Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

Optimality and duality in nonlinear programming involving semilocally B-preinvex and related functions

A nonlinear programming problem is considered where the functions involved are η-semidifferentiable. Fritz John and Karush–Kuhn–Tucker types necessary optimality conditions are obtained. Moreover, a result concerning sufficiency of optimality conditions is given. Wolfe and Mond–Weir types duality results are formulated in terms of η-semidifferentials. The duality results are given using concepts of generalized semilocally B-preinvex functions.     

A homotopy method for nonlinear semidefinite programming

In this paper, for solving the nonlinear semidefinite programming problem, a homotopy is constructed by using the parameterized matrix inequality constraint. Existence of a smooth path determined by the homotopy equation, which starts from almost everywhere and converges to a Karush–Kuhn–Tucker point, is proven under mild conditions. A predictor-corrector algorithm is given for numerically tracing the smooth path. Numerical tests with nonlinear semidefinite programming formulations of several control design problems with the data contained in COMPl _e ib are done. Numerical results show that the proposed algorithm is feasible and applicable.     

Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints

4OR - In this paper, the class of differentiable semi-infinite multiobjective programming problems with vanishing constraints is considered. Both Karush–Kuhn–Tucker necessary optimality...     

A complementarity partition theorem for multifold conic systems

Consider a homogeneous multifold convex conic system $$Ax = 0, \quad x\in K_1\times \cdots \times K_r$$ and its alternative system $$A^t y \in K_1^*\times \cdots \times K_r^*$$ , where K ₁,..., K _r are regular closed convex cones. We show that there is a canonical partition of the index set {1,...,r} determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman–Tucker Theorem for linear programming.     

Pseudoinvexity,optimality conditions and efficiency in multiobjective problems; duality

In this paper, we establish characterizations for efficient solutions to multiobjective programming problems, which generalize the characterization of established results for optimal solutions to scalar programming problems. So, we prove that in order for Kuhn–Tucker points to be efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we obtain characterizations for efficient solutions by using Fritz–John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results. We study the dual problem and establish weak, strong and converse duality results.     

Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems

We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufficient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x* to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisfied at x*.     

Optimality and invexity in optimization problems in Banach algebras (spaces)

In this paper we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-(p,r)-invexity. In fact, this paper focuses on the optimality conditions for optimization problems in Banach algebras, regarding the generalized KT-(p,r)-invexity notion and Kuhn–Tucker points.     

Spectral estimates for unreduced symmetric KKT systems arising from Interior Point methods

We consider symmetrized Karush–Kuhn–Tucker systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods. Their coefficient matrices usually have 3 × 3 block structure, and under suitable conditions on both the quadratic programming problem and the solution, they are nonsingular in the limit. We present new spectral estimates for these matrices: the new bounds are established for the unpreconditioned matrices and for the matrices preconditioned by symmetric positive definite augmented preconditioners. Some of the obtained results complete the analysis recently given by Greif, Moulding, and Orban in [SIAM J. Optim., 24 (2014), pp. 49‐83]. The sharpness of the new estimates is illustrated by numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity

In the paper, we consider a class of nonsmooth multiobjective programming problems in which involved functions are locally Lipschitz. A new concept of invexity for locally Lipschitz vector-valued functions is introduced, called V-r-invexity. The generalized Karush–Kuhn–Tuker necessary and sufficient optimality conditions are established and duality theorems are derived for nonsmooth multiobjective programming problems involving V-r-invex functions (with respect to the same function η).     

Nonlinear programming with E-preinvex and local E-preinvex functions

In this paper, we extend the class of E-convex sets, E-convex and E-quasiconvex functions introduced by [Youness, E.A., 1999. E-convex sets, E-convex functions and E-convex programming. Journal of Optimization Theory and Applications 102, 439–450], respectively by [Syau, Yu-Ru, Lee, E. Stanley, 2005. Some properties of E-convex functions. Applied Mathematics Letters 18, 1074–1080] to E-invex set, E-preinvex, E-prequasiinvex and corresponding local concepts. Some properties of these classes are studied. As an application of our results, we consider the nonlinear programming problem for which, we establish that, under mild conditions, a local minimum is a global minimum.     

Some Inexact Hybrid Proximal Augmented Lagrangian Algorithms

In this work, Solodov–Svaiter's hybrid projection-proximal and extragradient-proximal methods [16,17] are used to derive two algorithms to find a Karush–Kuhn–Tucker pair of a convex programming problem. These algorithms are variations of the proximal augmented Lagrangian. As a main feature, both algorithms allow for a fixed relative accuracy of the solution of the unconstrained subproblems. We also show that the convergence is Q-linear under strong second order assumptions. Preliminary computational experiments are also presented.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

The optimization technique for solving a class of non-differentiable programming based on neural network method

In this paper, the optimization techniques for solving a class of non-differentiable optimization problems are investigated. The non-differentiable programming is transformed into an equivalent or approximating differentiable programming. Based on Karush–Kuhn–Tucker optimality conditions and projection method, a neural network model is constructed. The proposed neural network is proved to be globally stable in the sense of Lyapunov and can obtain an exact or approximating optimal solution of the original optimization problem. An example shows the effectiveness of the proposed optimization techniques.     

Cohomology theories in triangulated categories

Let C be a triangulated category with a proper class E of triangles.We prove that there exists an Avramov–Martsinkovsky type exact sequence in C,which connects E-cohomology,E-Tate cohomology and E-Gorenstein cohomology.     

Nondifferentiable multiobjective programming under generalized d-univexity

In this paper, we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [Numer. Funct. Anal. Optim. 22 (2001) 775], d-invex functions in Antczak [Europ. J. Oper. Res. 137 (2002) 28] and univex functions in Bector et al. [Univex functions and univex nonlinear programming, Proc. Admin. Sci. Assoc. Canada, 1992, p. 115]. By utilizing the new concepts, we derive a Karush–Kuhn–Tucker sufficient optimality condition and establish Mond–Weir type and general Mond–Weir type duality results for the nondifferentiable multiobjective programming problem.     

On efficient applications of G-Karush-Kuhn-Tucker necessary optimality theorems to multiobjective programming problems

We prove a slightly modified G-Karush-Kuhn-Tucker necessary optimality theorem for multiobjective programming problems, which was originally given by Antczak (J Glob Optim 43:97–109, 2009), and give an example showing the efficient application of (modified) G-Karush-Kuhn-Tucker optimality theorem to the problems.     

Optimality of E-pseudoconvex multi-objective programming problems without constraint qualifications

In this note, we consider the optimality criteria of multi-objective programming problems without constraint qualifications involving generalized convexity. Under the E-pseudoconvexity assumptions, the unified necessary and sufficient optimality conditions are established for weakly efficient and efficient solutions, respectively, in multi-objective programming problems.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

On a New Class of Vector Variational Control Problems

Abstract

In this paper, we introduce geodesic (strongly) b-V-KT-pseudoinvex multidimensional control problems. This new class of multiobjective variational control problems, involving multiple integral cost functionals, is described such that every geodesic Kuhn–Tucker point is geodesic efficient solution. In addition, to illustrate the effectiveness of our main result, the paper is completed with an application.