Higher Order Sufficient Optimality Conditions Research Articles

New Properties of Higher-Order Radial Sets and Higher-Order Radial Derivatives and Applications to Optimality Conditions

: Several new properties are obtained for higher-order upper radial sets and corresponding derivatives introduced in [J. Glob. Optim. DOI 10.1007/s10898-012-9861- z]. Higher-order sufficient optimality conditions and higher-order necessary optimality conditions are established for weak efficient solutions of set-valued optimization problems. Several corollaries are provided to show that obtained results include a few existing ones. In addition, one removes deficiencies contained of two results in [J. Glob. Optim. DOI 10.1007/s10898-012-9861-z].

Higher order optimality and duality in fractional vector optimization over cones

In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

Generalized higher-order cone-convex functions and higher-order duality in vector optimization

In this paper, we introduce a new class of higher-order cone-convex, \((K_1, K_2)\)-pseudoconvex and quasiconvex functions which encapsulates several already known functions. Higher-order sufficient optimality conditions have been established for a vector optimization problem over cones by using these functions, under weaker conditions on multipliers as compared to other papers in this domain. Wolfe type and Mond–Weir type higher-order duals are formulated and corresponding duality results are established. A number of previously studied problems appear as special cases of our primal-dual models. In case of nonlinear programming problem, our higher-order duals reduce to the corresponding higher-order duals given by Mangasarian (J Math Anal Appl 51:607–620, 1975) and Mond and Zhang (Generalized convexity, generalized monotonicity: recent results. Kluwer, Dordrecht, pp 357–372, 1998).

Generalized Higher Order (φ, η, ω, π, ρ, θ, m)-Invexities in Parametric Optimality Conditions for Discrete Minmax Fractional Programming

First several new classes of higher order (φ, η, ω, π, ρ, θ, m)-invexities are introduced, and then a set of higher-order parametric necessary optimality conditions and several sets of higher order sufficient optimality conditions for a discrete minmax fractional programming problem applying various higher order (φ, η, ω, π, ρ, θ, m)-invexity constraints are established. The obtained results are new and generalize a wide range of results in the literature.

Higher-order cone-pseudoconvex, quasiconvex and other related functions in vector optimization

Recently Bhatia (Optim. Lett. doi:10.1007/s11590-010-0248-0, 2010) introduced higher-order cone-convex functions and used them to obtain higher-order sufficient optimality conditions and duality results for a vector optimization problem over cones. The concepts of higher-order (strongly) cone-pseudoconvex and cone-quasiconvex functions were also defined by Bhatia (Optim. Lett. doi:10.1007/s11590-010-0248-0, 2010). In this paper we introduce the notions of higher-order naturally cone-pseudoconvex, strictly cone-pseudoconvex and weakly cone-quasiconvex functions and study various interrelations between the above mentioned functions. Higher-order sufficient optimality conditions have been established by using these functions. Generalized Mond–Weir type higher-order dual is formulated and various duality results have been established under the conditions of higher-order strongly cone-pseudoconvexity and higher-order cone quasiconvexity.

Higher order duality in vector optimization over cones

In this paper higher order cone convex, pseudo convex, strongly pseudo convex, and quasiconvex functions are introduced. Higher order sufficient optimality conditions are given for a weak minimum, minimum, strong minimum and Benson proper minimum solution of a vector optimization problem. A higher order dual is associated and weak and strong duality results are established under these new generalized convexity assumptions.