A Structural Interaction Framework for Optimization Problems

We consider optimization problems under the structural interaction representa-tionZ(x) = F (x)R(x),where F and R are structural interaction functions. If F and R are differentiable,then∇Z(x) = R(x)∇F (x) + F (x)∇R(x),which characterizes the interaction-induced gradient structure. We establish astructural sufficiency principle: if Z(x) ≥ 0 for all x and Z(x∗) = 0 for some x∗,then x∗ is a global minimizer, and this optimality follows directly from the inter-action structure without convexity assumptions. The framework remains valid fornon-differentiable F and R via generalized derivatives, providing a unified approachto optimization based on structural interaction.