math.OCApr 7, 2026

Intrinsic perturbation scale for certified oracle objectives with epigraphic information

Authors: Karim Bounja, Boujemaâ Achchab, Abdeljalil Sakat

Organizations: Laboratory for Analysis and Modeling of Systems and Decision Support (LAMSAD), Hassan 1st University of Settat, Settat 26000, Morocco

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Abstract

We introduce a natural displacement control for minimizer sets of oracle objectives equipped with certified epigraphic information. Formally, we replace the usual local uniform value control of objective perturbations - uncertifiable from finite pointwise information without additional structure - by the strictly weaker requirement of a cylinder-localized vertical epigraphic control, naturally provided by certified envelopes. Under set-based quadratic growth (allowing nonunique minimizers), this yields the classical square-root displacement estimate with optimal exponent 1/2, without any extrinsic assumption.

Figures & tables

Figure 1: Localized vertical epigraphic gauge on the cylinder CR,MC_{R,M} . At fixed (x0,t)(x_{0},t) , the quantities dFv(x0,t)d_{F}^{\mathrm{v}}(x_{0},t) and dF~v(x0,t)d_{\widetilde{F}}^{\mathrm{v}}(x_{0},t) are the vertical distances to epi⁡(F)\operatorname{epi}(F) and epi⁡(F~)\operatorname{epi}(\widetilde{F}) ; their difference is the local vertical discrepancy contributing to GR,M(F,F~)\mathcal{G}_{R,M}(F,\widetilde{F}) .
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