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We consider the problem of minimization of a convex function on a simple setwith convex non-smooth inequality constraint and describe first-order methodsto solve such problems in different situations: smooth or non-smooth objectivefunction; convex or strongly convex objective and constraint; deterministic orrandomized information about the objective and constraint. We hope that it isconvenient for a reader to have all the methods for different settings in oneplace. Described methods are based on Mirror Descent algorithm and switchingsubgradient scheme. One of our focus is to propose, for the listed differentsettings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule.This means that neither stepsize nor stopping rule require to know theLipschitz constant of the objective or constraint. We also construct MirrorDescent for problems with objective function, which is not Lipschitzcontinuous, e.g. is a quadratic function. Besides that, we address the problemof recovering the solution of the dual problem.

Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints