feasible iterates, such as feasible active set methods and interior point methods; see, for example, [@NicholasPhilippe2002; @NocedalWright2006; @Robert1996]. It is worth mentioning that, if the exact projection is used, then Subgradient-InexP method becomes the projected subgradient method considered in [@AlberIusemSolodov1998]. Several methods similar to the projected subgradient method have been studied in different papers, see [@GoffinKiwiel1999; @Mainge2008]. However, as far as we know, none of them use the concept of feasible inexact projection.

The main tool used in our analysis of Subgradient-InexP method is a version of the inequality obtained in [@CorreaLemarecha1993 Lemma 1.1]; see also a variant of it in [@nedic_bertsekas2001 Lemma 2.1]. By using this inequality, we establish asymptotic convergence results and iteration-complexity bounds for the sequence generated by our method employing the well known exogenous stepsizes, Polyak’s stepsizes, and dynamic stepsizes. We point out that these stepsizes have been discussed extensively in the related literature, including [@AlberIusemSolodov1998; @GoffinKiwiel1999; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @NedicBertsekas2010; @xmwang2018], where many of our results were inspired. Let us describe the results in the present and their relationship with the literature on the subject. With respect to the exogenous stepsize we establish convergence results without any compactness assumption, existence of a solution, and the iteration-complexity 