Role of Exponential Type Random Invexities for Asymptotically Sufficient Efficiency Conditions in Semi-Infinite Multi-Objective Fractional Programming

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1 University of South Florida Scholar Commons Mathematics and Statistics Faculty Publications Mathematics and Statistics 206 Role of Exponential Type Random Invexities for Asymptotically Sufficient Efficiency Conditions in Semi-Infinite Multi-Objective Fractional Programming Ram U. Verma University of North Texas Youngsoo Seol University of South Florida Follow this and additional works at: Part of the Physical Sciences and Mathematics Commons Scholar Commons Citation Verma, Ram U. and Seol, Youngsoo, "Role of Exponential Type Random Invexities for Asymptotically Sufficient Efficiency Conditions in Semi-Infinite Multi-Objective Fractional Programming" 206. Mathematics and Statistics Faculty Publications This Article is brought to you for free and open access by the Mathematics and Statistics at Scholar Commons. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications by an authorized administrator of Scholar Commons. For more information, please contact

2 Verma and Seol SpringerPlus 2065:476 DOI 0.86/s RESEARCH Open Access Role of exponential type random invexities for asymptotically sufficient efficiency conditions in semi infinite multi objective fractional programming Ram U. Verma and Youngsoo Seol 2* *Correspondence: 2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33260, USA Full list of author information is available at the end of the article Abstract First a new notion of the random exponential Hanson Antczak type α, β, γ, ξ, η, ρ, h,,, θ-v-invexity is introduced, which generalizes most of the existing notions in the literature, second a random function h,, of the second order is defined, and finally a class of asymptotically sufficient efficiency conditions in semiinfinite multi-objective fractional programming is established. Furthermore, several sets of asymptotic sufficiency results in which various generalized exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-invexity assumptions are imposed on certain vector functions whose components are the individual as well as some combinations of the problem functions are examined and proved. To the best of our knowledge, all the established results on the semi-infinite aspects of the multi-objective fractional programming are new, which is a significantly new emerging field of the interdisciplinary research in nature. We also observed that the investigated results can be modified and applied to several special classes of nonlinear programming problems. Keywords: Semi-infinite programming, Multi-objective fractional programming, Generalized random α, β, γ, ξ, η, ρ, h,,, θ-invex, Infinitely many equality and inequality constraints, Parametric sufficient efficiency conditions Mathematics Subject Classification: 90C29, 90C30, 90C32, 90C34, 90C46 Background Based on the work of Antczak 2005 on the V-r-invex functions, Zalmai 203a generalized and investigated some multi-parameter generalizations of the parametrically sufficient efficiency results under various Hanson Antczak-type generalized α, β, γ, ξ, ρ, θ-v-invexity assumptions for the semi-infinite multi-objective fractional programming problems. Recently, Verma 203a, 204 has explored and investigated some results on the multi-objective fractional programming based on new ǫ-optimality conditions, and second-order, η, ρ, θ-invexities for parameter-free ǫ-efficiency conditions. Based on the on-going research advances in several areas of multi-objective programming, we observe that the field of the semi-infinite nonlinear multi-objective fractional programming problems seems to be still less explored compared to the 206 The Authors. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

3 Verma and Seol SpringerPlus 2065:476 Page 2 of 20 general area of mathematical programming. For more details, we refer the readers to Antczak 2005, 2009, Ben-Israel and Mond 986, Brosowski 982, Chen and Hu 2009, Craven 98, Daum and Werner 20, Ergenç et al. 2004, Fiacco and Kortanek 983, Giorgi and Guerraggio 996, Giorgi and Mititelu 993, Glashoff and Gustafson 983, Goberna and López 998, 200, Gribik 979, Gustafson and Kortanek 983, Hanson 98, Hanson and Mond 982, Henn and Kischka 976, Hettich 976, Hettich and Kortanek 993, Hettich and Zencke 982, Jess et al. 200, Jeyakumar and Mond 992, Kanniappan and Pandian 996, López and Still 2007, Martin 985, Miettinen 999, Mititelu 2004, 2007, Mititelu and Postolachi 20, Mititelu and Stancu-Minasian 993, Mond and Weir 98, Neralić and Stein 2004, Pini and Singh 997, Reemtsen and Rückmann 998, Reiland 990, Sawaragi et al. 986, Verma 203a, b, 204, 206, Weber 2002, Weber et al. 2008a, b, 2009, Weber and Tezel 2007, White 982, Winterfeld 2008, Yu 985, Zalmai 998, 203a, b, c. In this paper, we plan to introduce the new notion of the random exponential Hanson Antczak type α, β, γ, ξ, η, ρ, h,,, θ-v-invexity, which generalizes most of the existing notions in the literature, and then establish some results on random function h,, to the context of a class of asymptotically sufficient efficiency conditions in semiinfinite multi-objective fractional programming. Now we consider the following semi-infinite multi-objective fractional programming problem based on the random exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-invexity: P Minimize ϕx = ϕ x,..., ϕ p x f x = g x,..., f px g p x subject to G j x, t 0 for all t T j, j q, H k x, s = 0 for all s S k, k r, x X, 2 where p, q, and r are positive integers, X is a nonempty open convex subset of R n n-dimensional Euclidean space for each j q {, 2,..., q} and k r, T j and S k are compact subsets of complete metric spaces for each i p, f i and g i are real-valued functions defined on X, for each j q, G j, t is a real-valued function defined on X for all t T j, for each k r, H k, s is a real-valued function defined on X for all s S k, for each j q and k r, G j x, and H k x, are continuous real-valued functions defined, respectively, on T j and S k for all x X, and for each i p, g i x >0 for all x satisfying the constraints of P. As a matter of fact, all the parametric sufficient efficiency results established in this paper regarding problem P can easily be modified and restated for each one of the following seven special classes of nonlinear programming problems; P P2 P3 Minimize x F Minimize x F f x,..., f p x ; f x g x ; Minimize f x, x F

