A Class of Invariant Programming Problems: A Convexity-Free and KKT-Independent Framework for Optimality

In this paper we introduce a class of invariant programming problems in math-ematical programming. The proposed framework is based on an invariant identityrelation connecting the objective structure with a reference expression through a non-negative remainder termF (x) − Φ(x, u) = R(x, u), R(x, u) ≥ 0.This relation provides a structural mechanism for establishing optimality withoutrelying on convexity assumptions or classical Karush–Kuhn–Tucker (KKT) optimalityconditions.Within this framework we derive necessary and sufficient optimality conditionsand establish unified duality results. The invariant programming approach offersan alternative viewpoint for studying optimality in mathematical programming andsuggests several directions for future research in optimization theory