In this paper, we study approximation algorithms for several classes of DR-submodular optimization problems, where DR is short for diminishing return. Following a newly introduced algorithm framework for zeroth-order stochastic approximation methods, we first propose algorithms {\bf CG-ZOSA} and {\bf RG-ZOSA} for smooth DR-submodular optimization based on the coordinate-wise gradient estimator and the randomized gradient estimator, respectively. Our theoretical analysis proves that \rm{\bf{CG-ZOSA}} can reach a solution whose expected objective value exceeds $(1-e^{-1}-\epsilon^{2})$OPT$-\epsilon$ after $\mathcal{O}(\epsilon^{-2})$ iterations and $\mathcal{O}(N^{2/3}d\epsilon^{-2})$ oracle calls, where $d$ represents the problem dimension. On the other hand, \rm{\bf{RG-ZOSA}} improves the approximation ratio to $(1-e^{-1}-\epsilon^{2}/d)$ while maintaining the same overall oracle complexity. For non-smooth up-concave maximization problems, we propose a novel auxiliary function based on a smoothed objective function and introduce the \rm{\bf{NZOSA}} algorithm. This algorithm achieves an approximation ratio of $(1-e^{-1}-\epsilon \ln \epsilon^{-1}- \epsilon^{2}\ln \epsilon^{-1})$ with $\mathcal{O}(d\epsilon^{-2})$ iterations and $\mathcal{O}(N^{2/3}d^{3/2} \epsilon^{-3})$ oracle calls. We also extend \rm{\bf{NZOSA}} to handle a class of robust DR-submodular maximization problems. To validate the effectiveness of our proposed algorithms, we conduct experiments on both synthetic and real-world problems. The results demonstrate the superior performance and efficiency of our methods in solving DR-submodular optimization problems.

Euler's elastica model has been widely used in image processing. Since it is a challenging nonconvex and nonsmooth optimization model, most existing algorithms do not have convergence theory for it. In this paper, we propose a penalty relaxation algorithm with mathematical guarantee to find a stationary point of Euler's elastica model. To deal with the nonsmoothness of Euler's elastica model, we first introduce a smoothing relaxation problem, and then propose an exact penalty method to solve it. We establish the relationships between Euler's elastica model, the smoothing relaxation problem, and the penalty problem in theory regarding optimal solutions and stationary points. Moreover, we propose an efficient block coordinate descent algorithm to solve the penalty problem by taking advantage of convexity of its subproblems. We prove global convergence of the algorithm to a stationary point of the penalty problem. Finally we apply the proposed algorithm to denoise the optical coherence tomography images with real data from an optometry clinic and show the efficiency of the method for image processing using Euler's elastica model.