In this paper,we mainly study the existence of solutions to sparsity constrained optimization(SCO).Based on the expressions of tangent cone and normal cone of sparsity constraint,we present and characterize two first-...In this paper,we mainly study the existence of solutions to sparsity constrained optimization(SCO).Based on the expressions of tangent cone and normal cone of sparsity constraint,we present and characterize two first-order necessary optimality conditions for SCO:N-stationarity and T-stationarity.Then we give the second-order necessary and sufficient optimality conditions for SCO.At last,we extend these results to SCO with nonnegative constraint.展开更多
This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices.We first benefit from a convex optimization which develops l1-norm penalty to encourage ...This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices.We first benefit from a convex optimization which develops l1-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property.For the proposed estimator,we then prove that with high probability,the Frobenius norm of the estimation rate can be of order O(√((slgg p)/n))under a mild case,where s and p denote the number of nonzero entries and the dimension of the population covariance,respectively and n notes the sample capacity.Finally,an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem,and merits of the approach are also illustrated by practicing numerical simulations.展开更多
基金supported in part by the National Natural Science Foundation of China(Nos.11431002,71271021).
文摘In this paper,we mainly study the existence of solutions to sparsity constrained optimization(SCO).Based on the expressions of tangent cone and normal cone of sparsity constraint,we present and characterize two first-order necessary optimality conditions for SCO:N-stationarity and T-stationarity.Then we give the second-order necessary and sufficient optimality conditions for SCO.At last,we extend these results to SCO with nonnegative constraint.
基金The work was supported in part by the National Natural Science Foundation of China(Nos.11431002,11171018,71271021,11301022).
文摘This paper aims at achieving a simultaneously sparse and low-rank estimator from the semidefinite population covariance matrices.We first benefit from a convex optimization which develops l1-norm penalty to encourage the sparsity and nuclear norm to favor the low-rank property.For the proposed estimator,we then prove that with high probability,the Frobenius norm of the estimation rate can be of order O(√((slgg p)/n))under a mild case,where s and p denote the number of nonzero entries and the dimension of the population covariance,respectively and n notes the sample capacity.Finally,an efficient alternating direction method of multipliers with global convergence is proposed to tackle this problem,and merits of the approach are also illustrated by practicing numerical simulations.