4 Verma and Seol SpringerPlus 2065:476 Page 3 of 20 where F assumed to be nonempty is the feasible set of P, that is, F ={x X : G j x, t 0 for all t T j, j q, H k x, s = 0 for all s S k, k r}; 3 f x P4 Minimize g x,..., f px g p x subject to G j x 0, j q, H k x = 0, k r, x X, 4 where f i and g i, i p, are as defined in the description of P, G j, j q, and H k, k r, are real-valued functions defined on X; P5 P6 P7 Minimize x G Minimize x G f x,..., f p x ; f x g x ; Minimize f x, x G where G is the feasible set of P4, that is, G = { x X : G j x 0, j q, } H k x = 0, k r. 5 Furthermore, we introduce the random function h,, defined on the probability space, F, P, which is the second order and define some new types of invexities regarding randomness and provide some asymptotic sufficient efficiency results for problem P under various generalized α, β, γ, ξ, η, ρ, h,,, θ-invexity assumptions with the random function h,,. The rest of the paper is organized as follows. Some introductory and basic concepts are introduced and studied in Preliminaries section along with introduction of the exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-invexities under the random function h,,, which generalizes HAα, β, γ, ξ, η, ρ, h,, θ-v-invexities. In Asymptotic sufficiency conditions section, we discuss some sufficient efficiency conditions where we formulate and prove several sets of sufficiency criteria under a variety of the exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-invexities with the random function h,, that are placed on certain vector-valued functions whose entries consist of the individual as well as some combinations of the problem functions. Concluding remarks section concludes the paper with final remarks on the obtained results and their future applications to other fields of research. Preliminaries In this section, we first introduce the concepts of the general probability theory and the exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-invexities under the random function h,,, and then recall some other related auxiliary results instrumental to the problem on hand.

5 Verma and Seol SpringerPlus 2065:476 Page 4 of 20 Random variables and probability theory In this subsection, we review the fundamental concepts of the probability theory on which the function h,, can be defined. We first define here the probability space and filtered probability space as the followings; Definition A probability space is a triple, F, P, where a is a set of all events which is called sample space. Elements of are denoted by ω and are sometimes called outcomes. b F is a σ-algebra or σ-field, i.e., a nonempty collection of subsets of that satisfy i if A F then A c F, and ii if A i F is a countable sequence of sets then i A i F. c P: F [0, ] is a function with P = and such that if E, E 2,... F are disjoint, P E j = PE j. 6 j= j= Definition 2 Let, F, P be a probability space. A filtration on, F, P is an increasing family F t t 0 of σ-algebra of F. In other words, for each t, F t is a σ-algebra included in F and if s t, F s F t. A probability space, F, P endowed with a filtration F t t 0 is called a filtrated probability space. Filtration have been a feature of the theory in the literature of the stochastic processes and mathematical fiance and advanced probability field such as stochastic control theory, martingales, semi-martingales, stopping times or Markov processes. In this paper, we restrict the concept filtrated probability space to investigate some results regarding the function h as defined by the martingale processes or Markov chain in the future work. In the followings, we define the random variables and study the concepts of the first and second moments of random variables. Definition 3 A random variable X is a measurable function from a probability space, F, P to the reals, i.e., it is a function X :, 7 such that for every Borel set B, X B ={ω : Xω B} F. 8 Furthermore, we define a function on Borel sets by µ X B = P{X B} =P[X B]. Let X be a random variable. We define the expectation of X, denoted by EX, by EX = XdP, 9 0

