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By applying some theorems of Levy and Mordukhovich (Math Program 99:311--327, 2004) and other related results, we estimate the Fr\'echetcoderivative and the Mordukhovich coderivative of the stationary point set mapof a smooth parametric optimization problem with one smooth functionalconstraint under total perturbations. From the obtained formulas we derivenecessary and sufficient conditions for the local Lipschitz-like property ofthe stationary point set map. This leads us to new insights into the precedingdeep investigations of Levy and Mordukhovich in the above-cited paper and ofQui (J Optim Theory Appl 161: 398--429, 2014; J Glob Optim 65: 615--635, 2016).

Sensitivity Analysis of a Stationary Point Set Map under Total Perturbations. Part 1: Lipschitzian Stability