Invexity-Based Duality for Composite Vector Optimization over Cones

The vector optimization problems in which the objective and/or constraint functions are formed as compositions of two or more functions are known as composite vector optimization problems.  This structure arises naturally in several settings such as ex-act penalty reformulations of constrained optimization problems and problems where an outer function measures or orders the output of an inner mapping . The system-atic study of composite optimization was initiated by Jeyakumar [5] in 1991. He took up a nonsmooth optimization problem where the objective function and the constraint functions are compositions of locally Lipschitz and Gaˆteaux differentiable functions and presented necessary and sufficient optimality conditions. Jeyakumar and Yang [6] con-sidered multi-objective optimization problems where the objective and the constraints are compositions of convex and locally Lipschitz functions and Gaˆteaux differentiable functions and established optimality and duality results. Bo¸t et all [1], [2] extended this to a convex, cone-ordered setting via Fenchel–Lagrangian duality, working with cone-increasing convex outer functions composed with cone-convex inner mappings. Tang and Zhao [12] later addressed a nonsmooth, cone-constrained multiobjective composite problem, introducing a generalized null space condition and cone-generalized invexity to obtain necessary and sufficient optimality conditions. More recently, Chuong [16], [17] developed optimality and duality theory for nonsmooth composite vector and conic vector optimization problems using limiting subdifferential calculus. Saadati and Oveisiha [18] extended the composite framework to uncertain (robust) multiobjective settings over As-plund spaces, formulating a Mond–Weir-type robust dual and establishing weak, strong, and converse duality.

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