6 Verma and Seol SpringerPlus 2065:476 Page 5 of 20 where the integral is the Lebesgue integral. In particular, the expectation of a random variable depends only on its distribution and not on the probability space on which it is defined. If X has a density F, then the measure µ X is the same as fxdx, so we can write E[X] = xf xdx, where again the expectation exists if and only if x f xdx <. 2 Furthermore, the second moment is defined by E[X 2 ]= x 2 f xdx, and Var[X] =E[X EX] =E[X 2 ] E[X] 2. 3 The following are the concepts of independence. Definition 4 a σ-algebra F, F 2,..., F n are independent if whenever A i F i for i =,..., n we have n n P A i = PA i. 4 b Random variables X, X 2,..., X n are independent if for every Borel sets B i for i =,..., n we have n n P {X i B i } = PX i B i. 5 The law of large numbers, which is a theorem proved about the mathematical model of probability, shows that this model is consistent with the frequency interpretation of probability and this theorem is the main idea to prove the main theorem in this paper. Theorem 5 Law of Large Numbers Let X, X 2,..., X n be a sequence of independent random variables with common distribution function. Set µ = E[X i ] < and σ 2 = Var[X i ], and for S n = n X i, we have a Weak law of large numbers lim P ω : S n ω n µ n ǫ = 0, ǫ >0. 6 b Strong law of large numbers S n ω P ω : lim = µ =. n n 7

7 Verma and Seol SpringerPlus 2065:476 Page 6 of 20 Deterministic cases Definition 6 Let f be a differentiable real-valued function defined on R n. Then f is said to be η-invex invex with respect to η at y R n if there exists a function η : R n R n R n such that for each x R n, f x f y f y, ηx, y, 8 where f y = f y/ y, f y/ y 2,..., f y/ y n is the gradient of f at y, and a, b denotes the inner product of the vectors a and b; f is said to be η-invex on R n if the above inequality holds for all x, y R n. Hanson 98 showed based on the role of the function η that for a nonlinear programming problem of the form Minimize f x subject to g i x 0, i m, x R n, where the differentiable functions f, g i : R n R, i m, are invex with respect to the function η : R n R n R n, the Karush Kuhn Tucker necessary optimality conditions are also sufficient. Let the function F = F, F 2,..., F N : R n R N be differentiable at x. The following generalizations of the notions of invexity, pseudoinvexity, and quasiinvexity for vectorvalued functions were originally proposed in Jeyakumar and Mond 992. Definition 7 The function F is said to be α, η-v-invex at x if there exist functions α i : R n R n R + \{0} 0,, i N, and η : R n R n R n such that for each x R n and i N, F i x F i x α i x, x F i x, ηx, x. 9 Definition 8 The function F is said to be β, η-v-pseudoinvex at x if there exist functions β i : R n R n R + \{0}, i N, and η : R n R n R n such that for each x R n, N N N F i x, ηx, x 0 β i x, x F i x β i x, x F i x. 20 Definition 9 The function F is said to be γ, η-v-quasiinvex at x if there exist functions γ i : R n R n R + \{0}, i N, and η : R n R n R n such that for each x R n, N N N γ i x, x F i x γ i x, x F i x F i x, ηx, x 0. 2 Recently, Antczak 2005 introduced the following exponential type of the class of V-invex functions. Definition 0 A differentiable function f : X R k is called strictly ζ i - r-invex with respect to η at u X if there exist functions η : X X R n and ζ i : X X R + \{0}, i k, such for each x X,

8 Verma and Seol SpringerPlus 2065:476 Page 7 of 20 r e rf ix > r e rf iu [ + rζ i x, u f i u, ηx, u] for r = 0, f i x f i u ζ i x, u f i u, ηx, u for r = As the exponential type of the class of functions was considered in Antczak 2005 for establishing some sufficiency and duality results for a nonlinear programming problem with differentiable functions, and their nonsmooth analogues were discussed in Antczak 2009, recently, Zalmai 203a introduced the Hanson Antczak type generalized HAα, β, γ, ξ, η, ρ, θ-v-invexity, an exponential type framework, and then applied to a set of problems on fractional programming. As a result, Zalmai further envisioned a vast array of interesting and significant classes of generalized convex functions. Now we are ready to present the exponential type HAα, β, γ, ξ, η, ρ, h,, θ-v-invexities that generalize and encompass most of the existing notions available in the current literature. Let the function F = F, F 2,..., F p : X R p be differentiable at x. Definition The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, θ-v-invex at x X if there exist functions α : X X R, β : X X R, γ i : X X R +, ξ i : X X R + \{0}, i p, z R n, η : X X R n, ρ i : X X R, i p, and θ : X X R n such that for all x X x = x and i p, αx, x γ ix, x e αx,x [F i x F i x ] > βx, x ξ i x, x z h i x, z, e βx,x ηx,x + ρ i x, x θx, x 2 if αx, x = 0 and βx, x = 0 for all x X, 24 αx, x γ ix, x e αx,x [F i x F i x ] > ξ i x, x z h i x, z, ηx, x + ρ i x, x θx, x 2 if αx, x = 0 and βx, x 0 for all x X, γ i x, x [ F i x F i x ] > βx, x ξ i x, x z h i x, z, e βx,x ηx,x + ρ i x, x θx, x 2 if αx, x 0 and βx, x = 0 for all x X, γ i x, x [ F i x F i x ] > ξ i x, x z h i x, z, ηx, x + ρ i x, x θx, x 2 if αx, x 0 and βx, x 0 for all x X, 27 where is a norm on R n and e βx,x η x,x e βx,x η x,x,..., e βx,x η n x,x, 28 with h : R n R n R n differentiable. Definition 2 The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, θ- V-pseudoinvex at x X if there exist functions α : X X R, β : X X R, γ :

