IN SEMIINFINITE FRACTIONAL PROGRAMMING. PART II: FIRST-ORDER DUALITY MODELS

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1 GENERALIZED α, β, γ, η, ρ, θ-invex FUNCTIONS IN SEMIINFINITE FRACTIONAL PROGRAMMING. PART II: FIRST-ORDER DUALITY MODELS G.J. ZALMAI Communicated by the former editorial board In this paper, we discuss a fairly large number of first-order duality results under various generalized α, β, γ, η, ρ, θ-invexity assumptions for a semiinfinite fractional programming problem. AMS 200 Subject Classification: 49N5, 90C30, 90C32, 90C34, 90C46. Key words: Semiinfinite programming, fractional programming, generalized α, β, γ, η, ρ, θ-invex functions, infinitely many equality and inequality constraints, first-order dual problems, duality theorems.. INTRODUCTION In this paper, we state and prove a fairly large number of parametric and parameter-free duality results under various generalized α, β, γ, η, ρ, θ- invexity assumptions for the following semiinfinite fractional programming problem: P Minimize ϕx = fx gx 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, where 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, f and g are real-valued functions defined on X, for each j q, z G j z, t is a real-valued function defined on X for all t T j, for each k r, z H k z, s is a real-valued function defined on X for all s S k, for each j q and k r, t G j x, t and s H k x, s are continuous real-valued functions defined, respectively, on T j and S k for all x X, and gx > 0 for all x satisfying the constraints of P. REV. ROUMAINE MATH. PURES APPL., ,, 35 62

2 36 G.J. Zalmai 2 The present study is essentially a continuation of the investigation initiated in the companion paper [2 where some information about fractional programming is presented, the current status of semiinfinite programming is briefly discussed and numerous key references are cited, an overview of the concept of α, β, γ, η, ρ, θ-invexity and some of its extensions is given, and a fairly large number of sets of global sufficient optimality results under various generalized α, β, γ, η, ρ, θ-invexity assumptions are established. For the necessary background material and preliminaries, the reader is referred to [2. Here we shall make use of the sufficient optimality results developed in [2 and a set of necessary optimality conditions, which will be recalled in the next section, and construct several first-order duality models for P and prove appropriate duality theorems. Second- and higher-order counterparts of these results are discussed in [3, 4. Although presently various duality results exist in the related literature for several classes of static and dynamic fractional optimization problems with a finite number of constraints, so far no such results are available for any kind of semiinfinite fractional programming problems involving generalized α, β, γ, η, ρ, θ-invex functions. To the best of our knowledge, all the duality results established in this paper are new in the area of semiinfinite programming. The rest of the paper is organized as follows. In Section 2, we recall an auxiliary result which will be needed in the sequel. We begin our discussion of duality for P in Section 3 where we formulate two parametric duality models with somewhat restricted constraint structures, and prove weak, strong, and strict converse duality theorems under appropriate generalized α, β, γ, η, ρ, θ- invexity assumptions. In Section 4, we present another pair of parametric duality models with relatively more general constraint structures that allow for a greater variety of generalized α, β, γ, η, ρ, θ-invexity hypotheses under which duality can be established. We continue our investigation of parametric duality for P in Section 5 where we consider two generalized duality models and prove several sets of duality results. These duality models are, in fact, two families of dual problems for P whose members can easily be identified by appropriate choices of certain sets and functions. The nonparametric counterparts of these generalized duality models are taken up in Section 6, and a number of duality results are established. Finally, in Section 7, we summarize our main results and also point out some additional research opportunities arising from certain modifications of the principal problem model P. 2. PRELIMINARIES For proving our duality theorems, we shall need a few basic definitions and auxiliary results. For the definitions of the new classes of generalized

3 3 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 37 α, β, γ, η, ρ, θ-invex functions that will be utilized in the statements and proofs of our duality theorems, the reader is referred to [2, and the two auxiliary results that will be needed in the sequel for proving strong and strict converse duality theorems are stated below. Theorem 2.. [5. Let x F and λ = fx /gx, let f and g be continuously differentiable at x, for each j q, let the function z G j z, t be continuously differentiable at x for all t T j, and for each k r, let the function z H k z, s be continuously differentiable at x for all s S k. If x is an optimal solution of P, if the generalized Guignard constraint qualification holds at x, and if the set cone{ G j x, t : t ˆT j x, j q} + span{ H k x, s : s S k, k r} is closed, then there exist 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 tm ˆT jm x, m ν0, ν ν0 indices k m, with k m r, together with ν ν0 points sm S km for m ν \ν0, and ν real numbers vm, with vm > 0 for m ν0, with the property that fx λ gx + ν0 vm G jm x, t m + m= ν m=ν 0 + v m H km x, s m = 0, where ˆT j x = {t T j : G j x, t = 0} and ν \ν0 is the complement of the set ν0 relative to ν. Eliminating λ and redefining the Lagrange multipliers v m, m ν, we obtain the following parameter-free version of Theorem 2.. Theorem 2.2. Let x F, let f and g be continuously differentiable at x, for each j q, let the function z G j z, t be continuously differentiable at x for all t T j, and for each k r, let the function z H k z, s be continuously differentiable at x for all s S k. If x is an optimal solution of P, if the generalized Guignard constraint qualification holds at x, and if the set cone{ G j x, t : t ˆT j x, j q} + span{ H k x, s : s S k, k r} is closed, then there exist 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 sm S km for m ν \ν0, and ν real numbers vm, with vm > 0 for m ν0, with the property that ν0 ν gx fx fx gx + vm G jm x, t m + vm H km x, s m = 0. m= m=ν0 + We shall call x a normal feasible solution of P if x satisfies all the constraints of P, if the generalized Guignard constraint qualification holds

