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Recently, there were introduced important classes of relatively smooth,relatively continuous, and relatively strongly convex optimization problems.These concepts have significantly expanded the class of problems for whichoptimal complexity estimates of gradient-type methods in high-dimensionalspaces take place. Basing on some recent works devoted to online optimization(regret minimization) problems with both relatively Lipschitz continuous andrelatively strongly convex objective function, we introduce algorithms forsolving the strongly convex optimization problem with inequality constraints inthe online setting. We propose a scheme with switching between productive andnonproductive steps for such types of problems and prove its convergence ratefor the class of relatively Lipschitz and strongly convex minimizationproblems. We also provide an experimental comparison between the proposedmethod and AdaMirr, recently proposed for relatively Lipschitz convex problems.

Online Optimization Problems with Functional Constraints under Relative Lipschitz Continuity and Relative Strong Convexity Conditions