sufficiency and duality in nonsmooth multiobjective optimization involving generalized f α π d type i functions in this paper new classes of generalized f a p d type i functions are introduced for a nonsmooth multiobjective programming problem based upon these generalized functions sufficient optimality conditions are established weak strong and strict converse duality theorems are also derived for wolfe and mond weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems

some nondifferentiable multiobjective programming under generalized d v type i univexity in this paper we introduce new classes of functions called d v type i univex by extending the definition of d v type i functions and consider a multiobjective optimization problem involving generalized d v type i univex functions a number of karush kuhn tucker type sufficient optimality conditions are obtained for a feasible solution to be a weak pareto efficient solution the mond weir type duality results are also presented the results obtained in this paper generalize and extend the previously known result in this area

invex equilibrium problems in this paper we introduce a new class of equilibrium problems known as invex equilibrium problems in the setting of invexity this class of equilibrium problems includes equilibrium problems variational inequalities and variational like inequalities as special cases we use the auxiliary principle technique to suggest and analyze some iterative schemes for solving invex equilibrium problems and study the convergence criteria of these methods under some mild conditions we also consider the concept of well posedness for invex equilibrium problems our results represent significant and important refinements of the previously known results

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

new iteration procedures with errors for multivalued φ strongly pseudocontractive and φ strongly accretive mappings abstract the purpose of this paper is to introduce and study a new class of ishikawa iteration processes with errors for multivalued mappings in real banach spaces some convergence theorems of ishikawa iteration sequences with errors for approximation of the fixed points of multivalued strongly pseudocontractive mappings and approximation of the solutions of nonlinear equations with strongly accretive mappings are obtained respectively our results improve and extend some recent results in this field

relationships between vector variational like inequality and optimization problems abstract in this paper we will establish the relationships between vector variational like inequality and optimization problems we will be able to identify the vector critical points the weakly efficient points and the solutions of the weak vector variational like inequality problem under conditions of pseudo invexity these conditions are more general those existing in the literature

efficiency through variational like inequalities with lipschitz functions in this work first we introduce several notions of invexity and pseudoinvexity for a locally lipschitz function by means of the generalized jacobian we study relationships between these concepts in particular the implications between preinvexity and invexity next we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite dimensional non necessarily pareto vector optimization problems with locally lipschitz objective functions through solutions of vector variational like inequality problems these conditions are stated via the generalized jacobian and under pseudoinvexity hypotheses and they show that a vector optimization problem can be reformulated as a vector variational like inequality problem this work extends and improves several previous papers where the objective function of the vector optimization problem is assumed to be differentiable or being locally lipschitz the authors consider the componentwise subdifferential based on the clarke s generalized gradients of the components of the objective function throughout the paper some simple examples are given in order to illustrate the main concepts and results

generalized mixed quasi variational like inequalities in this paper we use the auxiliary principle technique to suggest and analyze some iterative methods for solving generalized mixed quasi variational like inequalities we show that the convergence of the proposed method requires either pseudomonotonicity or partially relaxed strongly monotonicity under some mild condition as special cases one can obtain a number of new and known algorithms for solving variational inequalities and complementarity problems our results represent significant and important refinements of the previously known results

optimality and invexity in optimization problems in banach algebras spaces abstract 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

p r invex sets and functions abstract notions of invexity of a function and of a set are generalized the notion of an invex function with respect to can be further extended with the aid of p invex sets slight generalization of the notion of p invex sets with respect to leads to a new class of functions a family of real functions called in general p r pre invex functions with respect to without differentiability or p r invex functions with respect to in the differentiable case is introduced some geometric properties of these classes of functions are derived sufficient optimality conditions are obtained for a nonlinear programming problem involving p r invex functions with respect to

second order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization abstract in this paper we introduce a new second order directional derivative and a second order subdifferential of hadamard type for an arbitrary nondifferentiable function we derive second order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second order in unconstrained optimization in particular we obtain two results with strongly pseudoconvex functions we also compare our conditions with the results of the recently published paper bednarik and pastor 2008 and a lot of other works published in high level journals and prove that they are particular cases of our necessary and sufficient ones we prove that the necessary optimality conditions concern more functions than the conditions in terms of lower dini directional derivative even the optimality conditions with the last derivative can be applied to a function which does not belong to some special class at last we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone constrained vector optimization

nonsmooth multiobjective optimization using limiting subdifferentials in this study using the properties of limiting subdifferentials in nonsmooth analysis and regarding a separation theorem some weak pareto optimality necessary and sufficient conditions for nonsmooth multiobjective optimization problems are proved

