  We propose a new method for estimating the minimizer $\boldsymbol{x}^*$ and
the minimum value $f^*$ of a smooth and strongly convex regression function $f$
from the observations contaminated by random noise. Our estimator
$\boldsymbol{z}_n$ of the minimizer $\boldsymbol{x}^*$ is based on a version of
the projected gradient descent with the gradient estimated by a regularized
local polynomial algorithm. Next, we propose a two-stage procedure for
estimation of the minimum value $f^*$ of regression function $f$. At the first
stage, we construct an accurate enough estimator of $\boldsymbol{x}^*$, which
can be, for example, $\boldsymbol{z}_n$. At the second stage, we estimate the
function value at the point obtained in the first stage using a rate optimal
nonparametric procedure. We derive non-asymptotic upper bounds for the
quadratic risk and optimization error of $\boldsymbol{z}_n$, and for the risk
of estimating $f^*$. We establish minimax lower bounds showing that, under
certain choice of parameters, the proposed algorithms achieve the minimax
optimal rates of convergence on the class of smooth and strongly convex
functions.
