This paper studies a class of nonconvex composite optimization, whose objective is a summation of an average of nonconvex(weakly) smooth functions and a convex nonsmooth function, where the gradient of the former func...This paper studies a class of nonconvex composite optimization, whose objective is a summation of an average of nonconvex(weakly) smooth functions and a convex nonsmooth function, where the gradient of the former function has the H o¨lder continuity. By exploring the structure of such kind of problems, we first propose a proximal(quasi-)Newton algorithm wPQN(Proximal quasi-Newton algorithm for weakly smooth optimization) and investigate its theoretical complexities to find an approximate solution. Then we propose a stochastic variant algorithm wPSQN(Proximal stochastic quasi-Newton algorithm for weakly smooth optimization), which allows a random subset of component functions to be used at each iteration. Moreover, motivated by recent success of variance reduction techniques, we propose two variance reduced algorithms,wPSQN-SVRG and wPSQN-SARAH, and investigate their computational complexity separately.展开更多
By introducing some parameters and estimating the weight function,we obtain an extension of reverse Hilbert-type inequality with the best constant factor.As applications,we build its equivalent forms and some particul...By introducing some parameters and estimating the weight function,we obtain an extension of reverse Hilbert-type inequality with the best constant factor.As applications,we build its equivalent forms and some particular results.展开更多
In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the lit...In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11871453)The Major Key Project of PCL(Grant No.PCL2022A05).
文摘This paper studies a class of nonconvex composite optimization, whose objective is a summation of an average of nonconvex(weakly) smooth functions and a convex nonsmooth function, where the gradient of the former function has the H o¨lder continuity. By exploring the structure of such kind of problems, we first propose a proximal(quasi-)Newton algorithm wPQN(Proximal quasi-Newton algorithm for weakly smooth optimization) and investigate its theoretical complexities to find an approximate solution. Then we propose a stochastic variant algorithm wPSQN(Proximal stochastic quasi-Newton algorithm for weakly smooth optimization), which allows a random subset of component functions to be used at each iteration. Moreover, motivated by recent success of variance reduction techniques, we propose two variance reduced algorithms,wPSQN-SVRG and wPSQN-SARAH, and investigate their computational complexity separately.
文摘By introducing some parameters and estimating the weight function,we obtain an extension of reverse Hilbert-type inequality with the best constant factor.As applications,we build its equivalent forms and some particular results.
基金supported by NNSF of China(11571090)GCCHB(GCC2014052)
文摘In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.
