() CUBO A Mathematical Journal Vol.17, No¯ 01, (41–64). March 2015 Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks and efficiency conditions for multiobjective fractional programming problems Ram U. Verma Department of Mathematics, Texas State University, San Marcos, TX 78666, USA. verma99@msn.com ABSTRACT The parametrically generalized sufficient efficiency conditions for multiobjective frac- tional programming based on the hybrid (Φ, Ψ, ρ, ζ, θ)−invexities are developed and then efficient solutions to the multiobjective fractional programming problems are es- tablished. Plus, the obtained results on sufficient efficiency conditions are generalized to the case of the ǫ−efficient solutions. The results thus obtained generalize and unify a wider range of investigations on the theory and applications to the multiobjective fractional programming based on the hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks. RESUMEN Se desarrollan las condiciones de eficiencia suficiente generalizadas paramétricamente de programación multifraccional multiobjetivo basado en las invexidades-(Φ, Ψ, ρ, ζ, θ)− h́ıbridas y luego se establecen las soluciones eficientes a los problemas de programación fraccional multiobjetivo. Además, los resultados obtenidos sobre condiciones de efi- ciencia suficiente se generalizan al caso de soluciones ǫ−-eficientes. Los resultados obtenidos generalizan y unifican una amplia gama de investigaciones en la teoŕıa y aplicaciones de la programación fraccional multiobjetivo basado en el marco de trabajo de las invexidades-(Φ, Ψ, ρ, ζ, θ)−. Keywords and Phrases: Generalized invexity, Multiobjective fractional programming, Efficient solutions, ǫ−efficient solutions, Parametric sufficient efficiency conditions. 2010 AMS Mathematics Subject Classification: 90C32, 90C45. 42 Ram U. Verma CUBO 17, 1 (2015) 1 Introduction Among recent developments on higher order generalized invexties and duality models for math- ematical programming, we begin with the work of Kawasaki [5] on some second order necessary conditions of the Kuhn - Tucker type under new weaker constraint qualifications for twice contin- uously differentiable functions, while Mishra and Rueda [11] introduced higher order generalized invexity and duality models in mathematical programming. Mangasarian [8] focused on the second order duality for a conventional nonlinear programming problem, where the approach is based on constructing a second order dual problem by taking linear and quadratic approximations of the objective and constraint functions for an arbitrary but fixed point leading to the Wolfe dual model for the approximated problem, while letting the fixed point to vary. Verma [24] introduced and studied the second order (ρ, η, θ)−invexities to the context of parametrically sufficient optimality conditions in semiinfinite discrete minimax fractional programming. Zalmai and Zhang [37] have established a set of efficiency conditions and a fairly large number of global nonparametric sufficient efficiency results under various frameworks for generalized (η, ρ)−invexity for the semiinfinite dis- crete minimax fractional programming. Just recently, Verma [22] investigated a general framework for a class of (ρ, η, θ)−invex functions to examine some parametric sufficient efficiency conditions for multiobjective fractional programming problems for weakly ǫ−efficient solutions. Inspired by these research advances, we first introduce the hybrid (Φ, Ψ, ρ, ζ, θ)−invexities as well as the second order hybrid (Φ, Ψ, ρ, ζ, θ)−invexities, second, introduce some parametrically sufficient efficiency conditions for multiobjective fractional programming, and finally, explore the efficient solutions to multiobjective fractional programming problems. In addition, we generalize the obtained results based on the hybrid (Φ, Ψ, ρ, ζ, θ)−invexities regarding the efficient solutions to the multiobjective fractional programming problems to the case of the ǫ−efficient solutions to the multiobjective fractional programming problems. The results established in this communication, not only gener- alize (and unify) the results