on optimality conditions for multiobjective optimization problems in topological vector space abstract in this paper we are concerned with a differentiable multiobjective programming problem in topological vector spaces an alternative theorem for generalized k subconvexlike mappings is given this permits the establishment of optimality conditions in this context several generalized fritz john conditions in line to those in hu and ling y hu c ling the generalized optimality conditions of multiobjective programming problem in topological vector space j math anal appl 290 2004 363 372 are obtained and in the presence of the generalized slater s constraint qualification the karush kuhn tucker necessary optimality conditions

hemiequilibrium like problems abstract in this paper we introduce and consider a new class of equilibrium problems known as hemiequilibrium like problems this new class includes hemiequilibrium equilibrium like problems and several classes of variational inequalities as special cases a number of iterative methods for solving hemiequilibrium like problems are suggested and analyzed by using the auxiliary principle technique we also study the convergence analysis of these iterative methods under some mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

pseudolinear fuzzy mappings two classes of fuzzy mappings called pseudolinear and pseudolinear fuzzy mappings are introduced by relaxing the definitions of pseudo convex and pseudo invex fuzzy mappings first some characterizations of pseudolinear and pseudolinear fuzzy mappings are obtained then characterizations of the solution sets of pseudolinear and pseudolinear fuzzy programs are derived

some characterizations of strongly preinvex functions in this paper a new class of generalized convex function is introduced which is called the strongly preinvex function we study some properties of strongly preinvex function in particular we establish the equivalence among the strongly preinvex functions strongly invex functions and strongly monotonicity under some suitable conditions as special cases one can obtain several new and previously known results for preinvex invex functions

on minimax fractional optimality conditions with invexity abstract under different forms of invexity conditions sufficient kuhn tucker conditions and three dual models are presented for the minimax fractional programming

relationships between pre invex concepts abstract it is well known that some properties of convex optimization problems especially sufficient conditions for optimality and duality conditions are preserved when convexity is weakened to pre invexity however there are several distinct pre invex concepts b vex b pre invex pre univex p r pre invex and these are not equivalent the pre invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions

new efficiency conditions for multiobjective interval valued programming problems abstract in this paper we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval valued functions a new concept of fritz john and karush kuhn tucker type points is introduced for this mathematical programming problem based on the gh derivative concept the innovation and importance of these concepts are presented from a practical and computational point of view the problem is approached directly without transforming it into a real valued programming problem thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses we also provide necessary conditions for efficiency which have been inexistent in the relevant literature to date the identification of necessary conditions is important for the development of future computational optimization techniques in an interval valued environment we introduce new generalized convexity notions for gh differentiable interval valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts finally the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper the theoretical developments are illustrated and justified through several numerical examples

higher order duality in multiobjective fractional programming with square root term under generalized higher order f α β p σ d v type i univex functions in this paper a new generalized class of higher order f d v type i univex function is introduced with some examples for a differentiable multiobjective programming mp then multiobjective fractional programming problem mfp is considered in which the numerator and denominator of objective functions contain square root of positive semidefinite quadratic form and the necessary and sufficient conditions for efficient solution for mfp are established under generalized higher order f d v type i univex functions again higher order dual program is proposed for mfp and the duality results are established under generalized of higher order f d v type i univex functions also some computational work has been done to substantiate the analysis

non differentiable minimax fractional programming with generalized α univexity in this paper we study a non differentiable minimax fractional programming problem under the assumption of generalized a univex function in this paper we extend the concept of a invexity m a noor on generalized preinvex functions and monotonicities j inequalities pure appl math 5 2004 1 9 and pseudo a invexity s k mishra m a noor on vector variational like inequality problems j math anal appl 311 2005 69 75 to a univexity and pseudo a univexity from a view point of generalized convexity we also introduce the concept of strict pseudo a univex and quasi a univex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different form of dual problems the results in this paper extend a few known results in the literature

on p r invexity type nonlinear programming problems abstract mathematical programming problems involving p r invex functions with respect to are considered we introduce new classes of nonlinear programming problems called kt p r invex wd p r invex and hc p r invex problems where p r are some real numbers it is shown that for these types of problems kuhn tucker conditions are both necessary and sufficient for optimality furthermore these p r invexity type problems with r 0 are not sufficient for wolfe weak duality in this way it was shown that the optimization problems possessing some kind of invexity need not be equivalent to the class of optimization problems for which kuhn tucker necessary conditions for optimality are also sufficient and wolfe weak duality holds

