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Cristinca Fulga, V. Preda (2009)
Nonlinear programming with E-preinvex and local E-preinvex functionsEur. J. Oper. Res., 192
E. Youness (1999)
E-Convex Sets, E-Convex Functions, and E-Convex ProgrammingJournal of Optimization Theory and Applications, 102
S. Jaiswal, G. Panda (2010)
Duality results using higher order generalised E-invex functionsInt. J. Comput. Sci. Math., 3
Xinmin Yang (2001)
On E-Convex Sets, E-Convex Functions, and E-Convex ProgrammingJournal of Optimization Theory and Applications, 109
D. Duca, L. Lupsa (2006)
On the E-Epigraph of an E-Convex FunctionJournal of Optimization Theory and Applications, 129
Yu-Ru Syau, E. Lee (2005)
Some properties of E-convex functionsAppl. Math. Lett., 18
Xiusu Chen (2002)
Some properties of semi-E-convex functionsJournal of Mathematical Analysis and Applications, 275
Hindawi Publishing Corporation Advances in Operations Research Volume 2012, Article ID 175176, 11 pages doi:10.1155/2012/175176 Research Article Generalized Differentiable E-Invex Functions and Their Applications in Optimization S. Jaiswal and G. Panda Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India Correspondence should be addressed to G. Panda, geetanjali@maths.iitkgp.ernet.in Received 9 December 2011; Accepted 4 September 2012 Academic Editor: Chandal Nahak Copyright q 2012 S. Jaiswal and G. Panda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The concept of E-convex function and its generalizations is studied with differentiability assump- tion. Generalized differentiable E-convexity and generalized differentiable E-invexity are used to derive the existence of optimal solution of a general optimization problem. 1. Introduction E-convex function was introduced by Youness 1 and revised by Yang 2. Chen 3 intro- duced Semi-E-convex function and studied some of its properties. Syau and Lee 4 defined E-quasi-convex function, strictly E-quasi-convex function and studied some basic properties. Fulga and Preda 5 introduced the class of E-preinvex and E-prequasi-invex functions. All the above E-convex and generalized E-convex functions are defined without differentiability assumptions. Since last few decades, generalized convex functions like quasiconvex, pseudoconvex, invex, B-vex, p, r-invex, and so forth, have been used in nonlinear programming to derive the sufficient optimality condition for the existence of local optimal point. Motivated by earlier works on convexity and E-convexity, we have introduced the concept of differentiable E-convex function and its generalizations to derive sufficient optimality condition for the existence of local optimal solution of a nonlinear programming problem. Some preliminary definitions and results regarding E-convex function are discussed below, which will be needed in the sequel. Throughout this paper, we consider functions n n n E : R → R , f : M → R,and M are nonempty subset of R . Definition 1.1 see 1. M is said to be E-convex set if 1 − λEx λEy ∈ M for x, y ∈ M, λ ∈ 0, 1. 2 Advances in Operations Research Definition 1.2 see 1. f : M → R is said to be E-convex on M if M is an E-convex set and for all x, y ∈ M and λ ∈ 0, 1, f 1 − λEx λE y ≤ 1 − λfEx λf E y . 