Polyak's subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution. In this paper we extend the convergence properties of the Polyak's subgradient algorithm with a fixed target value to a more general case with variable target values. Then a target value updating scheme is provided which finds an optimal solution without prior knowledge of the optimal objective value. The convergence proof of the scheme is provided and computational results of the scheme are reported.