on general sufficient efficiency conditions for multiobjective fractional programming problems based on the hybrid invexity of functions, but also generalize second order invexity results in more general settings. There exists an enormous literature on higher order generalized invexity and duality models in mathematical programming. We consider, based on the generalized (Φ, Ψ, ρ, ζ, θ)−invexities of functions, the following multi- objective fractional programming problem: (P) Minimize ( f1(x) g1(x) , f2(x) g2(x) , · · ·, fp(x) gp(x) ) subject to x ∈ Q = {x ∈ X : Hj(x) ≤ 0, j ∈ {1, 2, · · ·, m}}, where X is an open convex subset of Ren (n-dimensional Euclidean space), fi and gi for i ∈ {1, ···, p} and Hj for j ∈ {1, · · ·, m} are real-valued functions defined on X such that fi(x) ≥ 0, gi(x) > 0 for i ∈ {1, · · ·, p} and for all x ∈ Q. Here Q denotes the feasible set of (P). CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 43 Next, we observe that problem (P) is equivalent to the nonfractional programming problem: (Pλ) Minimize ( f1(x) − λ1g1(x), · · ·, fp(x) − λpgp(x) ) subject to x ∈ Q with λ = ( λ1, λ2, · · ·, λp ) = ( f1(x ∗) g1(x ∗) , f2(x ∗) g2(x ∗) , · · ·, fp(x ∗) gp(x ∗) ) , where x∗ is an efficient solution to (P). We all can agree that general Mathematical programming problems offer a great opportunity for applications to other fields, for instance, applications to game theory, statistical analysis, engi- neering design (including design of control systems, design of earthquakes-resistant structures, digital filters, and electronic circuits), random graphs, boundary value problems, wavelet analy- sis, environmental protection planning, decision and management sciences, optimal control prob- lems, continuum mechanics, robotics, and beyond. For more details on generalized efficiency and ǫ−efficiency results and applications, we recommend the reader [1 - 40]. This submission is organized as follows: the introductory section deals with a brief historical devel- opment for the multiobjective fractional mathematical programming, while emphasizing the roles of the generalized first (and second) order (Φ, Ψ, ρ, ζ, θ)−invex functions as well as the first (and sec- ond) order generalized (Φ, Ψ, ρ, ζ, θ)−invex functions. In Section 2, the hybrid (Φ, Ψ, ρ, ζ, θ)−invex functions of higher orders are introduced, and Section 3 deals with sufficient efficiency conditions leading to the solvability of the problem (P) using the hybrid (Φ, Ψ, ρ, ζ, θ)−invexities. In Section 4, general remarks are presented. 2 Hybrid Invexities In this section, we introduce and develop some concepts and notations for the problem on hand. Let X be an open convex subset of Ren (n-dimensional Euclidean space). Let 〈·, ·〉 denote the inner product, and let z ∈ Ren . Suppose that f : X → Re is a real-valued twice continuously differentiable function defined on X, and that ▽f(y) and ∇2f(y) denote, respectively, the gradient and Hessian of f at y. Recall a function Ψ : Ren → Re is sublinear (super linear) if Ψ(x + y) ≤ (≥)Ψ(x) + Ψ(y) for all x, y ∈ Ren, and Ψ(ax) = aΨ(x) for all x ∈ Ren and a ∈ Re+ = [0, ∞). Let x∗ ∈ X. Definition 2.1. A twice differentiable function f : X → Re is said to be hybrid (Φ, Ψ, ρ, ζ, θ)−invex at x∗ of second order if there exists a function Φ : Re → Re such that for each x ∈ X, ρ : X×X → Re, Ψ : Ren → Re, ζ, θ : X × X → Ren and z ∈ Ren, Φ ( f(x) − f(x∗) ) ≥ Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2. 44 Ram U. Verma CUBO 17, 1 (2015) Definition 2.2. A twice differentiable function f : X → Re is said to be hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo- invex at x∗ of second order if there exists a function Φ : Re → Re and z ∈ Ren such that for each x ∈ X, ρ : X × X → Re, Ψ : Ren → Re, and ζ, θ : X × X → Ren, Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≥ 0 ⇒ Φ ( f(x) − f(x∗) ) ≥ 0. Definition 2.3. A twice differentiable function f : X → Re is said to be strictly hybrid (Φ, Ψ, ρ, ζ, θ)− pseudo-invex at x∗ of second order if there exists a function Φ : Re → Re and z ∈ Ren such that for each x ∈ X, ρ : X × X → Re, Ψ : Ren → Re, and ζ, θ : X × X → Ren, Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≥ 0 ⇒ Φ ( f(x) − f(x∗) ) > 0. Definition 2.4. A twice differentiable function f : X → Re is said to be prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ of second order if there if there exists a function Φ : Re → Re and z ∈ Ren such that for each x ∈ X, ρ : X × X → Re, Ψ : Ren → Re, and θ, ζ : X × X → Ren, Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 > 0 ⇒ Φ ( f(x) − f(x∗) ) ≥ 0. Definition 2.5. A twice differentiable function f : X → Re is said to be hybrid (Φ, Ψ, ρ, ζ, θ)−quasi- invex at x∗ of second order if there exists a function Φ : Re → Re such that for each x ∈ X, ρ : X × X → Re, Ψ : Ren → Re, and θ, ζ : X × X → Ren, Φ ( f(x) − f(x∗) ) ≤ 0 ⇒ Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≤ 0. Definition 2.6. A twice differentiable function f : X → Re is said to be strictly hybrid (Φ, Ψ, ρ, ζ, θ)− quasi-invex at x∗ of second order if there exist a function Φ : Re → Re, and z ∈ Ren such that for each x ∈ X, ρ : X × X → Re, Ψ : Ren → Re, and θ, ζ : X × X → Ren, Φ ( f(x) − f(x∗) ) ≤ 0 ⇒ Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 < 0. CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 45 Definition 2.7. A twice differentiable function f : X → Re is said to be prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)− quasi-invex at x∗ of second order if there exist a function Φ : Re → Re, and z ∈ Ren such that for each x ∈ X, ρ : X × X → Re, Ψ : Ren → Re, and θ, ζ : X × X → Ren, Φ ( f(x) − f(x∗) ) < 0 ⇒ Ψ ( 〈▽f(x∗) + 1 2 ∇2f(x∗)z, ζ(x, x∗)〉 ) + ρ(x, x∗)‖θ(x, x∗)‖2 ≤ 0. Definition 2.8. A point x∗ ∈ Q is an efficient solution to (P) if there exists no x ∈ Q such that fi(x) gi(x) ≤ fi(x ∗) gi(x ∗) ∀ i = 1, · · ·, p, fj(x) gj(x) < fj(x ∗) gj(x ∗) for some j ∈ {1, · · ·, p}. Next to this context, we have the following auxiliary problem: (Pλ̄) minimizex∈Q(f1(x) − λ̄1g1(x), · · ·, fp(x) − λ̄pgp(x)), subject to x ∈ Q, where λ̄i for i ∈ {1, · · ·, p} are parameters, and λ̄i = f(x ∗ ) gi(x ∗) . Next, we introduce the efficiency solvability conditions for (Pλ̄) problem. Definition 2.9. A point x∗ ∈ Q is an efficient solution to (Pλ̄) if there does not exist an x ∈ Q such that fi(x) − λ̄igi(x) ≤ fi(x ∗) − λ̄igi(x ∗) ∀ i = 1, · · ·, p, fj(x) − λ̄jgj(x) < fj(x ∗) − λ̄jgj(x ∗) for some j ∈ {1, · · ·, p}, where λ̄i = fi(x ∗ ) gi(x ∗) for i = 1, · · ·, p. Next, we recall the following result (Verma [24]) that is crucial to developing the results for the next section based on second order (Φ, Ψ, ρ, z, θ)−invexities. Theorem 2.1. Let x∗ ∈ F and λ∗ = max1≤i≤p fi(x ∗)/gi(x ∗), for each i ∈ p, let fi and gi be twice continuously differentiable at x∗, for each j ∈ q, let the function z → Gj(z, t) be twice continuously differentiable at x∗ for all t ∈ Tj, and for each k ∈ r, let the function z → Hk(z, s) be twice continuously differentiable at x∗ for all s ∈ Sk. If x ∗ is an efficient solution of (P), if the second 46 Ram U. Verma CUBO 17, 1 (2015) order generalized Guignard constraint qualification holds at x∗, and if for any critical direction y, the set cone ({ ( ∇Gj(x ∗, t), 〈y, ∇2Gj(x ∗, t)y〉 ) : t ∈ T̂j(x ∗), j ∈ q} ∪ {∇fi(x ∗) − λ∗i ∇gi(x ∗) : i ∈ p, i 6= i0}) + span({ ( ∇Hk(x ∗, s), 〈y, ∇2Hk(x ∗, s)y〉 ) : s ∈ Sk, k ∈ r}), where T̂j(x ∗) ≡ {t ∈ Tj : Gj(x ∗, t) = 0}) is closed, then there exist u∗ ∈ U ≡ {u ∈ Rp : u ≥ 0, ∑p i=1 ui = 1} and integers ν ∗ 0 and ν ∗, with 0 ≤ ν∗0 ≤ ν ∗ ≤ n + 1, such that there exist ν∗0 indices jm, with 1 ≤ jm ≤ q, together with ν ∗ 0 points tm ∈ T̂jm(x ∗), m ∈ ν∗0, ν ∗ − ν∗0 indices km, with 1 ≤ km ≤ r, together with ν ∗ − ν∗0 points sm ∈ Skm for m ∈ ν ∗\ν∗0, and ν ∗ real numbers v∗m, with v ∗ m > 0 for m ∈ ν ∗ 0, with the property that p∑ i=1 u∗i [∇fi(x ∗) − λ∗(∇gi(x ∗)] + ν ∗ 0∑ m=1 v∗m[∇Gjm(x ∗, tm) + ν ∗ ∑ m=ν∗ 0 +1 v∗m∇Hk(x ∗, sm) = 0, (2.1) 〈y, [ p∑ i=1 u∗i [∇ 2fi(x ∗ ) − λ∗∇2gi(x ∗ )] + ν ∗ 0∑ m=1 v∗m∇ 2Gjm(x ∗, tm) + ν ∗ ∑ m=ν∗ 0 +1 v∗m∇ 2Hk(x ∗, sm) ] y〉 ≥ 0, (2.2) u∗i [fi(x ∗) − λ∗gi(x ∗)] = 0, i ∈ p, (2.3) where ν \ ν0 is the complement of the set ν0 relative to the set ν. 3 Efficiency Conditions for Problem (P) This section deals with some parametrically sufficient