9 Verma and Seol SpringerPlus 2065:476 Page 8 of 20 X X R +, ξ i : X X R + \{0}, i p, z R n, η : X X R n, ρ : X X R, and θ : X X R n such that for all x X x = x, p βx, x z h i x, z, e βx,x ηx,x ρx, x θx, x 2 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] > 0 if αx, x = 0 and βx, x = 0 for all x X, p z h i x, z, ηx, x ρx, x θx, x 2 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] > 0 if αx, x = 0 and βx, x 0 for all x X, p βx, x z h i x, z, e βx,x ηx,x ρx, x θx, x 2 γx, x ξ i x, x [ F i x F i x ] > 0 if αx, x 0 and βx, x = 0 for all x X, p z h i x, z, ηx, x ρx, x θx, x 2 γx, x ξ i x, x [ F i x F i x ] > 0 if αx, x 0 and βx, x 0 for all x X with h : R n R n R n differentiable. The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, θ-v-pseudoinvex on X if it is strictly HAα, β, γ, ξ, η, ρ, h,, θ-v-pseudoinvex at each point x X. Definition 3 The function F is said to be prestrictly HAα, β, γ, ξ, η, ρ, h,, θ-vquasiinvex at x X if there exist functions α : X X R, β : X X R, γ : X X R +, ξ i : X X R + \{0}, i p, η : X X R n, ρ : X X R, and θ : X X R n such that for all x X, αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] < 0 p βx, x z h i x, z, e βx,x ηx,x ρx, x θx, x 2 if αx, x = 0 and βx, x = 0 for all x X, 33

10 Verma and Seol SpringerPlus 2065:476 Page 9 of 20 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] < 0 p z h i x, z, ηx, x ρx, x θx, x 2 if αx, x = 0 and βx, x 0 for all x X, 34 γx, x ξ i x, x [ F i x F i x ] < 0 p βx, x z h i x, z, e βx,x ηx,x ρx, x θx, x 2 if αx, x 0 and βx, x = 0 for all x X, 35 γx, x ξ i x, x [ F i x F i x ] < 0 p z h i x, z, ηx, x ρx, x θx, x 2 if αx, x 0 and βx, x 0 for all x X. 36 with h : R n R n R n differentiable. Example 4 The function F is said to be strictly HAα, β, γ, ξ, η, ρ, θ-v-invex at x X if there exist functions α : X X R, β : X X R, γ i : X X R +, ξ i : X X R + \{0}, i p, z R n, η : X X R n, ρ i : X X R, i p, and θ : X X R n such that for all x X x = x and i p, αx, x γ ix, x e αx,x [F i x F i x ] > βx, x ξ i x, x f x, e βx,x ηx,x + ρ i x, x θx, x 2 if αx, x = 0 and βx, x = 0 for all x X. 37 We also noticed that for the proofs of the sufficient efficiency theorems, sometimes it may be more appropriate to apply certain alternative but equivalent forms of the above definitions based on considering the contrapositive statements. For example, the exponential type HAα, β, γ, ξ, η, ρ, h,, θ-v-quasiinvexity when αx, x = 0 and βx, x = 0 for all x X can be defined in the following equivalent way: The function F is an exponential type HAα, β, γ, ξ, η, ρ, h,, θ-v-quasiinvex at x X if there exist functions α : X X R, β : X X R, γ : X X R +, ξ i : X X R + \{0}, i p, η : X X R n, ρ : X X R, and θ : X X R n such that for all x X, βx, x p z h i x, z, e βx,x ηx,x > ρx, x θx, x 2 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] > 0, 38