1.1 Definition 1.3 see 3.Let M be an E-convex set. f is said to be semi-E-convex on M if for x, y ∈ M and λ ∈ 0, 1, f λEx 1 − λE y ≤ λfx 1 − λf y . 1.2 n n n Definition 1.4 see 5. M is said to be E-invex with respect to η : R ×R → R if for x, y ∈ M and λ ∈ 0, 1, Ey ληEx,Ey ∈ M. n n n Definition 1.5 see 6.Let M be an E-invex set with respect to η : R × R → R .Also f : M → R is said to be E-preinvex with respect to η on M if for x, y ∈ M and λ ∈ 0, 1, f E y λη E x ,E y ≤ λf E x 1 − λ f E y . 1.3 n n n Definition 1.6 see 7.Let M be an E-invex set with respect to η : R × R → R .Also f : M → R is said to be semi-E-invex with respect to η at y ∈ M if f E y λη Ex,E y ≤ λfx 1 − λf y 1.4 for all x ∈ M and λ ∈ 0, 1. n n Definition 1.7 see 7.Let M be a nonempty E-invex subset of R with respect to η : R × n n n n n R → R , E : R → R .Let f : M → R and EM be an open set in R . Also f and E are differentiable on M. Then, f is said to be semi-E-quasiinvex at y ∈ M if fx ≤ f y ∀x ∈ M ⇒ ∇ f ◦ E y η Ex,E y ≤ 0, 1.5 or 1.6 ∇ f ◦ E y η Ex,E y > 0 ∀x ∈ M ⇒ fx >f y . Lemma 1.8 see 1. If a set M ⊆ R is E-convex, then EM ⊆ M. Lemma 1.9 see 5. If M is E-invex, then EM ⊆ M. Lemma 1.10 see 5. If {M } is a collection of E-invex sets and M ⊆ R , for all i ∈ I,then i i i∈I ∩ M is E-invex. i∈I i Advances in Operations Research 3 2. E-Convexity and Its Generalizations with Differentiability Assumption E-convexity and convexity are different from each other in several contests. From the pre- vious results on E-convex functions, as discussed by our predecessors, one can observe the following relations between E-convexity and convexity. 1 All convex functions are E-convex but all E-convex functions are not necessarily convex. In particular, E-convex function reduces to convex function in case Ex x for all x in the domain of E. n n 2 A real-valued function on R may not be convex on a subset of R ,but E-convex on that set. 3 An E-convex function may not be convex on a set M but E-convex on EM. 4 It is not necessarily true that if M is an E-convex set then EM is a convex set. In this section we study E-convex and generalized E-convex functions with differen- tiability assumption. 2.1. Some New Results on E-Convexity with Differentiability E-convexity at a point may be interpreted as follows. n n n Let M be a nonempty subset of R , E : R → R .Afunction f : M → R is said to be E-convex at x ∈ M if M is an E-convex set and fλEx 1 − λEx ≤ λ f ◦ E x 1 − λ f ◦ E x 2.1 for all x ∈ N x and λ ∈ 0, 1, where N x is δ-neighborhood of x, for small δ> 0. δ δ It may be observed that a function may not be convex at a point but E-convex at that point with a suitable mapping E. 2 2 2 Example 2.1. Consider M {x, y ∈ R | y ≥ 0}. E : R → R is Ex, y 0,y and 3 2 fx, y x y .Also f is not convex at −1, 1. For all x, y ∈ N −1, 1, δ> 0, and λ ∈ 0, 1,fλEx, y1−λE−1, 1−λf◦Ex, y−1−λf◦E−1, 1 −λ1−λy−1 ≤ 0. Hence, f is E-convex at −1, 1. n n n Proposition 2.2. Let M ⊆ R , E : R → R be a homeomorphism. If f : M → R attains a local minimum point in the neighborhood of Ex,thenitis E-convex at x. Proof. Suppose f has a local minimum point in a neighborhood N Ex of Ex for some x ∈ M, > 0. This implies f is