In this paper, we present primal-dual algorithms for online problems with non-convex objectives. Problems with convex objectives have been extensively studied in recent years where the analyses rely crucially on the convexity and the Fenchel duality. However, problems with non-convex objectives resist against current approaches and non-convexity represents a strong barrier in optimization in general and in the design of online algorithms in particular. In our approach, we consider configuration linear programs with the multilinear extension of the objectives. We follow the multiplicative weight update framework in which a novel point is that the primal update is defined based on the gradient of the multilinear extension. We introduce new notions, namely (local) smoothness, in order to characterize the competitive ratios of our algorithms. The approach leads to competitive algorithms for several problems with convex/non-convex objectives.
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