Continuous-time optimization problems involving invex functions

In [D.H. Martin, The essence of invexity, J. Optim. Theory Appl. 47 (1985) 65-76] Martin introduced the notions of KKT-invexity and WD-invexity for mathematical programming problems. These notions are relaxations of invexity. In this work we generalize these concepts for continuous-time nonlinear optimization problems. We prove that the notion of KKT-invexity is a necessary and sufficient condition for global optimality of a Karush-Kuhn-Tucker point and that the notion of WD-invexity is a necessary and sufficient condition for weak duality.     

On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data

In this paper, we consider a nondifferentiable multiobjective semi-infinite optimization problem. We introduce a qualification condition and derive strong Karusk Kuhn Tucker(KKT) necessary conditions. Then a sufficient optimality condition is proved under invexity assumptions.     

Generalized convexity in non-regular programming problems with inequality-type constraints

Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKT-invexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKT-invexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.     

Generalized invexity for nonsmooth vector-valued mappings

We define four types of invexity for Lipschitz vector-valued mappings from R^p to R^q that generalize previous definitions of invexity in the differentiable setting. After establishing relationships between the various definitions, we show the importance of the concept of nonsmooth invexity in the field of optimization. In particular, we obtain conditions sufficient for optimality in unconstrained and cone-constrained nondifferentiable programming that are weaker than previous conditions presented in the literature; we also obtain weak and strong duality results.     

One class of separable optimization problems: solution method,application

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ? ⁿ ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.     

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).     

Optimization with generalized invexity

Constrained optimization is studied, with nonsmooth (Lipschitz) functions in abstract spaces and cone-constraints. Some more general Lagrangian necessary conditions are obtained, using strict minimum and approximation methods. These conditions are sufficient for a minimum under generalized invex assumptions. A characterization is obtained for generalized invexity, generalizing a known result for differentiable functions. Generalized invexity happens exactly when the generalized Wolfe and Lagrangian dual problems coincide.     

Optimality conditions in optimization problems with convex feasible set using convexificators

In this paper, we consider a nonsmooth optimization problem with a convex feasible set described by constraint functions which are neither convex nor differentiable nor locally Lipschitz necessarily. Utilizing upper regular convexificators, we characterize the normal cone of the feasible set and derive KKT type necessary and sufficient optimality conditions. Under some assumptions, we show that the set of KKT multipliers is bounded. We also characterize the set of optimal solutions and introduce a linear approximation corresponding to the original problem which is useful in checking optimality. The obtained outcomes extend various results existing in the literature to a more general setting.     

Optimality conditions for vector equilibrium problems with constraint in Banach spaces

In this paper, vector equilibrium problems with constraint in Banach spaces are investigated. Kuhn–Tucker-like conditions for weakly efficient solutions are given by using the Gerstewitz’s function and nonsmooth analysis. Moreover, the sufficient conditions of weakly efficient solutions are established under the assumption of generalized invexity. As applications, necessary conditions of weakly efficient solutions for vector variational inequalities with constraint and vector optimization problems with constraint are obtained.     

Optimality and duality for nonsmooth multiobjective fractional programming with mixed constraints

We consider nonsmooth multiobjective fractional programming problems with inequality and equality constraints. We establish the necessary and sufficient optimality conditions under various generalized invexity assumptions. In addition, we formulate a mixed dual problem corresponding to primal problem, and discuss weak, strong and strict converse duality theorems. This research was partially supported by Project no. 850203 and Center of Excellence for Mathematics, University of Isfahan, Iran.     

Necessary and sufficient optimality conditions for fractional multi-objective problems

Using a special scalarization, we give necessary optimality conditions for fractional multiobjective optimization problems. Under a generalized invexity, sufficient optimality conditions are also given. All over the article, the data are assumed to be continuous but not necessarily Lipschitz.     

The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function

In this paper, we study the Karush–Kuhn–Tucker optimality conditions in a class of nonconvex optimization problems with an interval-valued objective function. Firstly, the concepts of preinvexity and invexity are extended to interval-valued functions. Secondly, several properties of interval-valued preinvex and invex functions are investigated. Thirdly, the KKT optimality conditions are derived for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiablity and Hukuhara differentiablity. Finally, the relationships between a class of variational-like inequalities and the interval-valued optimization problems are established.     

Optimality and Duality for Invex Nonsmooth Multiobjective programming problems

We consider nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, we introduce a Wolfe-type dual and Mond–Weir-type dual and establish duality relations under various (generalized) invexity and regularity conditions.     

Convexificators and boundedness of the Kuhn–Tucker multipliers set

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded.     

Infine Functions, Nonsmooth Alternative Theorems and Vector Optimization Problems

In this paper we introduce a new notion of infine nonsmooth functions and give several characterizations of infineness property. We prove alternative theorems with mixed constraints (i.e., inequality and equality constraints) being described by invex-infine nonsmooth functions. We establish a necessary and sufficient condition for a solution of a vector optimization problem involving mixed constraints to be a properly efficient solution.     

Optimality conditions in nonconvex optimization via weak subdifferentials

In this paper we study optimality conditions for optimization problems described by a special class of directionally differentiable functions. The well-known necessary and sufficient optimality condition of nonsmooth convex optimization, given in the form of variational inequality, is generalized to the nonconvex case by using the notion of weak subdifferentials. The equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented.     

KKT conditions for weak compact convex sets,theorems of the alternative,and optimality conditions

We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak^? compact convex sets. Using them, we extend several existing theorems of the alternative in terms of weak^? compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints. Furthermore, several new equivalent optimality conditions for optimization problems with inequality constraints are obtained.     

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

On necessary and sufficient optimality conditions for multicriterial optimization problems

In this paper we derive first order necessary and sufficient optimality conditions for nonsmooth optimization problems with multiple criteria. These conditions are given for different optimality notions (i.e. weak, Pareto- and proper minimality) and for different types of derivatives of nonsmooth objective functions (locally Lipschitz continuous and quasidifferentiable) mappings. The conditions are given, if possible, in terms of a derivative and a subdifferential of those mappings.     

Banach空间中的一类非光滑齐次优化问题

本文主要研究了一类非光滑齐次优化问题(HOP).通过运用Clarke次微分的广义欧拉恒等式获得了使得(HOP)问题的最优解成为KKT点的充分条件并给出了(HOP)问题与(HOP)问题的KKT点及最优解之间的等价刻画.本文的结果是文[1]中已有结果的推广.文中还举例说明了结果的正确性.