a characterization of pseudoinvexity for the efficiency in non differentiable multiobjective problems duality abstract we prove that in order for the kuhn tucker or fritz john points to be efficient solutions it is necessary and sufficient that the non differentiable multiobjective problem functions belong to new classes of functions that we introduce here kt pseudoinvex ii or fj pseudoinvex ii respectively we illustrate it by examples these characterizations generalize recent results given for the differentiable case we study the dual problem and establish weak strong and converse duality results

symmetric duality for a class of nonlinear fractional programming problems abstract a pair of symmetric dual nonlinear fractional programming problems is presented and duality theorems are established under pseudoinvexity pseudoincavity and invexity incavity respectively type assumptions on the kernel function special cases are particularly discussed to show that this paper extends some work appeared in this area

kuhn tucker sufficiency for global minimum of multi extremal mathematical programming problems abstract the kuhn tucker sufficiency theorem states that a feasible point that satisfies the kuhn tucker conditions is a global minimizer for a convex programming problem for which a local minimizer is global in this paper we present new kuhn tucker sufficiency conditions for possibly multi extremal nonconvex mathematical programming problems which may have many local minimizers that are not global we derive the sufficiency conditions by first constructing weighted sum of square underestimators of the objective function and then by characterizing the global optimality of the underestimators as a consequence we derive easily verifiable kuhn tucker sufficient conditions for general quadratic programming problems with equality and inequality constraints numerical examples are given to illustrate the significance of our criteria for multi extremal problems

second order symmetric duality in multiobjective programming involving generalized cone invex functions abstract in this paper cone second order pseudo invex and strongly cone second order pseudo invex functions are defined a pair of mond weir type second order symmetric dual multiobjective programs is formulated over arbitrary cones weak strong and converse duality theorems are established under aforesaid generalized invexity assumptions a second self duality theorem is also given by assuming the functions involved to be skew symmetric

hemivariational like inequalities in this paper we introduce and consider a new class of variational inequalities known as the hemivariational like inequalities it is shown that the hemivariational like inequalities include hemivariational inequalities variational like inequalities and the classical variational inequalities as special cases the auxiliary principle is used to suggest and analyze some iterative methods for solving hemivariational like inequalities under mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

on variational problems characterization of solutions and duality in this paper we introduce a new class of pseudoinvex functions for variational problems using this new concept we obtain a necessary and sufficient condition for a critical point of the variational problem to be an optimal solution illustrated with an example also weak strong and converse duality are established

a note on invex problems with nonnegative variable we provide sufficient optimality conditions involving invexity for mathematical programming problems with nonnegative variable we also show that upon certain assumptions on the scaling function these problems are convex transformable hence certain associated convex problems should be solved some applications are shown

characterization of efficient solutions of multi objective e convex programming problems in this paper we shall interest with characterization of efficient solutions for special classes of problems these classes are called e convex programming problems in which the decision spaces effected by an operator e duality gap in multi objective programming problems can t be occur in our considered class in the end of paper an illustrative example was presented

g pre invex functions in mathematical programming in the present paper we introduce the concept of g pre invex functions with respect to h defined on an invex set with respect to h 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

existence and iterative approximations of solutions for mixed quasi variational like inequalities in banach spaces abstract in this paper we introduce and investigate a new class of mixed quasi variational like inequalities in reflexive banach spaces by applying a minimax inequality due to ding tan and a lemma due to chang we establish some existence and uniqueness results of solution for the mixed quasi variational like inequality next by using a kkm theorem due to fan and an auxiliary principle technique due to cohen we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms our results extend improve and unify several known results in the literature

necessary and sufficient conditions in constrained optimization additional conditions are attached to the kuhn tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization under appropriate constraint qualifications necessary and sufficient conditions are also given for optimality of the dual problem duality and converse duality are treated accordingly

optimality conditions for an isolated minimum of order two in c1 constrained optimization abstract in this paper we obtain first and second order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint first order sufficient conditions are derived in terms of generalized convex functions in the necessary conditions we suppose that the data are continuously differentiable a notion of strongly kt invex inequality constrained problem is introduced it is shown that each kuhn tucker point is an isolated global minimizer of order two if and only if the problem is strongly kt invex the article could be considered as a continuation of i ginchev v i ivanov second order optimality conditions for problems with c 1 data j math anal appl 340 2008 646 657

