Law-Based Optimization under Zulfia Mapping: KKT Comparison and Illustrative Example

Abstract

We introduce a law-based optimization framework derived from the Zulfia mapping,which decomposes a mapping into directional and induced components to characterizeoptimization behavior without classical constraint qualifications. The mapping is definedas Z(x, u) = η(x, u) + ∆(x, u), where η is the directional component and ∆ is the inducedcomponent. This decomposition yields three governing laws that regulate the optimizationprocess:• Law of Directional Generation: D(x, u) = ∇f (u)TZ(x, u);• Law of Inducement: I(x, u) = Φ(x, u, ∆(x, u));• Law of Stability: limk→∞ ∥Z(x, uk)∥ = 0.These laws generate optimization directions, induced feasibility-constraint communica-tion, and stabilization behavior directly from the mapping decomposition. Classical opti-mality conditions emerge as limiting realizations of this structure. A numerical illustrationand a comparative analysis with KKT conditions are provided, demonstrating that Zul-fia mapping generalizes classical invexity via the induced correction ∆(x, u) and providesglobal optimality without constraint qualifications.