and properties of it have been discovered throughout the years, resulting in a wide literature on the subject; see, for exemple [@AlberIusemSolodov1998; @Yunier2013; @Bertsekas1999; @GoffinKiwiel1999; @KiwielBook1985; @NedicBertsekas2010] and the references therein.

The aim of this paper is to present an inexact version of the projected subgradient method, which consists in using an inexact projection instead of the exact one, for minimizing a convex function $f: \mathbb{R}^n \to \mathbb{R}$ onto a closed and convex subset $C$ of $\mathbb{R}^n$. The proposed method, that we call [*Subgradient-InexP method*]{}, generates a sequence $\{x_k\}$ where each iteration consists of two stages. The first stage performs a step from the current iterate $x_k$ in the opposite direction of a $\epsilon$-subgradient of $f$ at $ x_k $ and the second inexactly projects the resulting vector onto the feasible set $C$. From the theoretical point of view, considering methods that use inexact projections are particularly interesting for the following reasons. Even when the projection onto a convex set is an easy problem, iterative methods provide only approximated solutions with small errors, due to round-off errors in floating-point arithmetics. Therefore, the study of inexact methods gives theoretical support for real computational implementations of exact schemes. On the other hand, 