convex on N Ex.Thatis, fλz 1 − λEx ≤ λfz 1 − λfEx ∀z ∈ N Ex. 2.2 n n Since E : R → R is a homeomorphism, so inverse of the neighborhood N Ex is a neighborhood of x say N x for some δ> 0. Hence, there exists x ∈ N x such that Ex δ δ z, Ex ∈ N Ex. Replacing z by Ex in the above inequality, we conclude that f is E- convex at x. 4 Advances in Operations Research In the above discussion, it is clear that if a local minimum exists in a neighborhood of Ex, then f is E-convex at x. But it is not necessarily true that if f is E-convex at x then Ex is local minimum point. Consider the above example where f is E-convex at −1, 1 but E−1, 1 is not local minimum point of f. Theorem 2.3. Let M be an open E-convex subset of R , f and E are differentiable functions, and let E be a homeomorphism. Then, f is E-convex at x ∈ M if and only if 2.3 f ◦ E x ≥ f ◦ E x ∇ f ◦ E x Ex − Ex for all Ex ∈ N Ex where N Ex is -neighborhood of Ex, > 0. Proof. Since M is an E-convex set, by Lemma 1.8, EM ⊆ M.Also, EM is an open set as E is a homeomorphism. Hence, there exists > 0 such that Ex ∈ N Ex for all x ∈ N x, δ> 0, very small. So, f is differentiable on EM. Using expansion of f at Ex in the neighborhood N Ex, fEx λz − Ex fEx λ∇fEx z − Ex 2.4 αEx,λz − Exλ z − Ex , where z ∈ N Ex and lim αEx,λz − Ex 0. Since f is E-convex at x ∈ M,sofor λ → 0 all x ∈ N x, λ ∈ 0, 1, x x, λ f ◦ E x − f ◦ E x ≥ fEx λEx − Ex − f ◦ E x. 2.5 Since E is a homeomorphism, there exists x ∈ N x such that Ex z. Replacing z by Ex in 2.4 and using above inequality, we get f ◦ E x − f ◦ E x ≥ ∇ f ◦ E x Ex − Ex 2.6 αEx,λEx − Ex Ex − Ex , where lim αEx,λEx − Ex 0. Hence, 2.3 follows. λ → 0 The converse part follows directly from 2.4. It is obvious that if Ex is a local minimum point of f, then ∇f ◦ Ex 0. The following result proves the sufficient part for the existence of local optimal solution, proof of which is easy and straightforward. We leave this to the reader. Corollary 2.4. Let M ⊆ R be an open E-convex set, and let f be a differentiable E-convex function n n at x.If E : R → R is a homeomorphism and ∇f ◦ Ex 0,then Ex is the local minimum of f. Advances in Operations Research 5 2.2. Some New Results on Generalized E-Convexity with Differentiability Here, we introduce some generalizations of E-convex function like semi-E-convex, E-invex, semi-E-invex, E-pseudoinvex, E-quasi-invex and so forth, with differentiability assumption and discuss their properties. 2.2.1. Semi-E-Convex Function Chen 3 introduced a new class of semi-E-convex functions without differentiability assumption. Semi-E-convexity at a point may be understood as follows: f : M → R is said to be semi-E-convex at x ∈ M if M is an E-convex set and fλEx 1 − λEx ≤ λfx 1 − λfx 2.7 for all x ∈ N x and λ ∈ 0, 1, where N x is δ-neighborhood of x. δ δ The following result proves the necessary and sufficient condition for the existence of asemi-E-convex function at a point. n n Theorem 2.5. Suppose f : M → R and E : R → R are differentiable functions. Let E be a homeomorphism and let x be a fixed point of E. Then, f is semi-E-convex at x ∈ M if and only if 2.8 fx ≥ fx ∇ f ◦ E x Ex − Ex for all Ex ∈ N Ex, very small > 0. Proof. Proceeding as in Theorem 2.3, we get