Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

区间规划问题的最优性条件

区间规划是带有区间参数的规划问题,是一种更易于求解实际问题的柔性规划。它是确定性优化问题的延伸,有区间线性规划和区间非线性规划两种形式。本文讨论了目标函数是区间函数的区间非线性问题。给出了区间规划问题最优性必要条件的较简单证明方法,并利用LU最优解的概念,在一类广义凸函数－(p,r)－ρ－(η,θ)－不变凸函数定义下讨论了最优性充分条件。     

群体多目标决策联合有效解类的不变凸充分条件

对于群体多目标决策问题，文[1]引进它的联合有效解类的概念，并给出这类解的最优性必要条件，在对于问题的目标函数和约束函数附加凸性的条件下，文[2]又给出了联合有效解类的最优性充分条件，本文进一步在目标函数和约束函数具不变凸和不变广义 凸的情况下，分别给出了联合有效解类的若干最优性充分条件。     

The Karush–Kuhn–Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions

The KKT conditions in multiobjective programming problems with interval-valued objective functions are derived in this paper. Many concepts of Pareto optimal solutions are proposed by considering two orderings on the class of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we are able to consider the continuity and differentiability of an interval-valued function. The KKT optimality conditions can then be naturally elicited.     

Invex Functions and Generalized Convexity in Multiobjective Programming

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.     

The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems

This paper considers optimization problems with fuzzy-valued objective functions. For this class of fuzzy optimization problems we obtain Karush–Kuhn–Tucker type optimality conditions considering the concept of generalized Hukuhara differentiable and pseudo-invex fuzzy-valued functions.     

The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function

The KKT conditions in an optimization problem with interval-valued objective function are derived in this paper. Two solution concepts of this optimization problem are proposed by considering two partial orderings on the set of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we derive the KKT optimality conditions.     

ON SUFFICIENCY AND DUALITY OF SOLUTIONS FOR QUASI Bs-INVEX SEMI-INFINITE PROGRAMMING

1　IntroductionRecently,various kinds of generalized convex functions were introduced.Bector andSingh[1 ] introduced a class of functions which called B-vex function.Bector,Suneja,andLalitha[2 ] introduced quasi B-vex function,pseudo B-vex function,B-invex function,quasi B-invex function,and pseudo B-invex function.We[3] extended invex function[4] ,gave thedefinitions of the symmetricη-function,symmetricη-pseudoconvex function,symmetricη-quasiconvex function for symmetric differentiable…     

Riemann流形上ρ-(η,d)-B不变凸的向量变分不等式及向量优化问题

该文研究了Riemann流形上的优化问题.首先,利用广义方向导数在Riemann流形上引入ρ-(η,d)-B不变凸函数、ρ-(η,d)-B伪不变凸函数和ρ-(η,d)-B拟不变凸函数.其次,讨论了变分不等式的解与Riemann流形上向量优化问题解之间的关系.最后,建立了优化问题的Kuhn-Tucker充分条件.     

Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma

In this paper, we present Lagrange multiplier necessary conditions for global optimality that apply to non-convex optimization problems beyond quadratic optimization problems subject to a single quadratic constraint. In particular, we show that our optimality conditions apply to problems where the objective function is the difference of quadratic and convex functions over a quadratic constraint, and to certain class of fractional programming problems. Our necessary conditions become necessary and sufficient conditions for global optimality for quadratic minimization subject to quadratic constraint. As an application, we also obtain global optimality conditions for a class of trust-region problems. Our approach makes use of outer-estimators, and the powerful S-lemma which has played key role in control theory and semidefinite optimization. We discuss numerical examples to illustrate the significance of our optimality conditions. The authors are grateful to the referees for their useful comments which have contributed to the final preparation of the paper.     

G-pre-invex functions in mathematical programming

In the present paper, we introduce the concept of G-pre-invex functions with respect to η defined on an invex set with respect to η. These function unify the concepts of nondifferentiable convexity, pre-invexity and r-pre-invexity. Furthermore, relationships of G-pre-invex functions to various introduced earlier pre-invexity concepts are also discussed. Some (geometric) properties of this class of functions are also derived. Finally, optimality results are established for optimization problems under appropriate G-pre-invexity conditions.     

Duality Theory for Optimization Problems with Interval-Valued Objective Functions

A solution concept in optimization problems with interval-valued objective functions, which is essentially similar to the concept of nondominated solution in vector optimization problems, is introduced by imposing a partial ordering on the set of all closed intervals. The interval-valued Lagrangian function and interval-valued Lagrangian dual function are also proposed to formulate the dual problem of the interval-valued optimization problem. Under this setting, weak and strong duality theorems can be obtained.     

Duality and saddle-point type optimality for interval-valued programming

In this paper, Mond-Weir’s type dual in programming problem with an interval-valued objective function and interval-valued inequality constrict conditions is formulated. Duality theorems are established under suitable conditions. A real-valued Lagrangian function for the interval-valued programming is defined. Further, the saddle point of Lagrangian function is also defined and saddle point optimality conditions are presented.     

Invex multifunctions and duality

For optimization problems with multifunction objective and constraints, duality theorems are proved for analogs of the Wolfe and Mond–Weir dual problems, assuming that the multifunctions satisfy a generalization of the invex property for functions. Several characterizations of generalized invexity are obtained.     

On interval-valued nonlinear programming problems

The Wolfe's duality theorems in interval-valued optimization problems are derived in this paper. Four kinds of interval-valued optimization problems are formulated. The Karush-Kuhn-Tucker optimality conditions for interval-valued optimization problems are derived for the purpose of proving the strong duality theorems. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.     

Second-order Kuhn-Tucker invex constrained problems

A new notion of a second-order KT invex problem (P) with inequality constraints is introduced in this paper. This class of problems strictly includes the KT invex ones. Some properties of the second-order KT invex problems are presented. For example, (P) is second-order KT invex if and only if each point, which satisfies the second-order Kuhn-Tucker necessary optimality conditions, is a global minimizer. A problem with quasiconvex data is (second-order) KT invex if and only if it is (second-order) KT pseudoconvex.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

Exactness Property of the Exact Absolute Value Penalty Function Method for Solving Convex Nondifferentiable Interval-Valued Optimization Problems

In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given.     

Separation Functions and Optimality Conditions in Vector Optimization

In this paper, we propose weak separation functions in the image space for general constrained vector optimization problems on strong and weak vector minimum points. Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions. By virtue of such nonlinear separation functions, we derive Lagrangian-type sufficient optimality conditions in a general context. Especially for nonconvex problems, we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions, and we further deduce Karush–Kuhn–Tucker necessary conditions in terms of Clarke subdifferentials.     

Optimality and duality in constrained interval-valued optimization

Fritz John and Karush–Kuhn–Tucker necessary conditions for local LU-optimal solutions of the constrained interval-valued optimization problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators are established. Under suitable assumptions on the generalized convexity of objective and constraint functions, sufficient conditions for LU-optimal solutions are given. The dual problems of Mond–Weir and Wolfe types are studied together with weak and strong duality theorems for them.