11 Verma and Seol SpringerPlus 2065:476 Page 0 of 20 where h : R n R n R n is differentiable. In the sequel, we shall also need a consistent notation for vector inequalities. For a, b R m, the following order notation will be used: a b if and only if a i b i for all i m; a b if and only if a i b i for all i m, but a = b; a > b if and only if a i > b i for all i m; and a b is the negation of a b. Consider the multi-objective problem P Minimize x F Fx = F x,..., F p x, where F i, i p, are real-valued functions defined on R n. An element x F is said to be an efficient Pareto optimal, non-dominated, non-inferior solution of P if there exists no x F such that Fx Fx. In the area of multiobjective programming, there exist several versions of the notion of efficiency most of which are discussed in Miettinen 999, Verma 204, White 982, Yu 985. However, throughout this paper, we shall deal exclusively with the efficient solutions of P in the sense defined above. For the purpose of comparison with the sufficient efficiency conditions that will be proposed and discussed in this paper, we next recall a set of necessary efficiency conditions for P. Theorem 5 Zalmai 203a Let x F, let λ = ϕx, for each i p, let f i and g i be continuously differentiable at x, for each j q, let the function G j, t be continuously differentiable at x for all t T j, and for each k r, let the function H k, s be continuously differentiable at x for all s S k. If x is an efficient solution of P, if the generalized Guignard constraint qualification holds at x, and if for each i 0 p, the set cone{ G j x, t : t ˆT j x, j q} { i x λ i g ix : i p, i = i 0 }+span{ H k x, s : s S k, k r} is closed, then there exist u U and integers ν0 and ν, with 0 ν0 ν n +, such that there exist ν0 indices j m, with j m q, together with ν0 points t m ˆT jm x, m ν0, ν ν0 indices k m, with k m r, together with ν ν0 points s m S km for m ν \ν0, and ν real numbers vm, with v m > 0 for m ν 0, with the property that u [ i fi x λ i g ix ] + ν0 vm G ν j m x, t m + vm H k m x, s m = 0, m=ν m= where conev is the conic hull of the set V R n i.e., the smallest convex cone containing V, spanv is the linear hull of V i.e., the smallest subspace containing V, ˆT j x ={t T j : G j x, t = 0}, U ={u R p : u > 0, p u i = }, and ν \ν0 is the complement of the set ν0 relative to the set ν. Random cases Let the function h, z, ω hx, z, ω = f x f xz, z, 40

12 Verma and Seol SpringerPlus 2065:476 Page of 20 be defined on the probability space, F, P where ω and z is a direction. We start with the assumptions we will use throughout the paper. Assumption 6 A h,, ω is an i.i.d. sequence identically and independently distributed sequence. A2 E[h,, ω] <. Notice that A implies that z h,, ω is i.i.d. and is a function of ω and random variable. In the view of A2, E[ z h,, ω] <. The following are the new definitions related with randomness which will be used for the main results. Definition 7 Let f be a differentiable real-valued function defined on R n. Then f is said to be random η-asymptotic invex invex with respect to η at y if there exists a function η : R n n R n such that for each ω n, f ω f y f y, E[ηω, y], 4 where f y = f y/ y, f y/ y 2,..., f y/ y n is the gradient of f at y, and a, b denotes the inner product of the vectors a and b. Definition 8 The function F is said to be randomα, η-v-asymptotic-invex at x if there exist functions α i : R n R n R + \{0} 0,, i N, and η : R n n R n such that for each ω n and i N, F i x F i x α i x, x F i x, E[ηω, x ]. 42 Definition 9 The function F is said to be random β, η-v-asymptotic-pseudoinvex at x if there exist functions β i : R n R n R + \{0}, i N, and η : R n n R n such that for each ω n, N N N F i x, E[ηω, x ] 0 β i x, x F i x β i x, x F i x. 43 Definition 20 The function F is said to be randomγ, η-v-asymptotic-quasiinvex at x if there exist functions γ i : R n R n R + \{0}, i N, and η : R n n R n such that for each ω n, N N N γ i x, x F i x γ i x, x F i x F i x, E[ηω, x ] Next, we define the exponential type Hanson Antczak type generalized HAα, β, γ, ξ, η, ρ, h,, ω, θ-v-asymptotic-invexities under the random function h,, ω. Let the function F = F, F 2,..., F p : X R p be differentiable at x.