efficiency conditions for problem (P) under the hybrid frameworks for (Φ, Ψ, ρ, ζ, θ)−invexities. We begin with real-valued functions Ei(., x ∗, u∗) and Bj(., v) defined by Ei(x, x ∗, u∗) = ui[fi(x) − ( fi(x ∗) gi(x ∗) ) gi(x)], i ∈ {1, · · ·, p} and Bj(., v) = vjHj(x), j = 1, · · ·, m. CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 47 Theorem 3.1. Let x∗ ∈ Q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗ ) gi(x ∗) ≥ 0, gi(x ∗) > 0 and Hj for j ∈ {1, · · ·, m} be twice continuously differentiable at x∗ ∈ Q, and let there exist u∗ ∈ U = {u ∈ Rep : u > 0, Σ p i=1 ui = 1} and v ∗ ∈ Rem+ such that Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + Σmj=1v ∗ j ▽ Hj(x ∗) = 0, (3.1) 〈 ζ(x, x∗), [ p∑ i=1 u∗i [∇ 2fi(x ∗) − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)] + m∑ j=1 v∗j ∇ 2Hj(x ∗) ] z 〉 ≥ 0, (3.2) and v∗j Hj(x ∗) = 0, j ∈ {1, · · ·, m}. (3.3) Suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): (i) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ with Φ̄(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are hybrid (Φ̃, Ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for Φ̃ in- creasing with Φ̃(0) = 0, and Ψ sublinear. (ii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (Φ̃, Ψ, ρ, ζ, θ)−quasi- invex at x∗ for Φ̃ increasing with Φ̃(0) = 0, and Ψ sublinear. (iii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are prestrictly hybrid (Φ, Ψ, ρ, η, θ)−quasi-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (Φ̃, Ψ, ρ̄, ζ, θ)−quasi- invex at x∗ for Φ̃ increasing with Φ̃(0) = 0, and Ψ sublinear. (iv) For each i ∈ {1, ···, p}, fi is hybrid (Φ, Ψ, ρ1, ζ, θ)−invex and −gi is hybrid (Φ, Ψ, ρ2, ζ, θ)−invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, Hj(. , v∗) ∀ j ∈ {1, · · ·, m} is hybrid (Φ̄, Ψ, ρ3, ζ, θ)−quasi- invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, Ψ sublinear, and Σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for ρ∗ = Σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2) and for φ(x ∗) = fi(x ∗ ) gi(x ∗) . Then x∗ is an efficient solution to (P). 48 Ram U. Verma CUBO 17, 1 (2015) Proof. If (i) holds, and if x ∈ Q, then using the sublinearity of Ψ, it follows from (3.1) and (3.2) that Ψ (〈 Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗ )z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗ )z], ζ(x, x∗) 〉) + Ψ ( 〈Σmj=1v ∗ j ▽ Hj(x ∗) + 1 2 ∇2Hj(x ∗)z, ζ(x, x∗) 〉) ≥ 0. (3.4) Since v∗ ≥ 0, x ∈ Q and (3.3) holds, we have Σmj=1v ∗ j Hj(x) ≤ 0 = Σ m j=1v ∗ j Hj(x ∗ ), and in light of the hybrid (Φ̃, Ψ, ρ̄, ζ, θ)−quasi-invexity of Bj(., v ∗) at x∗, and assumptions on Φ̃, we find Φ̃ ( Σmj=1v ∗ j Hj(x) − Σ m j=1v ∗ j Hj(x ∗) ) ≤ 0, which results in Ψ ( 〈▽Hj(x ∗) + 1 2 ∇2Hj(x ∗)z, ζ(x, x∗)〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 ≤ 0. (3.5) It follows from (3.4) and (3.5) that Ψ (〈 Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)z], ζ(x, x∗) 〉) ≥ ρ̄(x, x∗)‖θ(x, x∗)‖2 ≥ −ρ(x, x∗)‖θ(x, x∗)‖2. (3.6) Since ρ(x, x∗) ≥ 0, applying the hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.6) and assump- tions on Φ, we have Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(g ∗) )gi(x)] − Σ p i=1 u∗i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) )gi(x ∗ )] ) ≥ 0, which implies Σ p i=1u ∗ i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)]) = 0. Thus, we have Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ 0. (3.7) CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 49 Since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ Q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) ) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) ) < 0 for some j ∈ {1, · · ·, p}. Hence, x∗ is an efficient solution to (P). Next, If (ii) holds, and if x ∈ Q, then using the sublinearity of Ψ, it follows from (3.1) and (3.2) that Ψ ( 〈Σ p i=1u ∗ i [▽fi(x ∗ ) − ( fi(x ∗) gi(x∗) ) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)z], ζ(x, x∗) 〉) + Ψ ( 〈Σmj=1v ∗ j ▽ Hj(x ∗ ) + 1 2 ∇2Hj(x ∗ )z, ζ(x, x∗) 〉) ≥ 0. (3.8) Since v∗ ≥ 0, x ∈ Q and (3.3) holds, we have Σmj=1v ∗ j Hj(x) ≤ 0 = Σ m j=1v ∗ j Hj(x ∗), which results (using assumptions on Φ̃) in Φ̃ ( Σmj=1v ∗ j Hj(x) − Σ m j=1v ∗ j Hj(x ∗) ) ≤ 0. Now, in light of the strictly hybrid (Φ̃, Ψ, ρ̄, ζ, θ)−quasi-invexity of Bj(., v ∗) at x∗, we find Ψ ( 〈▽Hj(x ∗) + 1 2 ∇2Hj(x ∗)z, η(x, x∗)〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 < 0. (3.9) It follows from (3.8) and (3.9) that Ψ ( 〈Σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)z], ζ(x, x∗)〉 > ρ̄(x, x∗)‖θ(x, x∗)‖2 > −ρ(x, x∗)‖θ(x, x∗)‖2. (3.10) 50 Ram U. Verma CUBO 17, 1 (2015) As a result, since ρ(x, x∗) ≥ 0, applying the prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.10) and assumptions on Φ, we have Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(g ∗) )gi(x)] − Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)] ) ≥ 0, which implies Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)]) = 0. Thus, we have Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ 0. (3.11) Since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ Q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) ) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) ) < 0 for some j ∈ {1, · · ·, p}. Hence, x∗ is an efficient solution to (P). The proof applying (iii) is similar to that of (ii), and we just need to include the proof using (iv) as follows: since x ∈ Q, it follows that Hj(x) ≤ Hj(x ∗), which implies Φ̄ ( Hj(x) − Hj(x ∗) ) ≤ 0. Then applying the hybrid (Φ̄, Ψ, ρ3, ζ, θ)−quasi-invexity of Hj at x ∗ and v∗ ∈ Rm+ , we have Ψ ( 〈Σmj=1v ∗ j ▽ Hj(x ∗), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), Σmj=1v ∗ j ∇ 2Hj(x ∗)z 〉) ≤ −Σmj=1v ∗ j ρ3‖θ(x, x ∗)‖2. Since u∗ ≥ 0 and fi(x ∗ ) gi(x ∗) ≥ 0, it follows from the hybrid (Φ, Ψ, ρ3, ζ, θ)−invexity assumptions that CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 51 Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ) = Φ ( Σ p i=1 u∗i {[fi(x) − fi(x ∗)] − ( fi(x ∗) gi(x ∗) )[gi(x) − gi(x ∗)]} ) ≥ Ψ ( Σ p i=1 u∗i {〈▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗), ζ(x, x∗)〉} + 1 2 〈ζ(x, x∗), Σ p i=1u ∗ i [∇ 2fi(x ∗ )z − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗ )z〉] ) + Σ p i=1 u∗i [ρ1 + φ(x ∗)ρ2]‖θ(x, x ∗)‖2 ≥ −Ψ ( [ 〈Σmj=1v ∗ j ▽ Hj(x ∗), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), Σmj=1v ∗ j ∇ 2Hj(x ∗)z 〉 ] ) + Σ p i=1 u∗i [ρ1 + φ(x ∗)ρ2]‖θ(x, x ∗)‖2 ≥ (Σmj=1v ∗ j ρ3 + Σ p i=1 u∗i [ρ1 + φ(x ∗ )ρ2])‖θ(x, x ∗ )‖2 = (Σmj=1v ∗ j ρ3 + ρ ∗)‖θ(x, x∗)‖2 ≥ 0, where φ(x∗) = fi(x ∗ ) gi(x ∗) and ρ∗ = Σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2). This implies that Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ) ≥ 0. Theorem 3.2. Let x∗ ∈ Q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗ ) gi(x ∗) ≥ 0, gi(x ∗) > 0 and Hj for j ∈ {1, · · ·, m} be continuously differentiable at x∗ ∈ Q, and let there exist u∗ ∈ U = {u ∈ Rep : u > 0, Σ p i=1 ui = 1} and v ∗ ∈ Rem+ such that 〈 Σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗ )] + Σmj=1v ∗ j ▽ Hj(x ∗ ), z) 〉 ≥ 0 (3.12) and v∗j Hj(x ∗) = 0, j ∈ {1, · · ·, m}. (3.13) Suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): 52 Ram U. Verma CUBO 17, 1 (2015) (i) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are first-order hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are first-order hybrid (Φ̄, Ψ, ρ̄, ζ, θ)−quasi- invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, and Ψ sublinear. (ii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are first-order hybrid prestrictly (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are first-order strictly hybrid (Φ̄, Ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, and Ψ sublinear. (iii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are first-order prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)−quasi-invex at x∗ Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are first-order strictly hybrid (Φ̄, Ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, and Ψ sublinear. and z (iv) For each i ∈ {1, · · ·, p}, fi is first-order hybrid (Φ, Ψ, ρ1, ζ, θ)−invex and −gi is first-order hybrid (Φ, Ψ, ρ2, ζ, θ)−invex at x ∗ for Φ(a) ≥ 0 ⇒ a ≥ 0. Hj(. , v∗) ∀ j ∈ {1, · · ·, m} is hybrid (Φ̄, Ψ, ρ̄3, ζ, θ)−quasi-invex at x ∗, and Σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for Φ̄ increasing