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The objective of this paper is to improve the subgradient optimization method which is used to solve non-differentiable optimization problems in the Lagrangian dual problem. One of the main drawbacks of the subgradient method is the tuning process to determine the sequence of step-lengths to update successive iterates. In this paper, we propose a modified subgradient optimization method with various step size rules to compute a tuning-free subgradient step-length that is geometrically motivateddoi:10.3329/dujs.v61i2.17059 fatcat:vdkfh3o45nfjvmbgkhbrp5qlk4