13 Verma and Seol SpringerPlus 2065:476 Page 2 of 20 Definition 2 The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, ω, θ-random V-asymptotic-invex at x X if there exist functions α : X X R, β : X X R, γ i : X X R +, ξ i : X X R + \{0}, i p, z R n, η : X X R n ρ i : X X R, i p, and θ : X X R n such that for all x X x = x and i p, αx, x γ ix, x e αx,x [F i x F i x ] > βx, x ξ i x, x E [ z h i x, z, ω ], e βx,x ηx,x + ρ i x, x θx, x 2 if αx, x = 0 and βx, x = 0 for all x X, 45 αx, x γ ix, x e αx,x [F i x F i x ] > ξ i x, x E [ z h i x, z, ω ], ηx, x + ρ i x, x θx, x 2 if αx, x = 0 and βx, x 0 for all x X, 46 γ i x, x [ F i x F i x ] > βx, x ξ i x, x E [ z h i x, z, ω ], e βx,x ηx,x + ρ i x, x θx, x 2 if αx, x 0 and βx, x = 0 for all x X, 47 γ i x, x [ F i x F i x ] > ξ i x, x E [ z h i x, z, ω ], ηx, x + ρ i x, x θx, x 2 if αx, x 0 and βx, x 0 for all x X, 48 where is a norm on R n and e βx,x η x,x e βx,x η x,x,..., e βx,x η n x,x, 49 with h : R n R n n R n differentiable and random function. The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, ω, θ-random V-invex x X if the expectation is dropped in the above definition. Definition 22 The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, ω, θ -random V-asymptotic-pseudoinvex at x X if there exist functions α : X X R, β : X X R, γ : X X R +, ξ i : X X R + \{0}, i p, z R n, η : X X R n, ρ : X X R, and θ : X X R n such that for all x X x = x, p βx, x E [ z h i x, z, ω ], e βx,x ηx,x ρx, x θx, x 2 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] > 0 if αx, x = 0 and βx, x = 0 for all x X, 50 p E [ z h i x, z, ω ], ηx, x ρx, x θx, x 2 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] > 0 if αx, x = 0 and βx, x 0 for all x X, 5

14 Verma and Seol SpringerPlus 2065:476 Page 3 of 20 p βx, x E [ z h i x, z, ω ], e βx,x ηx,x ρx, x θx, x 2 γx, x ξ i x, x [ F i x F i x ] > 0 if αx, x 0 and βx, x = 0 for all x X, p E [ z h i x, z, ω ], ηx, x ρx, x θx, x 2 γx, x ξ i x, x [ F i x F i x ] > 0 if αx, x 0 and βx, x 0 for all x X with h : R n R n n R n differentiable and random function. The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, ω, θ-random V-pseudoinvex x X if the expectation is dropped in the above definition. Definition 23 The function F is said to be prestrictly HAα, β, γ, ξ, η, ρ, h,, ω, θ -random V-asymptotic-quasiinvex at x X if there exist functions α : X X R, β : X X R, γ : X X R +, ξ i : X X R + \{0}, i p, η : X X R n, ρ : X X R, and θ : X X R n such that for all x X, αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] < 0 p βx, x E [ z h i x, z, ω ], e βx,x ηx,x ρx, x θx, x 2 if αx, x = 0 and βx, x = 0 for all x X, 54 αx, x γx, x e αx,x p ξ ix,x [F i x F i x ] < 0 p E [ z h i x, z, ω ], ηx, x ρx, x θx, x 2 if αx, x = 0 and βx, x 0 for all x X, 55 γx, x ξ i x, x [ F i x F i x ] < 0 p βx, x E [ z h i x, z, ω ], e βx,x ηx,x ρx, x θx, x 2 if αx, x 0 and βx, x = 0 for all x X, 56

15 Verma and Seol SpringerPlus 2065:476 Page 4 of 20 γx, x ξ i x, x [ F i x F i x ] < 0 p E [ z h i x, z, ω ], ηx, x ρx, x θx, x 2 if αx, x 0 and βx, x 0 for all x X. 57 with h : R n R n n R n differentiable and random function. The function F is said to be strictly HAα, β, γ, ξ, η, ρ, h,, ω, θ-random V-quasiinvex x X if the expectation is dropped in the above definition. Asymptotic sufficiency conditions In this section, we present several sets of asymptotic sufficiency results in which various generalized exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-invexity assumptions are imposed on certain vector functions whose components are the individual as well as some combinations of the problem functions. Let the function E i, λ, u : X R be defined, for fixed λ and u, on X by E i z, λ, u = u i [f i z λ i g i z], i p. Theorem 24 Let x F, let λ = ϕx, let the functions f i, g i, i p, G j, t, and H k, s be differentiable at x for all t T j and s S k, j q, k r, and assume that Assumption 6 is satisfied and that there exist u U and integers ν 0 and ν, with 0 ν 0 ν n +, such that there exist ν 0 indices j m, with j m q, together with ν 0 points t m ˆT jm x, m ν 0, ν ν 0 indices k m, with k m r, together with ν ν 0 points s m S km, m ν\ν 0, and ν real numbers vm with v m > 0 for m ν 0, with d = min{u i, v m }, and with the property that 58 d [E [ z h i x, z, ω ] λe [ z κ i x, z, ω ]] + d E [ z ψ jm x, t m, z, ω ] + d E [ z km x, s m, z, ω ] = or + u [ [ i E z h i x, z, ω ] λ i E[ z κ i x, z, ω ]] + ν m=ν 0 + v m E[ z km x, s m, z, ω ] = 0. ν 0 vm E[ z ψ jm x, t m, z, ω ] m= 60 Assume, furthermore, that either one of the following three sets of conditions holds under the random function h: a i f i is exponential type HAα, β, γ, ξ, η, ρ, h,, ω, θ-random V-invex at x, g i is exponential type HAα, β, γ, ξ, η, ρ, κ,, ω, θ-random V-invex at x, and γx, x >0 for all x F; ii v G j, t,..., vν 0 G jν0, t ν 0 is exponential type HAα, β, ˆγ, π, η, ˆρ, ψ,, ω, θ-random V-invex at x ;