exact penalty functions method for mathematical programming problems involving invex functions in this paper some new results on the exact penalty function method are presented simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method the equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption furthermore the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved

sufficiency and duality in nondifferentiable multiobjective programming involving generalized type i functions abstract recently hachimi and aghezzaf defined generalized f d type i functions a new class of functions that unifies several concepts of generalized type i functions in this paper the generalized f d type i functions are extended to nondifferentiable functions by utilizing the new concepts we obtain several sufficient optimality conditions and prove mixed type and mond weir type duality results for the nondifferentiable multiobjective programming problem

nonsmooth invex functions and sufficient optimality conditions in this paper the various definitions of nonsmooth invex functions are gathered in a general scheme by means of the concept of k directional derivative characterizations of nonsmooth k invexity are derived as well as results concerning constrained optimization without any assumption of convexity of the k directional derivatives

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

nondifferentiable multiobjective programming under generalized di invexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce new concepts of d i invexity and generalized d i invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction d i new fritz john type necessary and karush kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be weakly efficient efficient or properly efficient moreover we prove weak strong converse and strict duality results for a mond weir type dual under various types of generalized d i invexity assumptions

duality with invexity for a class of nondifferentiable static and continuous programming problems abstract the concept of invexity has allowed the convexity requirements in a variety of mathematical programming problems to be weakened we extend the duality theorems for a class of static nondifferentiable problems with wolfe type and mond weir type duals and further extend these for the continuous analogues

symmetric duality for multiobjective variational problems with invexity abstract we formulate a pair of multiobjective nonlinear symmetric dual variational problems for the single objective problems our problems become the symmetric dual pair of i smart and b mond j math anal appl 152 1990 536 545 under invexity assumptions we establish the weak strong converse and self duality theorems for our variational problems by using the concept of efficiency also we give the static case of our symmetric duality results

wolfe type duality involving b η invex functions for a minmax programming problem abstract a sufficient optimality theorem is proved for a certain minmax programming problem under the assumptions of proper b invexity conditions on the functions involved in the objective and in the constraints next a dual is presented for such a problem and duality theorems relating the primal and the dual are proved the dual for a minimax generalized fractional programming problem is presented as special case of the main problem considered in the paper

mixed type duality for nondifferentiable multiobjective fractional programming under generalized d ρ φ θ type 1 univex function in this paper we introduced a new generalized class of d r h q type 1 univex function with an example we introduced the mixed duality problem mfxd of the nondifferentiable multiobjective fractional programming problem mfp where the objective and constraint functions involved are directionally differentiable also we established the duality results under generalized class of d r h q type 1 univex function

optimality and duality for generalized fractional programming involving nonsmooth f ρ convex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing f r convex functions

on fuzzy generalized convex mappings and optimality conditions for fuzzy weakly univex mappings in this paper we first introduce the weakly invex fuzzy mappings based on weakly differentiable functions and discuss the relationships between several kinds of fuzzy generalized convex mappings then a kind of fuzzy weakly univex functions are introduced and some properties of them are investigated furthermore optimality and duality results are derived for a class of generalized convex optimization problems with fuzzy weakly univex functions

nondifferentiable multiobjective programming under generalized d univexity abstract 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

weakly efficient solutions and pseudoinvexity in multiobjective control problems abstract in this paper we introduce new pseudoinvexity conditions on functionals involved in a multiobjective control problem called w kt pseudoinvexity and w fj pseudoinvexity we prove that all kuhn tucker or fritz john points are weakly efficient solutions if and only if these conditions are fulfilled we relate weakly efficient solutions to optimal solutions of weighting control problems we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

on sufficiency and duality for nonsmooth multiobjective programming problems involving generalized v r invex functions abstract in this paper a class of nonsmooth multiobjective programming problems with inequality constraints is considered we introduce the concepts of v r pseudo invex strictly v r pseudo invex and v r quasi invex functions in which the involved functions are locally lipschitz based upon these generalized v r invex functions sufficient optimality conditions for a feasible point to be an efficient or a weakly efficient solution are derived appropriate duality theorems are proved for a mond weir type dual program of a nonsmooth multiobjective programming under the aforesaid functions

multi objective infinite programming using the theory of abstract optimization problems in infinite dimensional spaces we provide necessary optimality conditions of first and second order for weakly efficient solutions of the multi objective infinite programming problem sufficient conditions are given under invexity assumptions we generalize the notion of kkt invexity for the multi objective infinite problem and show that this notion is a necessary and sufficient condition for every vector kkt solution to be a weakly efficient solution moreover we develop a theorem of the alternative useful for proving some of our results