with Φ̄(0) = 0, ρ∗ = Σ p i=1u ∗ i (ρ1 + φ(x ∗)ρ2) for φ(x ∗) = fi(x ∗ ) gi(x ∗) , and Ψ sublineaer. Then x∗ is an efficient solution to (P). Proof. Although the proof is similar to that of Theorem 3.1), we include for the sake of the completeness. If we consider (i), then proceeding as in Theorem 3.1 (and using the first-order hybrid (Φ, Ψ, ρ, ζ, θ)−invexity assumptions instead), we arrive at Ψ ( 〈Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)], ζ(x, x∗)〉 ) ≥ ρ(x, x∗)‖θ(x, x∗)‖2. (3.14) Since ρ(x, x∗) ≥ 0, applying the hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.14) and assump- tions on Φ, we have Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(g ∗) )gi(x)] − Σ p i=1 u∗i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) )gi(x ∗ )] ) ≥ 0, which implies Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) )gi(x ∗)]) = 0. CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 53 Thus, we have Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) )gi(x)] ≥ 0. (3.15) Since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ Q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) ) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) ) < 0 for some j ∈ {1, · · ·, p}. Hence, x∗ is an efficient solution to (P). Theorem 3.3. Let x∗ ∈ Q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗ ) gi(x ∗) ≥ 0, gi(x ∗) > 0 and Hj for j ∈ {1, · · ·, m} be twice continuously differentiable at x∗ ∈ Q, and let there exist u∗ ∈ U = {u ∈ Rep : u > 0, Σ p i=1 ui = 1} and v ∗ ∈ Rem+ such that Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) ) ▽ gi(x ∗)] + Σmj=1v ∗ j ▽ Hj(x ∗) = 0 (3.16) 〈 ζ(x, x∗), [ p∑ i=1 u∗i [∇ 2fi(x ∗) − ( fi(x ∗) gi(x ∗) )∇2gi(x ∗)] + m∑ j=1 v∗j ∇ 2Hj(x ∗) ] z 〉 ≥ 0, (3.17) and v∗j Hj(x ∗) = 0, j ∈ {1, · · ·, m}. (3.18) Suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): (i) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are hybrid (ρ, ζ, θ)−pseudo-invex at x∗, and Bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are hybrid (ρ, ζ, θ)−quasi-invex at x∗. (ii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, ···, p} are prestrictly hybrid (ρ, ζ, θ)−pseudo-invex at x∗, and Bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are hybrid (ρ, ζ, θ)−strictly-quasi-invex at x∗. (iii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are strictly hybrid (ρ, ζ, θ)−pseudo-invex at x∗, and Bj(. , v ∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (ρ, ζ, θ)−quasi-invex at x∗. (iv) For each i ∈ {1, · · ·, p}, fi is hybrid (ρ1, ζ, θ)−invex and −gi is (ρ2, ζ, θ)−invex at x ∗. Hj(. , v ∗) ∀ j ∈ {1, · · ·, m} is hybrid (ρ3, ζ, θ)−quasi-invex at x ∗, and Σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for ρ∗ = Σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2) and for φ(x ∗) = fi(x ∗ ) gi(x ∗) . 54 Ram U. Verma CUBO 17, 1 (2015) Then x∗ is an efficient solution to (P). Proof. The proof is similar to that of Theorem 3.1 based on the second order hybrid (ρ, ζ, θ)− invexity assumptions. We observe that Theorem 3.1 can be further generalized to the case of the ǫ−Efficient condi- tions based on the hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks. As a matter of fact, we generalize the ǫ−efficient solvability conditions for problem (P) based on the work of Verma [22], and Kim, Kim and Lee [6], where they have investigated the ǫ−efficiency as well as the weak ǫ−efficiency conditions for multiobjective fractional programming problems under constraint qualifications. To the best of our knowledge, the results established in this communication (Theorem 3.1 and Theo- rem 3.4) generalize and unify most of the results on the multiobjective fractional programming to the context of the generalized invexities in the literature. We recall some auxiliary concepts (for the hybrid (Φ, Ψ, ρ, ζ, θ)−invexity) crucial to the problem on hand. Definition 3.1. A point x∗ ∈ Q is an ǫ−efficient solution to (P) if there does not exist an x ∈ Q such that fi(x) gi(x) ≤ fi(x ∗) gi(x ∗) − ǫi ∀ i = 1, · · ·, p, fj(x) gj(x) < f(jx ∗) gj(x ∗) − ǫj for some j ∈ {1, · · ·, p}, where ǫi=(ǫ1, · · ·, ǫp) is with ǫi ≥ 0 for i = 1, · · ·, p. For ǫ = 