arXiv subjects
Publications and source records attributed to Junpeng Zhou.
Very recently, Alon and Frankl, and Gerbner studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. We consider the analogous Turán problems on hypergraphs with bounded matching number, and we obtain some exact results.
An $r$-uniform hypergraph ($r$-graph) is linear if any two edges intersect at most one vertex. For a graph $F$, a hypergraph $H$ is Berge-$F$ if there is a bijection $ϕ:E(F)\rightarrow E(H)$ such that $e\subseteq ϕ(e)$ for all $e$ in $E(F)$. In this paper, a kind of stability result for Berge-$K_{3,t}$ linear $r$-graphs is established. Based on this stability result, an upper bound for the linear Turán number of Berge-$K_{3,t}$ is determined. For an $r$-graph $H$, let $\mathcal{A}(H)$ be the adjacency tensor of $H$. The spectral radius of $H$ is the spectral radius of the tensor $\mathcal{A}(H)$. Some bounds for the maximum spectral radius of connected Berge-$K_{3,t}$-free linear $r$-graphs are obtained.
In this paper, we consider a class of structured fractional programs, where the numerator part is the sum of a block-separable (possibly nonsmooth nonconvex) function and a locally Lipschitz differentiable (possibly nonconvex) function, while the denominator is a convex (possibly nonsmooth) function. We first present a novel reformulation for the original problem and show the relationship between optimal solutions, critical points and KL exponents of these two problems. Inspired by the reformulation, we propose a flexible framework of multi-proximity gradient algorithms (MPGA), which computes the proximity operator with respect to the Fenchel conjugate associated with the convex denominator of the original problem rather than evaluating its subgradient as in the existing methods. Also, MPGA employs a nonmonotone linear-search scheme in its gradient descent step, since the smooth part in the numerator of the original problem is not globally Lipschitz differentiable. Based on the framework of MPGA, we develop two specific algorithms, namely, cyclic MPGA and randomized MPGA, and establish their subsequential convergence under mild conditions. Moreover, the sequential convergence of cyclic MPGA with the monotone line-search (CMPGA_ML) is guaranteed if the extended objective associated with the reformulated problem satisfies the Kurdyka-Łojasiewicz (KL) property and some other mild assumptions. In particular, we prove that the corresponding KL exponents are 1/2 for several special cases of the fractional programs, and so, CMPGA_ML exhibits a linear convergence rate. Finally, some preliminary numerical experiments are performed to demonstrate the efficiency of our proposed algorithms.