the following relation from the expansion of f at Ex in the neighborhood N Ex, where x is the fixed point of E. Since E is a homeomorphism, there exists > 0 such that Ex ∈ N Ex for all x ∈ N x, very small δ> 0: fEx λEx − Ex f ◦ E x λ ∇ f ◦ E x Ex − Ex 2.9 αEx,λEx − Exλ Ex − Ex , where Ex ∈ N Ex, lim αEx,λEx − Ex 0. Since f is semi-E-convex at λ → 0 x ∈ M,and x is a fixed point of E,so, forall x ∈ N x, λ ∈ 0, 1, x x, δ / λ fx − fx ≥ fEx λEx − Ex − f ◦ E x. 2.10 Using 2.9, the above inequality reduces to fx − fx ≥ ∇ f ◦ E x Ex − Ex 2.11 α E x ,λ E x − E x E x − E x , 6 Advances in Operations Research where lim αEx,λEx − Ex 0. Hence Inequality 2.8 follows for all Ex ∈ λ → 0 N Ex. Conversely, suppose Inequality 2.8 holds at the fixed point x of E for all Ex ∈ N Ex.Using 2.9 and Ex x in 2.8, we can conclude that f is semi-E-convex at x ∈ M. 2.2.2. Generalized E-Invex Function The class of preinvex functions defined by Ben-Israel and Mond is not necessarily differen- tiable. Preinvexity, for the differential case, is a sufficient condition for invexity. Indeed, the converse is not generally true. Fulga and Preda 5 defined E-invex set, E-preinvex function, and E-prequasiinvex function where differentiability is not required Section 1. Chen 3 introduced semi-E-convex, semi-E-quasiconvex, and semi-E-pseudoconvex functions without differentiability assumption. Jaiswal and Panda 7 studied some generalized E- invex functions and applied these concepts to study primal dual relations. Here, we define some more generalized E-invex functions with and without differentiability assumption, which will be needed in next section. First, we see the following lemma. n n n n Lemma 2.6. Let M be a nonempty E-invex subset of R with respect to η : R × R → R .Also f : M → R are differentiable on M. EM is an open set in R .If f is E-preinvex on M then f ◦ Ex ≥ f ◦ Ey∇f ◦ Ey ηEx,Ey for all x, y ∈ M. Proof. If EM is an open set, f and E are differentiable on M, then f ◦ E is differentiable on M. From Taylor’s expansion of f at Ey for some y ∈ M and λ> 0, f E y λη Ex,E y f ◦ E y λ ∇ f ◦ E y η Ex,E y 2.12 λ η Ex,E y α E y ,λη Ex,E y , where Ex Ey, lim αEy,ληEx,Ey 0. / λ → 0 If f is E-preinvex on M with respect to η Definition 1.5, then as λ → 0 , the above inequality reduces to f ◦Ex ≥ f ◦Ey∇f ◦Ey ηEx,Ey for all x, y ∈ M. As a consequence of the above lemma, we may define E-invexity with differentiability assumption as follows. n n n n with respect to η : R × R → R . Definition 2.7. Let M be a nonempty E-invex subset of R Also f : M → R are differentiable on M. EM is an open set in R . Then, f is E-invex on M if f ◦ Ex ≥ f ◦ Ey∇f ◦ Ey ηEx,Ey for all x, y ∈ M. From the above discussions on E-invexity and E-preinvexity, it is true that E-preinvexity with differentiability is a sufficient condition for E-invexity. Also a function which is not E-convex may be E-invex with respect to some η. This may be verified in the following example. Advances in Operations Research 7 2 2 2 Example 2.8. M {x, y ∈ R | x, y > 0}, E : R → R is Ex, y 0,y and f : M → R is 2 2 defined by fx, y −x − y ,and 2 2 x y 1 1 ⎪ , ,x 0,y 0, / / 2 2 2x 2y 2 2 ⎪ y ⎪ 1 0, ,x 0,y 0, ⎨ 2 2 / 2y η x ,y , x ,y 2.13 1 1 2 2 ⎪ 2 ⎪ , 0 ,x 0,y 0, 2 / 2 2x ⎪ 2 0, 0 , otherwise. n n n n Definition 2.9. Let M be a nonempty E-invex subset of R with respect to η : R × R → R , let EM be an open set in R . Suppose f and E are differentiable on M. Then, f is said to be E-quasiinvex on M if f ◦ E x ≤ f ◦ E y ∀x, y ∈ M ⇒ ∇ f ◦ E y η Ex,E y ≤ 0 2.14 or 2.15 ∇ f ◦ E y η Ex,E y > 0 ⇒ f ◦ E x > f ◦ E y . A function may not be E-invex with respect to some η but E-quasiinvex with respect to same η. This may be justified in the following example. 2 2 2 Example 2.10. Consider M {x, y ∈ R | x, y < 0}, E : R → R is Ex, y 0,y,and f : 3 3 2 2 2 M → R is fx, y x y , η : R ×R → R is ηx ,y , x ,y x −x ,y −y .Now for 1 1 2 2 1 2 1 2 all x ,y , x ,y ∈ M, f ◦Ex ,y −f ◦Ex ,y −∇f ◦Ex ,y ηEx ,y ,Ex ,y 1 1 2 2 1 1 2 2 2 2 1 1 2 2 3 3 2 y y − 3y y − y , which is not always positive. Hence, f is not E-invex with respect to 1 2 2 2 η on M. If we assume that f ◦ Ex ,y ≤ f ◦ Ex ,y for all x ,y , x ,y ∈ M, then 1 1 2 2 1 1 2 2 ∇f ◦ Ex ,y ηEx ,y ,Ex ,y 3y y − y ≤ 0. Hence, f is E-quasiinvex with 2 2 1 1 2 2 1 2 respect to same η on M. n n n n Definition 2.11. Let M be a nonempty E-invex subset of R with respect to η : R × R → R , let EM be an open set in R . Suppose f and E are differentiable on M. Then, f is said to be E-pseudoinvex on M if 2.16 ∇ f ◦ E y η Ex,E y ≥ 0 ∀x, y ∈ M ⇒ f ◦ E x ≥ f ◦ E y or f ◦ E x < f ◦ E y ∀x, y ∈ M ⇒ ∇ f ◦ E y η E x ,E y < 0. 2.17 8 Advances in Operations Research A function may not be E-invex with respect to some η but E-pseudoinvex with respect to same η. This can be verified in the following example. 2 2 2 Example 2.12. Consider M {x, y ∈ R | x, y > 0}. E : R → R is Ex, y 0,y and 2 2 2 2 2 f : M → R is fx, y −x − y . For η : R × R → R , defined by ηx ,y , x ,y 1 1 2 2 x − x ,y − y ,and forall x ,y , x ,y ∈ M, y y , f ◦ Ex ,y − f ◦ Ex ,y − 1 2 1 2 1 1 2 2 1 2 1 1 2 2 T 2 ∇f ◦ Ex ,y ηEx ,y ,Ex ,y −y − y < 0. Hence, f is not E-invex with respect 2 2 1 1 2 2 2 1 to η on M.If ∇f ◦Ex ,y ηEx ,y ,Ex ,y ≥ 0, then f ◦Ex ,y −f ◦Ex ,y − 2 2 1 1 2 2 1 1 2 2 f0,y − f0,y y y y − y ≥ 0. Hence, f is E-pseudoinvex with respect to η on M. 1 2 2 1 2 1 If a function f : M → R is semi-E-invex with respect to η at each point of an E-invex set M, then f is said to be semi-E-invex with respect to η on M.Semi-E-invex functions and some of its generalizations are studied in 7. Here, we discuss some more results on generalized semi-E-invex functions. Proposition 2.13. If f : M → R is semi-E-invex on an E-invex set M,then fEy ≤ fy for each y ∈ M. Proof. Since f is semi-E-invex on M ⊆ R and M is an E-invex set so for x, y ∈ M and λ ∈ 0, 1, we have EyληEx,Ey ∈ M and fEyληEx,Ey ≤ λfx1−λfy. In particular, for λ 0, fEy ≤ fy for each y ∈ M. An E-invex function with respect to some η may not be semi-E-invex with respect to same η may be verified in the following example. Example 2.14. Consider the previous example where M {x, y ∈ R | x, y < 0}, 2 2 3 3 2 2 2 E : R → R is Ex, y 0,y and f : M → R is fx, y x y ,η : R × R → R is ηx ,y , x ,y x − x ,y − y . It is verified that f is E-invex with respect to η on 1 1 2 2 1 2 1 2 M.But fE2, 0 >f2, 0. From Proposition 2.13 it can be concluded that f is not semi- E-invex with respect