0, Definition 3.1 reduces to the case that x∗ ∈ Q is an efficient solution to (P). Next, we start with real-valued functions Ei(., x ∗, u∗) and Bj(., v), respectively, defined by Ei(x, x ∗, u∗) = ui[fi(x) − ( fi(x ∗) gi(x ∗) − ǫi ) gi(x)], i ∈ {1, · · ·, p} and Bj(., v) = vjHj(x), j = 1, · · ·, m. CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 55 Theorem 3.4. Let x∗ ∈ Q, let fi, gi for i ∈ {1, · · ·, p} with fi(x ∗) ≥ ǫigi(x ∗), gi(x ∗) > 0 and Hj for j ∈ {1, · · ·, m} be twice continuously differentiable at x ∗ ∈ Q, and let there exist u∗ ∈ U = {u ∈ Rep : u > 0, Σ p i=1 ui = 1}, v ∗ ∈ Rem+ and z ∈ Re n such that Σ p i=1 u∗i [∇fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi ) ▽ gi(x ∗ )] + Σmj=1v ∗ j ▽ Hj(x ∗ ) = 0, (3.19) 〈 ζ(x, x∗), [ p∑ i=1 u∗i [∇ 2fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi ) ∇2gi(x ∗ )] + m∑ j=1 v∗j ∇ 2Hj(x ∗ ) ] z 〉 ≥ 0, (3.20) and v∗j Hj(x ∗ ) = 0, j ∈ {1, · · ·, m}. (3.21) Suppose, in addition, that any one of the following assumptions holds (for ρ(x, x∗) ≥ 0): (i) Ei(. ; x ∗, u∗) ∀ i ∈ {1, ···, p} are hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v ∗) ∀ j ∈ {1, · · ·, m} arehybrid (Φ̃, Ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for Φ̃ increasing with Φ̃(0) = 0, and Ψ sublinear. (ii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (Φ, Ψ, ρ, ζ, θ)−quasi- invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, and Ψ sublinear. (iii) Ei(. ; x ∗, u∗) ∀ i ∈ {1, · · ·, p} are strictly hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Bj(. , v∗) ∀ j ∈ {1, · · ·, m} are strictly hybrid (Φ̄, Ψ, ρ̄, ζ, θ)−quasi-invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, and Ψ sublinear. (iv) For each i ∈ {1, · · ·, p}, fi is hybrid (Φ, Ψ, ρ1, ζ, θ)−invex and −gi is (Φ, Ψ, ρ2, ζ, θ)−invex at x∗ for Φ(a) ≥ 0 ⇒ a ≥ 0, and Hj(. , v∗) ∀ j ∈ {1, · · ·, m} is hybrid (Φ̄, Ψ, ρ3, ζ, θ)−quasi- invex at x∗ for Φ̄ increasing with Φ̄(0) = 0, Ψ sublinear, and Σmj=1v ∗ j ρ3 + ρ ∗ ≥ 0 for ρ∗ = Σ p i=1 u∗i (ρ1 + φ(x ∗)ρ2), where φ(x ∗) = fi(x ∗ ) gi(x ∗) − ǫi. Then x∗ is an ǫ−efficient solution to (P). Proof. If (i) holds, and if x ∈ Q, then it follows using the sublinearity of Ψ from (3.1) and (3.2) 56 Ram U. Verma CUBO 17, 1 (2015) that Ψ ( 〈Σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z], ζ(x, x∗) 〉) + Ψ ( 〈Σmj=1v ∗ j ▽ Hj(x ∗ ) + 1 2 ∇2Hj(x ∗ )z, ζ(x, x∗) 〉) ≥ 0. (3.22) Since v∗ ≥ 0, x ∈ Q and (3.3) holds, we have Σmj=1v ∗ j Hj(x) ≤ 0 = Σ m j=1v ∗ j Hj(x ∗), which implies Σmj=1v ∗ j Hj(x) − Σ m j=1v ∗ j Hj(x ∗) ≤ 0, so in light of the hybrid (Φ̃, Ψ, ρ̄, ζ, θ)−quasi-invexity of Bj(., v ∗) at x∗, and assumptions on Φ̃, it results in Φ̃ ( Σmj=1v ∗ j Hj(x) − Σ m j=1v ∗ j Hj(x ∗ ) ) ≤ 0, which implies Ψ ( 〈▽Hj(x ∗), ζ(x, x∗)〉 + 1 2 〈ζ(x, x∗), ∇2Hj(x ∗)z〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 ≤ 0. (3.23) It follows from (3.22) and (3.23) that Ψ ( 〈Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗)], z〉 + 1 2 〈 ζ(x, x∗), p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z] 〉) ≥ ρ̄(x, x∗)‖θ(x, x∗)‖2 ≥ −ρ(x, x∗)‖θ(x, x∗)‖2. (3.24) As a result, since ρ(x, x∗) ≥ 0, applying the hybrid (Φ, Ψ, ρ, ζ, θ)− pseudo-invexity at x∗ to (3.24) and assumptions on Φ, we have Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] −Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗)] ) ≥ 0, CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 57 which implies Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗)] ≥ Σ p i=1u ∗ i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗ )] −Σ p i=1 u∗i ǫigi(x ∗) = 0. Thus, we have Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ 0. (3.25) Since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ Q such that ∑p i=1 fi(x)∑p i=1 gi(x) − ( fi(x ∗) gi(x ∗) − ǫi) ≤ 0 ∀ i = 1, · · ·, p, ∑p j=1 fj(x) ∑p j=1 gj(x) − ( fj(x ∗) gj(x ∗) − ǫj) < 0 for some j ∈ {1, · · ·, p}. Hence, x∗ is an ǫ−efficient solution to (P). If (ii) holds, and if x ∈ Q, then it follows from (3.1) and (3.2) that Ψ (〈 Σ p i=1 u∗i [▽fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗ )] + 1 2 p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z], ζ(x, x∗) 〉) + Ψ (〈 Σmj=1v ∗ j ▽ Hj(x ∗ ) + 1 2 ∇2Hj(x ∗ )z, ζ(x, x∗) 〉) ≥ 0. (3.26) Since v∗ ≥ 0, x ∈ Q and (3.3) holds, we have Σmj=1v ∗ j Hj(x) ≤ 0 = Σ m j=1v ∗ j Hj(x ∗), or Σmj=1v ∗ j Hj(x) − Σ m j=1v ∗ j Hj(x ∗ ) ≤ 0, 58 Ram U. Verma CUBO 17, 1 (2015) which implies based on assumptions on Φ̃ that Φ̃ ( Σmj=1v ∗ j Hj(x) − Σ m j=1v ∗ j Hj(x ∗) ) ≤ 0. Next, in light of the strict (Φ̃, Ψ, ρ̄, ζ, θ)−quasi-invexity of Bj(., v ∗) at x∗ with Φ̃ increasing and Φ̃(0) = 0, we find Ψ ( 〈▽Hj(x ∗), ζ(x, x∗)〉 + 1 2 〈ζ(x, x∗), ∇2Hj(x ∗)z〉 ) + ρ̄(x, x∗)‖θ(x, x∗)‖2 < 0. (3.27) It follows from (3.26) and (3.27) that Ψ ( 〈Σ p i=1 u∗i [▽fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗)], ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), p∑ i=1 u∗i [∇ 2fi(x ∗)z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗)z] 〉) > ρ̄(x, x∗)‖θ(x, x∗)‖2 > −ρ(x, x∗)‖θ(x, x∗)‖2. (3.28) As a result, since ρ(x, x∗) ≥ 0, applying the prestrictly hybrid (Φ, Ψ, ρ, ζ, θ)−pseudo-invexity at x∗ to (3.28) and assumptions on Φ, we have Φ ( Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] − Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) ) − ǫi)gi(x ∗)] ) ≥ 0, which implies Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ Σ p i=1 u∗i [fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗)] ≥ Σ p i=1 u∗i [fi(x ∗ ) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x ∗ )] − Σ p i=1 u∗i ǫigi(x ∗ ) = 0. Thus, we have Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ 0. (3.29) Since u∗i > 0 for each i ∈ {1, · · ·, p}, we conclude that there does not exist an x ∈ Q such that fi(x) gi(x) − ( fi(x ∗) gi(x ∗) − ǫi) ≤ 0 ∀ i = 1, · · ·, p, fj(x) gj(x) − ( fj(x ∗) gj(x ∗) − ǫj) < 0 for some j ∈ {1, · · ·, p}. CUBO 17, 1 (2015) Hybrid (Φ, Ψ, ρ, ζ, θ)−invexity frameworks . . . 59 Hence, x∗ is an ǫ−efficient solution to (P). The proof applying (iii) is similar to that of (ii), and we just need to include the proof using (iv) as follows: since x ∈ Q, it follows that Hj(x) ≤ Hj(x ∗). Then applying the (Φ̄, Ψ, ρ3, ζ, θ)−quasi- invexity of Hj at x ∗ and v∗ ∈ Rm+ , we have Ψ (〈 Σmj=1v ∗ j ▽ Hj(x ∗ ), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), Σmj=1v ∗ j ∇ 2Hj(x ∗ )z 〉) ≤ −Σmj=1v ∗ j ρ3‖θ(x, x ∗)‖2. Since u∗ ≥ 0 and fi(x ∗) ≥ ǫigi(x ∗), it follows from (Φ, Ψ, ρ3, ζ, θ)−invexity assumptions that Φ ( Σ p i=1u ∗ i [fi(x) − ( fi(x ∗) gi(x∗) − ǫi)gi(x)] ) = Φ ( Σ p i=1 u∗i {[fi(x) − fi(x ∗)] − ( fi(x ∗) gi(x ∗) − ǫi)[gi(x) − gi(x ∗)] + ǫigi(x ∗)} ) ≥ Ψ ( Σ p i=1 u∗i {〈▽fi(x ∗) − ( fi(x ∗) gi(x ∗) − ǫi) ▽ gi(x ∗), ζ(x, x∗)〉} + 1 2 〈ζ(x, x∗), Σ p i=1 u∗i [∇ 2fi(x ∗ )z − ( fi(x ∗) gi(x ∗) − ǫi)∇ 2gi(x ∗ )z〉] ) + [ρ1 + φ(x ∗)ρ2]‖θ(x, x ∗)‖2 + Σ p i=1 u∗i ǫigi(x ∗) ≥ −Ψ ( [ 〈Σmj=1v ∗ j ▽ Hj(x ∗), ζ(x, x∗)〉 + 1 2 〈 ζ(x, x∗), Σmj=1v ∗ j ∇ 2Hj(x ∗)z 〉 ] ) + Σ p i=1 u∗i [ρ1 + ( fi(x ∗) gi(x ∗) − ǫi)ρ2]‖θ(x, x ∗)‖2 + Σ p i=1 u∗i ǫigi(x ∗) ≥ (Σmj=1v ∗ j ρ3 + Σ p i=1 u∗i [ρ1 + ( fi(x ∗) gi(x ∗) − ǫi)ρ2])‖θ(x, x ∗ )‖2 + Σ p i=1 u∗i ǫigi(x ∗ ) = (Σmj=1v ∗ j ρ3 + ρ ∗)‖θ(x, x∗)‖2 + Σ p i=1 u∗i ǫigi(x ∗) ≥ (Σmj=1v ∗ j ρ3 + ρ ∗)‖θ(x, x∗)‖2 ≥ 0. Therefore, we have Σ p i=1 u∗i [fi(x) − ( fi(x ∗) gi(x ∗) − ǫi)gi(x)] ≥ 0. (3.30) 60 Ram U. Verma CUBO 17, 1 (2015) Thus, we conclude that there does not exist an x ∈ Q such that ∑p i=1 fi(x)∑p i=1 gi(x) − ( fi(x ∗) gi(x ∗) − ǫi) ≤ 0 ∀ i = 1, · · ·, p, ∑p j=1 fj(x) ∑p j=1 gj(x) − ( fj(x ∗) gj(x ∗) − ǫj) < 0 for some j ∈ {1, · · ·, p}. Hence, x∗ is an ǫ−efficient solution to (P). 4 Concluding Remarks We observe that the obtained results in this communication can be generalized to the case of multiobjective fractional programming with generalized hybrid invex functions of higher orders (including the exponential type generalized invexities), for instance, based on the work of Mishra and Rueda [11], Mishra, Laha and Verma [13], and Zalmai and Zhang [37] to the case of the efficiency as well as to the ǫ−efficiency conditions relating to the minimax fractional programming problems involving generalized invex functions. Received: May 2014. Accepted: October 2014. References [1] A. Ben-Israel and B. Mond, What is the invexity? Journal of Australian Mathematical Society Ser. 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