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This paper is devoted to studying the stationary solutions of a generalconstrained optimization problem through its associated unconstrained penalizedproblems. We aim to answer the question, "what do the stationary solutions of apenalized unconstrained problem tell us about the solutions of the originalconstrained optimization problem?". We answer the latter question byestablishing relationships between global (local) minimizers and stationarypoints of the two optimization problems. Given the strong connection betweenstationary solutions between problems, we introduce a new approximate$\varepsilon$-stationary solution for the constrained optimization problems. Wepropose an algorithm to compute such an approximate stationary solution for ageneral constrained optimization problem, even in the absence of Clarkeregularity. Under reasonable assumptions, we establish the rate$O(\varepsilon^{-2})$ for our algorithm, akin to the gradient descent methodfor smooth minimization. Since our penalty terms are constructed by the powersof the distance function, our stationarity analysis heavily depends on thegeneralized differentiation of the distance function. In particular, wecharacterize the (semi-)differentiability of the distance function$\mbox{dist}(. ;X)$ defined by a Fr\'echet smooth norm, in terms of thegeometry of the set $X$. We show that $\mbox{dist} (. ;X)$ issemi-differentiable if and only if $X$ is geometrically derivable. The latteropens the door to design optimization algorithms for constrained optimizationproblems that suffer from Clarke irregularity in their objectives andconstraint functions.

Penalty methods to compute stationary solutions in constrained optimization problems