On Sufficiency of the Generated-Direction Optimization Conditions (GDO)

We introduce a class of constrained nonlinear programs, the Generated-DirectionOptimization Problems (GDOPs), defined through a general directional mapping D(x, u)with invertible linear part A and the GDM-gradient ∇D = A−⊤∇. In parallel withHanson’s sufficiency theorem for invex problems, three results describe the class:(1) At a GDO point of a GDOP the point is a global minimizer; a gain version givesquadratic growth in the generated distance and uniqueness;(2) A GDOP at u is a problem that is invex at u for the normalized kernel ηD = A−1D,equivalently Zulfiable with transformation mapping ηD, and the class is invariant underD ↦ → BD for every invertible B;(3) Problems that become convex after a smooth change of variables x = ψ(y) areGDOPs with D(y, v) = ψ(y) − ψ(v), although they need not be convex in y.Examples, a comparison with Hanson’s framework, and future research directions aregiven.