to same η. Also, using Definition 1.6, for all x ,y , x ,y ∈ M, λ ∈ 1 1 2 2 0, 1,fEx ,y ληEx ,y ,Ex ,y − λfx ,y − 1 − λfx ,y −y λy /2y 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 λx y 1 − λx y , which is not always negative. Hence, f is not semi-E-invex with 1 1 2 2 respect to η on M. 3. Application in Optimization Problem In this section, the results of previous section are used to derive the sufficient optimality condition for the existence of solution of a general nonlinear programming problem. Consider a nonlinear programming problem P min fx 3.1 subject to gx ≤ 0, m n where f : M → R, g : M → R , M ⊆ R ,g g ,g ,...,g . M {x ∈ M : g x ≤ 0,i i 1 2 m i 1,...,m} is the set of feasible solutions. Theorem 3.1 sufficient optimality condition. Let M be a nonempty open E-convex subset of n m n n R ,f : M → R, g : M → R , and E : R → R are differentiable functions. Let E be Advances in Operations Research 9 a homeomorphism and let x be a fixed point of E.If f and g are semi-E-convex at x ∈ M and x, y ∈ M × R satisfies ∇ f ◦ E x y, g ◦ E x 0, 3.2 y, g ◦ E x 0,y ≥ 0, then x is local optimal solution of P. Proof. Since f and g are semi-E-convex at x ∈ M by Theorem 2.5, fx − fx ≥ ∇ f ◦ E x Ex − Ex ∀Ex ∈ N Ex, 3.3 gx − gx ≥ ∇ g ◦ E x Ex − Ex ∀Ex ∈ N Ex. Adding the above two inequalities, we have T T 3.4 fx − fx y gx − gx ≥∇ f ◦ E x y g ◦ E x Ex − Ex. If 3.2 hold, then fx − fx y gx ≥ 0 for all Ex ∈ N Ex. Since gx ≤ 0for all x ∈ M and y ≥ 0, so y gx ≤ 0. Hence, fx − fx ≥ 0 for all Ex ∈ N Ex. Since E is a homeomorphism, there exists δ> 0 such that x ∈ N x for all Ex ∈ N Ex, which means fx ≥ fx for all x ∈ N x. Hence, x is a local optimal solution of P. Also we see that a fixed point of E is a local optimal solution of P under generalized E-invexity assumptions. n n n n Lemma 3.2. Let M be a nonempty E-invex subset of R with respect to some η : R × R → R .Let g : M → R, i 1,...,m be semi-E-quasiinvex functions with respect to η on M. Then, M is an E-invex set. Proof. Let M {x ∈ M : g x ≤ 0},i 1,...,m. M ∩ M and M ⊆ M. Since g , i i i i i 1 i 1,...,m are semi-E-quasiinvex function on M, so for all x, y ∈ M and λ ∈ 0, 1,g Ey i i ληEx,Ey ≤ max{g x,g y}≤ 0. Hence, EyληEx,Ey ∈ M for all x, y ∈ M . i i i i So M is E-invex with respect to same η. From Lemma 1.10, M ∩ M is E-invex with i i i 1 respect to same η. n n n n Corollary 3.3. Let M be a nonempty E-invex subset of R with respect to some η : R × R → R . Let g : M → R, i 1,...,m, be semi-E-quasiinvex functions with respect to η on M.If x is a feasible solution of P,then Ex is also a feasible solution of P. Proof. Since x is a feasible solution of P,so x ∈ M ⇒ Ex ∈ EM . Since each g , i 1,...,m is semi-E-quasiinvex function on M, from Lemma 3.2, M is an E-invex set. Also EM ⊆ M . Hence, Ex ∈ M .Thatis, Ex is a feasible solution of P. Theorem 3.4 sufficient optimality condition. Let M be a nonempty E-invex subset of R with n n n n n n m respect to η : R × R → R .Let EM be an open set in R . Suppose f : R → R, g : R → R 10 Advances in Operations Research and E are differentiable functions on M.If f is E-pseudoinvex function with respect to η and for u ≥ 0,u g is semi-E-quasiinvex function with respect to the same η at x ∈ M ,where x is a fixed point of the map E and x, u ∈ M × R ,u ≥ 0 satisfies the following system: ∇ f ◦ E x u, g ◦ E x 0, 3.5 u, g ◦ E x 0, 3.6 then x is a local optimal solution of P. Proof. Suppose x, u ∈ M × R satisfies 3.5 and 3.6. For all y ∈ M , gy ≤ 0. Also, u ≥ 0. T T Hence, u gy ≤ 0 for all y ∈ M .From 3.6, u, g ◦ Ex 0, that is, u gEx 0. x is a fixed point of E that is Ex x.So u gx 0. Hence, T T u g y ≤ u gx ∀y ∈ M . 