bound, which are similar to the well known bound presented in [@AlberIusemSolodov1998; @nedic_bertsekas2001]. In particular, for $C = \mathbb{R}^n$, the convergence results merge into the ones presented in [@CorreaLemarecha1993] and the iteration-complexity bound into [@Nesterov2004 Theorem 3.2.2]. The asymptotic convergence result and the iteration-complexity bound obtained using Polyak’s stepsizes are similar to the corespondent ones in [@nedic_bertsekas2001; @Nesterov2014; @Polyak1969] and [@Nesterov2014], respectively. Regarding to the dynamic stepsize, we establish global convergence in objective values as address, for example, in [@GoffinKiwiel1999; @nedic_bertsekas2001]. In [@nedic_bertsekas2001rate Proposition 2.15], the authors presented the rate of convergence for another variant of subgradient method, known as incremental subgradient algorithms. This study allowed us to estimate an iteraction-complexity bound for the dynamic stepsize.

The organization of the paper is as follows. In Section \[sec:int.1\], we present some notation and basic results used in our presentation. In Section \[Sec:SubInexProj\] we describe the Subgradient-InexP method with different choices for the stepsize. The main results of the present paper, including the converge theorems and iteration-complexity, are presented in Section \[Sec:aca\]. Some numerical experiments are provided in Section \[Sec:NumExp\]. We conclude the paper with some remarks in Section \[Sec:Conclusions\].

Notation and definitions {#sec:int.1}
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In this section, we present some notations, 