In this paper, we study a class of online continuous optimization problems. At each round, the utility function is the sum of a weakly diminishing-returns (DR) submodular function and a concave function, certain cost associated with the action will occur, and the problem has total limited budget. Combining the two methods, the penalty function and Frank-Wolfe strategies, we present an online method to solve the considered problem. Choosing appropriate stepsize and penalty parameters, the performance of the online algorithm is guaranteed, that is, it achieves sub-linear regret bound and certain mild constraint violation bound in expectation.

In this paper, we mainly investigate the optimization model that minimizes the cost function such that the cover function exceeds a required threshold in the set cover problem, where the cost function is additive linear, and the cover function is non-monotone approximately submodular. We study the problem under streaming model and propose three bicriteria approximation algorithms. Firstly, we provide an intuitive streaming algorithm under the assumption of known optimal objective value. The intuitive streaming algorithm returns a solution such that its cover function value is no less than $\alpha \mathsf{}\mathrm{(}\mathrm{1}\mathrm{-}ϵ\mathrm{)}$ times threshold, and the cost function is no more than ${\mathrm{(}\mathrm{2}\mathrm{+}ϵ\mathrm{)}}^{\mathrm{2}}\mathrm{/}\mathrm{(}{ϵ}^{\mathrm{2}}\mathsf{}{\omega }^{\mathrm{2}}\mathrm{)}\mathrm{\cdot }\kappa$, where $\kappa$ is a value that we suppose for the optimal solution and $\alpha$ is the approximation ratio of an algorithm for unconstrained maximization problem that we can call directly. Next we present a bicriteria streaming algorithm scanning the ground set multi-pass to weak the assumption that we guess the optimal objective value in advance, and maintain the same bicriteria approximation ratio. Finally we modify the multi-pass streaming algorithm to a single-pass one without compromising the performance ratio. Additionally, we also propose some numerical experiments to test our algorithm’s performance comparing with some existing methods.