  We define and study a notion of minimal exponent for a locally complete
intersection subscheme $Z$ of a smooth complex algebraic variety $X$, extending
the invariant defined by Saito in the case of hypersurfaces. Our definition is
in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$. We show
that the minimal exponent describes how far the Hodge filtration and order
filtration agree on the local cohomology $H^r_Z({\mathcal O}_X)$, where $r$ is
the codimension of $Z$ in $X$. We also study its relation to the Bernstein-Sato
polynomial of $Z$. Our main result describes the minimal exponent of a higher
codimension subscheme in terms of the invariant associated to a suitable
hypersurface; this allows proving the main properties of this invariant by
reduction to the codimension $1$ case. A key ingredient for our main result is
a description of the Kashiwara-Malgrange $V$-filtration associated to any ideal
$(f_1,\ldots,f_r)$ in terms of the microlocal $V$-filtration associated to the
hypersurface defined by $\sum_{i=1}^rf_iy_i$.
