Let \(Z\) be a nonempty closed subset of a real normed space \(X\) and \(J:Z\to \mathbb R\) a lower semicontinuous function bounded from below. The paper is concerned with generic existence results for the perturbed optimization problem \(J\)-inf: find \(z_0\in Z\) such that \(\| x-z_0\| =\inf\{J(z)+\| x-z\| : z\in Z\}. \) Results of this kind were obtained by \textit{J. Baranger} [J. Math. Pur. Appl., IX. Sér. 52 (1973) (1974; Zbl 0274.46015)] in the case of uniformly convex spaces, and by the reviewer [J. Math. Anal. Appl. 243, No. 2, 344--356 (2000; Zbl 0957.49014)] in the case of reflexive Banach spaces having the Kadec-Klee property. The author gives an extension of this last result by proving some generic existence results in general Banach spaces, but with some supplementary conditions on the set \(Z,\) namely an extension of the notion of relatively weakly boundedly compact set. The proof is based on a result of \textit{W. J. Davis, T. Figiel, W. B. Johnson} and \textit{A. Pelczynski} [J. Funct. Anal. 17, 311--327 (1974; Zbl 0306.46020)], asserting that if \(K\) is a weakly compact subset of a Banach space \(X\) such that sp\((K)\) is dense in \(X,\) then there exists a reflexive Banach space \(Y\) and an injective continuous linear mapping \(T:Y\to X\) such that \(T(B_ Y) \supset K. \)