Optimality conditions via exact penalty functions

In this paper, we obtain optimality conditions for the problem with inequality, equality and closed set constraints in terms of the lower Hadamard derivative. The results are obtained applying exact penalty functions.

💡 Research Summary

The paper addresses a general nonlinear programming problem that involves inequality constraints, equality constraints, and a closed‑set constraint. The authors develop optimality conditions expressed through the lower Hadamard directional derivative and exploit exact penalty functions to bridge the original constrained problem with a simpler penalized formulation.

First, the authors introduce the Bouligand (contingent) tangent cone and the lower Hadamard derivative (f^{\downarrow H}(x;u;S)), defined as the lim‑inf of the difference quotient along feasible directions. This derivative is set to (+\infty) when the direction does not belong to the tangent cone, and it satisfies the basic property that if (x) is a local minimum of (f) on a set (S), then (f^{\downarrow H}(x;u;S)\ge 0) for all directions (u).

The central construction is the exact penalty function
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