Nonlinear Optimisation Under General Directional Mapping (GDM): Generated-Direction Optimality and Duality

We study constrained nonlinear optimisation when feasible directions are generated by ageneral directional mapping (GDM) D(x, u) with an invertible linear part A, in place of theplain displacement x − u. For a primal problem (P) and its Lagrangian dual (D), three theoremsdescribe the framework:(1) The Abadie, LICQ and MFCQ constraint qualifications are invariant under the mapping;(2) Generated-direction optimality (GDO) conditions, parallel to the KKT conditions, hold ata local minimizer, and their multipliers coincide with the KKT multipliers;(3) Weak duality holds, and for convex problems a GDO point gives zero duality gap, with itsmultipliers solving (D).A numerical example and readings for scaling, preconditioning and constrained mechanics aregiven, and second-order conditions and algorithms are listed as future research directions.