Sufficient optimality criteria and duality for multiobjective variational control problems with G-type I objective and constraint functions

In the paper, we introduce the concepts of G-type I and generalized G-type I functions for a new class of nonconvex multiobjective variational control problems. For such nonconvex vector optimization problems, we prove sufficient optimality conditions for weakly efficiency, efficiency and properly efficiency under assumptions that the functions constituting them are G-type I and/or generalized G-type I objective and constraint functions. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem is given and several duality results are established under (generalized) G-type I objective and constraint functions

Duality for multiobjective variational control problems with (Φ,ρ)-invexity

In this paper, Mond-Weir and Wolfe type duals for multiobjective variational control problems are formulated. Several duality theorems are established relating efficient solutions of the primal and dual multiobjective variational control problems under TeX-invexity. The results generalize a number of duality results previously established for multiobjective variational control problems under other generalized convexity assumptions

Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems

In this paper, a new class of generalized of nonconvex multitime multiobjective variational problems is considered. We prove the sufficient optimality conditions for efficiency and proper efficiency in the considered multitime multiobjective variational problems with univex functionals. Further, for such vector variational problems, various duality results in the sense of Mond-Weir and in the sense of Wolfe are established under univexity. The results established in the paper extend and generalize results existing in the literature for such vector variational problems

Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver

We consider a variant of inexact Newton Method, called Newton-MR, in which the least-squares sub-problems are solved approximately using Minimum Residual method. By construction, Newton-MR can be readily applied for unconstrained optimization of a class of non-convex problems known as invex, which subsumes convexity as a sub-class. For invex optimization, instead of the classical Lipschitz continuity assumptions on gradient and Hessian, Newton-MR's global convergence can be guaranteed under a weaker notion of joint regularity of Hessian and gradient. We also obtain Newton-MR's problem-independent local convergence to the set of minima. We show that fast local/global convergence can be guaranteed under a novel inexactness condition, which, to our knowledge, is much weaker than the prior related works. Numerical results demonstrate the performance of Newton-MR as compared with several other Newton-type alternatives on a few machine learning problems.Comment: 35 page

On the Necessity of the Sufficient Conditions in Cone-Constrained Vector Optimization

The object of investigation in this paper are vector nonlinear programming problems with cone constraints. We introduce the notion of a Fritz John pseudoinvex cone-constrained vector problem. We prove that a problem with cone constraints is Fritz John pseudoinvex if and only if every vector critical point of Fritz John type is a weak global minimizer. Thus, we generalize several results, where the Paretian case have been studied. We also introduce a new Frechet differentiable pseudoconvex problem. We derive that a problem with quasiconvex vector-valued data is pseudoconvex if and only if every Fritz John vector critical point is a weakly efficient global solution. Thus, we generalize a lot of previous optimality conditions, concerning the scalar case and the multiobjective Paretian one. Additionally, we prove that a quasiconvex vector-valued function is pseudoconvex with respect to the same cone if and only if every vector critical point of the function is a weak global minimizer, a result, which is a natural extension of a known characterization of pseudoconvex scalar functions.Comment: 12 page

Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d-r-type I functions

AbstractIn this paper, new classes of nondifferentiable functions constituting multiobjective programming problems are introduced. Namely, the classes of d-r-type I objective and constraint functions and, moreover, the various classes of generalized d-r-type I objective and constraint functions are defined for directionally differentiable multiobjective programming problems. Sufficient optimality conditions and various Mond–Weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type. Finally, it is showed that the introduced d-r-type I notion with r≠0 is not a sufficient condition for Wolfe weak duality to hold. These results are illustrated in the paper by suitable examples