16 Verma and Seol SpringerPlus 2065:476 Page 5 of 20 iii vν 0 + H k, ν0 + sν0+,..., vν H k ν, s ν is exponential type HAα, β, γ, δ, η, ρ,,, ω, θ-random V-invex at x ; iv ξ i = π k = δ l = σ for all i p, k ν 0, and l ν\ν 0 ; p v u i ρ i x, x + ν 0 m= ˆρ mx, x + ν m=ν0 + ρ mx, x 0 for all x F; b i f i is exponential type HAα, β, γ, ξ, η, ρ, h,, ω, θ-random V-asymptoticinvex at x, g i is exponential type HAα, β, γ, ξ, η, ρ, κ,, ω, θ-random V-asymptotic-invex at x, and γx, x >0 for all x F; ii v G j, t,..., vν 0 G jν0, t ν 0 is exponential type HAα, β, ˆγ, π, η, ˆρ, ψ,, ω, θ-random V-asymptotic-invex at x ; iii vν 0 + H k, ν0 + sν0+,..., vν H k ν, s ν is exponential type HAα, β, γ, δ, η, ρ,,, ω, θ-random V-asymptotic-invex at x ; iv ξ i = π k = δ l = σ for all i p, k ν 0, and l ν\ν 0 ; v p u i ρ i x, x + ν 0 m= ˆρ mx, x + ν m=ν0 + ρ mx, x 0 for all x F; c the function L, u, v, λ, t, s,..., L p, u, v, λ, t, s is exponential type HAα, β, γ, ξ, η, ρ, h,, ω, κ,, ω, ψ,, ω,,, ω, θ -random V-asymptoticpseudoinvex at x and γx, x >0 for all x F, where Then x is an efficient solution of P. L i z, u, v, λ, t, s = u i f i z λ i g ν 0 ν iz + vm G j m z, t m + vm H k m z, s m, i p. m= m=ν Proof a In view of our assumptions in i iv, we have αx, x γ ix, x e αx,x {f i x λ i g ix [f i x λ i g ix ]} βx, x σx, x [ E [ z h i x, z, ω ] λ i E[ z κ i x, z, ω ]], e βx,x ηx,x + ρ i x, x θx, x 2, i p, 62 αx, x ˆγ mx, x e αx,x [vm G jm x,tm vm G jm x,t m ] βx, x σx, x vm E[ z ψ jm x, t m, z, ω ], e βx,x ηx,x +ˆρ m x, x θx, x 2, m ν 0, αx, x γ mx, x e αx,x [vm H km x,sm vm H km x,s m ] βx, x σx, x vm E[ z km x, s m, z, ω ], e βx,x ηx,x + ρ m x, x θx, x 2, m ν\ν Multiplying 6 by u i and then summing over i p, summing 62 over m ν 0, and summing 63 over m ν\ν 0, and finally adding the resulting inequalities, we get

17 Verma and Seol SpringerPlus 2065:476 Page 6 of 20 { p αx, x u i γ ix, x e αx,x {f i x λ i g ix [f i x λ i g ix ]} ν 0 + ˆγ m x, x e αx,x [vm G jm x,tm vm G jm x,t m ] + m= ν m=ν 0 + γ m x, x e αx,x [vm H km x,sm vm H km x,s m ] βx, x σx, x ν 0 p + vm E[ z ψ jm x, t m, z, ω ] + m= ν m=ν 0 + u [ [ i E z h i x, z, ω ] λ i E[ z κ i x, z, ω ]] v m E[ z km x, s m, z, ω ], e βx,x ηx,x [ p + u i ρ i x, x + ν 0 m= Let l = min{p, ν}. Then the last term of above inequality ˆρ m x, x + m = ν 0 + ν ρ m x, x ] θx, x βx, x σx, p x u [ [ i E z h i x, z, ω ] λ i E[ z κ i x, z, ω ]] ν 0 + vm E[ z ψ jm x, t m, z, ω ] + m= ν m=ν 0 + v m E[ z km x, s m, z, ω ], e βx,x ηx,x + u i ρ i x, x + βx, x σx, x l ν 0 ν 0 m= ˆρ m x, x + ν m=ν 0 + ρ m x, x θx, x 2 u [ [ i E z h i x, z, ω ] λ i E[ z κ i x, z, ω ]] + vm l E[ z ψ jm x, t m, z, ω ] m= + ν vm l E[ z km x, s m, z, ω ], e βx,x ηx,x m=ν 0 + ν 0 ν + u i ρ i x, x + ˆρ m x, x + ρ m x, x θx, x 2. m= m=ν