3.7 Since u ≥ 0and u g is semi-E-quasiinvex function with respect to η at x, so the above inequal- ity implies ∇u gExη E y ,Ex ≤ 0. 3.8 From 3.5, ∇fEx −∇u gEx. Putting this value in the above inequality, we have ∇fExηEx,Ey ≥ 0. f is E-pseudoinvex at x with respect to η. Hence, ∇fExηEx,Ey ≥ 0 implies f E y ≥ fEx fx ∀y ∈ M . 3.9 Hence, x is the optimal solution P on EM . The following example justifies the above theorem. Example 3.5. Consider the optimization problem, 2 2 P min −x − y 3.10 2 2 subject to x y − 4 ≤ 0, 2 2 2 where M {x, y ∈ R | x, y > 0}. E : R → R is Ex, y 0,y. This is not a convex 2 2 2 programming problem. Consider η : R × R → R defined by ηx ,y , x ,y x − 1 1 2 2 1 2 2 2 x ,y − y . Here, M {x, y ∈ M : x y − 4 ≤ 0},and EM {0,y : y ≥ 0,y − 4 ≤ 0}. 2 1 2 The sufficient conditions 7-8 reduce to −2y u2y 0,u y − 4 0,u ≥ 0, 3.11 whose solution is y 2,u 1and E0, 2 0, 2. Advances in Operations Research 11 2 2 In Example 2.12, we have already proved that fx, y −x − y is E-pseudoinvex function with respect to η. Using Definition 1.7, one can verify that ugx, y is semi-E-quasi- 2 2 invex with respect to same η at 0, 2 ∈ M , where ugx, y x y −4. So 0, 2 is the optimal solution of P on EM . 4. Conclusion E-convexity and its generalizations are studied by many authors earlier without differentia- bility assumption. Here, we have studied the the properties of E-convexity, E-invexity, and their generalizations with differentiable assumption. From the developments of this paper, we conclude the following interesting properties. 1 A function may not be convex at a point but E-convex at that point with a suitable mapping E, and if a local minimum of f exists in a neighborhood of Ex, then f is E-convex at x. But it is not necessarily true that if f is E-convex at x then Ex is local minimum point. 2 From the relation between E-invexity and its generalizations, one may observe that a function which is not E-convex may be E-invex with respect to some η and E-preinvexity with differentiability is a sufficient condition for E-invexity. Moreover, a function may not be E-invex with respect to some η but E-quasi-invex with respect to same η, a function may not E-invex with respect to some η but E- pseudoinvex with respect to the same η and an E-invex function with respect to some η may not be semi-E-invex with respect to same η. Here, we have studied E-convexity for first-order differentiable functions. Higher- order differentiable E-convex functions may be studied in a similar manner to derive the necessary and sufficient optimality conditions for a general nonlinear programming prob- lems. References 1 E. A. 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Published: Oct 2, 2012
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