A practical smoothing method was analyzed and tested against state-of-the-art solvers for some non-smooth optimization problems in [BSS20a; BSS20b]. This method can be used to smooth the value functions and solution mappings of fully parameterized convex problems under mild conditions. In general, the smoothing of the value function lies from above the true value function and the smoothing of the solution mapping is an approximate selection over the true solution mapping. In this paper we say exactly when the smoothed value function is equal to the true value function and when the smoothed solution is an actual solution. Moreover, we describe a further application of this smoothing technique for building smooth approximations of optimal value reformulations of optimistic bilevel problems that converge epigraphically to the original bilevel problem. This new application motivates the characterization we prove.
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