Accelerating Level-Value Adjustment for the Polyak Stepsize

The Polyak stepsize formula has been widely used for convex optimization. However, stepsize computations require the generally unknown optimal value. Dynamic estimations of the optimal value are thus usually needed. In this paper, guided by a decision-based procedure through a novel easy-to-solve ``Polyak Stepsize Violation Detector'' (PSVD) linear constraint satisfaction problem, a series of level values is constructed to successively estimate the optimal value to guarantee convergence for subgradient as well as for approximate subgradient methods. Through a series of empirical tests of convex optimization problems with diverse characteristics, we illustrate the practical advantages of our approach as compared to existing methods.