {ρ(x), x∈Td} are independent and identically distributed (i.i.d.) vertex weights. We show that when dis large enough, there is a phase transition at λc(d) ∈ (0,∞) such that for λ<λc (d),the contact process dies out, and for λ>λc(d),the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at λe(d) such that for λ<λe(d),the contact process dies out at an exponential rate. Finally, we show that these two critical values have the same asymptotic behavior as dincreases.
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g-Good-neighbor conditional diagnosability of star graph networks under PMC model and MM* model
Shiying WANG, Zhenhua WANG, Mujiangshan WANG, Weiping HAN
Front. Math. China. 2017, 12 (5): 1221-1234.
https://doi.org/10.1007/s11464-017-0657-9
Diagnosability of a multiprocessor system is an important study topic. S. L. Peng, C. K. Lin, J. J. M. Tan, and L. H. Hsu [Appl. Math. Comput., 2012, 218(21): 10406–10412] proposed a new measure for fault diagnosis of the system, which is called the g-good-neighbor conditional diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a famous topological structure of interconnection networks, the n-dimensional star graph Sn has many good properties. In this paper, we establish the g-good-neighbor conditional diagnosability of Sn under the PMC model and MM∗ model.
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