18 Verma and Seol SpringerPlus 2065:476 Page 7 of 20 Let d = min{u i, v m } and by using the law of large number, we get βx, x σx, p x u [ [ i E z h i x, z, ω ] λ i E[ z κ i x, z, ω ]] ν 0 + vm E[ z ψ jm x, t m, z, ω ] m= ν + vm E[ z km x, s m, z, ω ], e βx,x ηx,x m=ν 0+ ν 0 ν + u i ρ i x, x + ˆρ m x, x + ρ m x, x θx, x 2 m= m=ν 0+ βx, x σx, x u [ [ i E z h i x, z, ω ] λ i l E[ z κ i x, z, ω ]] + ν 0 vm l E[ z ψ jm x, t m, z, ω ] m= + ν vm l E[ z km x, s m, z, ω ], e βx,x ηx,x m=ν 0+ ν 0 ν + u i ρ i x, x + ˆρ m x, x + ρ m x, x θx, x 2 m= m=ν 0+ βx, x σx, x d [E [ z h i x, z, ω ] λe [ z κ i x, z, ω ]] + d E [ z ψ jm x, t m, z, ω ] +d E [ z km x, s m, z, ω ], e βx,x ηx,x ν 0 ν + u i ρ i x, x + ˆρ m x, x + ρ m x, x θx, x 2. m= m=ν Now using 59 and the condition v, and noticing that σx, x >0, ϕx = λ ; x, x F, and G jm x, t m = 0 for all m ν 0, the above inequality reduces to αx, x u i γ ix, x e αx,x [f i x λ i gix] Since γx, x >0, even if we consider the both cases αx, x >0 and αx, x <0, it follows from the above inequality u i [f ix λ i g ix] Therefore, we conclude that x is an efficient solution of P. b Using the same idea in a and assumption 60, the proof follows. c Let x be an arbitrary feasible solution of P. From 60 we observe that p βx, x u [ [ i E fi x ] λ i E[ g i x ]] ν 0 + vm E[ G jm x, t m ] m= ν + vm E[ H km x, s m ], e βx,x ηx,x = 0, m=ν which in view of our HAα, β, γ, ξ, η, ρ, h,, ω, κ,, ω, ψ,, ω,,, ω, θ-random V-asymptotic-pseudoinvexity assumption implies that 7 αx, x γx, x e αx,x p ξ ix,x [L i x,u,v,λ, t, s L i x,u,v,λ, t, s] 0.

19 Verma and Seol SpringerPlus 2065:476 Page 8 of 20 We need to consider two cases: αx, x >0 and αx, x <0. If we assume that αx, x >0 and recall that γx, x >0, then the above inequality becomes e αx,x p ξ ix,x [L i x,u,v,λ, t, s L i x,u,v,λ, t, s], 72 which implies that ξ i x, x L i x, u, v, λ, t, s ξ i x, x L i x, u, v, λ, t, s. 73 Because x F, t m ˆT jm x, m ν 0, and λ i = ϕ i x, i p, the right-hand side of the above inequality is equal to zero, and hence we have Lx, u, v, λ, t, s 0. Next, as x F, and v m > 0, m ν 0, this inequality simplifies to u i ξ ix, x [f i x λ i g ix] 0. Since u > 0 and ξ i x, x >0, i p, the above inequality implies that f x λ g x,..., f p x λ p g px 0,...,0, which in turn implies that f x g x,..., f px λ g p x,..., λ p = ϕx. 76 Since x F was arbitrary, we conclude from this inequality that x is an efficient solution of P. On the other hand, we arrive at the same conclusion if we assume that αx, x <0. Concluding remarks In this paper, we have introduced several notions of random exponential type HAα, β, γ, ξ, η, ρ, h,,, θ-v-asymptotic invexities which generalize the Hanson Antczak type α, β, γ, ξ, η, ρ, h,, θ-v-invexity Zalmai 203a, while this generalizes most of the existing notions in the literature, and then applied to establish some results to the context of a class of asymptotically sufficient efficiency conditions in semi-infinite multi-objective fractional programming. Furthermore, the obtained results can be applied to generalize the related duality models and theorems in Zalmai 203b, c, and more. Our results also indicate a wide range of future applications to other problems arising from higher order random invexities and its variants. Authors contributions The authors have contributed equally to this research work, and approved the revised version. Both authors read and approved the final manuscript. Author details Department of Mathematics, University of North Texas, Denton, TX 7620, USA. 2 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33260, USA. Acknowledgements Authors are greatly indebted to the Reviewers for all valuable suggestions leading to the revised.

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