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

optimality and duality in nonsmooth multiobjective optimization involving generalized type i functions abstract a nonsmooth multiobjective optimization problem involving generalized type i vectorvalued functions is considered karush kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be an efficient or properly efficient solution duality theorems are proved for wolfe type and mond weir type duals under the generalized type i assumptions

multiobjective programming under d invexity abstract 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

the generalized quasi variational inequality problem with applications a new problem called the generalized quasi variational inequality problem is introduced this new problem extends all the existing variational inequality problems in finite dimensional spaces and enlarges the class of problems that can be approached by the variational inequality theory some existence results without convexity assumptions are presented applications to mathematical and equilibrium programming are given

optimality and duality in nondifferentiable multiobjective optimization involvingd type i and related functions abstract in this paper new classes of functions namely d type i d quasitype i and d pseudo type i are defined for a multiobjective nondifferentiable programming problem kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be a weak minimum for this problem two duals are formulated and various duality results are given by using the above defined classes of functions considering the concept of a weak minimum

generalized type i invexity and duality in multiobjective variational problems abstract in this paper we introduce new classes of generalized v type i invex functions for variational problems and consider a multiobjective variational problem vp a number of sufficiency results are established using lagrange multiplier conditions under various types of generalized v type i invexity requirements duality results are obtained for mond weir type duals under the above generalized v type i invexity assumptions and their generalizations

minimax fractional programming under nonsmooth generalized f ρ θ d univexity abstract in this paper we are concerned with a nondifferentiable minimax fractional problem with inequality constraints we introduce a new class of generalized convex function that is nonsmooth generalized f d univex function in the framework of the new concept we derive kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different types of dual problems

invariant monotone vector fields on riemannian manifolds abstract various concepts of invariant monotone vector fields on riemannian manifolds are introduced some examples of invariant monotone vector fields are given several notions of invexities for functions on riemannian manifolds are defined and their relations with invariant monotone vector fields are studied

generalized α invexity and nondifferentiable minimax fractional programming in this paper we study a nondifferentiable minimax fractional programming problem under the assumptions of a invex function in this paper we utilize the concept of a invexity m a noor on generalized preinvex functions and monotonicities j inequalities pure appl math 5 2004 1 9 and pseudo a invexity s k mishra m a noor on vector variational like inequality problems j math anal appl 311 2005 69 75 we also introduce the concept of strict pseudo a invex and quasi a invex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different dual problems the results in this paper extend several known results in the literature

the gap function for optimization problems in banach spaces abstract in this paper some properties of the gap function for multiple objective optimization problems in banach spaces are established

on ratio invexity in mathematical programming abstract the nonlinear fractional programming problem is considered the functions involved in the objective function and constraints are assumed to be invex and differentiable it is shown that the ratio of invex functions is invex sufficient optimality and duality theorems are presented for an invex fractional programming problem

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

d ρ η θ invexity in multiobjective optimization abstract in this paper a generalization of convexity is considered in the case of multiobjective optimization problems where the functions involved are non differentiable under d invexity assumptions on the functions involved weak strong and converse duality results are proved to relate weak pareto efficient solutions of the multiobjective programming problems pvp dvp and mwd we have also established the karush kuhn tucker sufficient optimality condition

on properties of preinvex functions abstract we consider in this paper the class of preinvex functions introduced by t weir and co workers 1988 j math anal appl 136 29 38 1988 bull austral math soc 38 177 189 under semi continuity conditions a determination of the satisfaction of preinvexity for a function can be achieved via an intermediate point preinvexity check a characterization of a preinvex function in terms of its relationship with an intermediate point preinvexity and prequasi invexity is provided

kt invex control problem abstract in this paper we introduce a new condition on functions of a control problem for which we define a kt invex control problem we prove that a kt invex control problem is characterized in order that a kuhn tucker point is an optimal solution we generalize optimality results of known mathematical programming problems we illustrate these results with examples

optimality conditions and duality for nondifferentiable multiobjective programming problems involving d r type i functions in 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

necessary and sufficient conditions for fuzzy optimality problems in this paper we define a new minimum concept for fuzzy optimization problems more general than those that exist in the literature we find necessary optimality conditions based on a new fuzzy stationary point definition and we prove that these conditions are also sufficient under new fuzzy generalized convexity notions