This paper proposes a novel technique called “successive stochastic smoothing” that optimizes nonsmooth and discontinuous functions while considering various constraints. Our methodology enables local and global optimization, making it a powerful tool for many applications. First, a constrained problem is reduced to an unconstrained one by the exact nonsmooth penalty function method, which does not require the selection of the penalty coefficient. This reduction is exact in the case of minimization of a lower semicontinuous function under convex constraints. Then the resulting objective penalty function is sequentially smoothed by the kernel method starting from relatively strong smoothing and with gradually vanishing degree of smoothing. The finite difference stochastic gradient descent with trajectory averaging minimizes each smoothed function locally. Finite differences over stochastic directions sampled from the kernel estimate the stochastic gradients of the smoothed functions. We investigate convergence rate of such stochastic finite-difference method on convex optimization problems. The “successive smoothing” algorithm uses the results of previous optimization runs to select the starting point for optimizing a consecutive, less smoothed function. Smoothing provides the “successive smoothing” method with some global properties.