Identity-Based Optimization: A Convexity-Free Framework for Optimality

Classical optimization theory derives optimality from the geometry of feasible setsand typically relies on convexity assumptions and Karush–Kuhn–Tucker (KKT) con-ditions. In this paper we introduce a different viewpoint by proposing the frameworkof identity-based optimization. Instead of deriving optimality from feasible-set geome-try, the proposed approach establishes optimality through a structural identity relationlinking the objective function with a reference expression.The fundamental relation considered in this work isF (x) − Φ(x, u) = R(x, u), R(x, u) ≥ 0where F (x) denotes the objective function, Φ(x, u) represents an invariant referenceexpression, and R(x, u) is a nonnegative remainder term. Optimality arises when theidentity closes, that is,R(x∗, u∗) = 0.Within this framework we establish identity-based optimality conditions and derivea unified duality relation connecting primal and dual structures. The proposed ap-proach provides a convexity-free and KKT-independent mechanism for optimality andsuggests a new structural viewpoint for optimization theory.