4 38 G.J. Zalmai 4 at x, and if the set cone{ G j x, t : t ˆT j x, j q} + span{ H k x, s : s S k, k r} is closed. The forms and features of these results will serve as our guide in this paper for constructing several parametric and nonparametric duality models for P and proving appropriate duality theorems. In the remainder of this paper, we assume that the functions f, g, G j, t, t T j, j q, and H k, s, s S k, k r, are continuously differentiable on the open set X. With regard to the choice of the type of generalized invex functions, specified in Definitions in [2, to be used in the statements and proofs of our duality theorems, we shall consistently use the cases in which the functions α and β are nonzero for all x, y X X. All the duality results established in this paper can be modified, restated, and proved for the other cases in a straightforward manner. 3. DUALITY MODEL I In this section, we consider two dual problems with relatively simple constraint structures and prove weak, strong, and strict converse duality theorems under appropriate α, β, γ, η, 0, θ-invexity conditions. More general duality models and results for P will be discussed in the subsequent sections. Let { H = y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s : y X; λ R; 0 ν 0 ν n + ; v R ν, v i > 0, i ν 0 ; J ν0 = j, j 2,..., j ν0, j i q; K ν\ν0 = k ν0 +,..., k ν, k i r; t = t, t 2,..., t ν 0, t i T ji ; s = s ν 0+,..., s ν, s i S ki }. Consider the following two problems: DI λ subject to sup y,v,λ,ν,ν 0,J ν0,k ν\ν0, t, s H 3. fy λ gy + ν 0 m= ν 0 v m G jm y, t m fy λgy + v m G jm y, t m + m= DI sup y,v,λ,ν,ν 0,J ν0,k ν\ν0, t, s H λ subject to 3.2 and ν m=ν 0 + ν m=ν 0 + v m H km y, s m = 0, v m H km y, s m 0;

5 5 Generalized α, β, γ, η, ρ, θ-invex functions. Part II fy λ gy + + ν m=ν 0 + ν 0 m= v m G jm y, t m v m H km y, s m, e βx,yηx,y 0 for all x F, where β is a function from X X to R and η is a function from X X to R n. The structures of the two problems designated above as DI and DI are based directly on the form and contents of the necessary optimality conditions of Theorem 2.. This is, of course, the standard method for constructing Wolfe-type dual problems. Comparing DI and DI, we see that DI is relatively more general than DI in the sense that any feasible solution of DI is also feasible for DI, but the converse is not necessarily true. Furthermore, we observe that 3. is a system of n equations, whereas 3.3 is a single inequality. Clearly, from a computational point of view, DI is preferable to DI because of the dependence of 3.3 on the feasible set of P. Despite these apparent differences, it turns out that the statements and proofs of all the duality theorems for P DI and P DI are almost identical and, therefore, we shall consider only the pair P DI. The next two theorems show that DI is a dual problem for P. Theorem 3. Weak Duality. Let x and y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s be arbitrary feasible solutions of P and DI, respectively, and assume that the Lagrangian-type function z Lz, v, λ, t, s = fz λgz + ν 0 m= v m G jm z, t m + ν m=ν 0 + v m H km z, s m is α, β, γ, η, 0, θ-pseudoinvex at y and γx, y > 0, where t t, t 2,..., t ν 0 and s s ν 0+, s ν 0+2,..., s ν. Then ϕx λ. Proof. From 3. we observe that fy λ gy + + ν 0 m= v m G jm y, t m ν m=ν 0 + v m H km y, s m, e βx,yηx,y = 0, which in view of our α, β, γ, η, 0, θ-pseudoinvexity assumption implies that αx, y γx, y e αx,y[lx,v,λ, t, s Ly,v,λ, t, s 0.

6 40 G.J. Zalmai 6 We need to consider two cases: αx, y > 0 and αx, y < 0. If we assume that αx, y > 0 and recall that γx, y > 0, then the above inequality becomes which implies that e αx,y[lx,v,λ, t, s Ly,v,λ, t, s, Lx, v, λ, t, s Ly, v, λ, t, s. In view of the dual feasibility of y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s and 3.2, the right-hand side of the above inequality is greater than or equal to zero, and hence we have Lx, v, λ, t, s 0. Inasmuch as x F, and v m > 0, m ν 0, this inequality simplifies to fx λgx 0. Hence, ϕx = fx/gx λ. If we assume that αx, y < 0, we arrive at the same conclusion. Theorem 3.2 Strong Duality. Let x be a normal optimal solution of P and assume that for each feasible solution y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s of DI, the conditions specified in Theorem 3. are satisfied. Then there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is an optimal solution of DI and ϕx = λ. Proof. Since x is a normal optimal solution of P, by Theorem 2., there exist v, λ, ν, ν 0, J ν 0, K ν \ν 0, t, and s such that w x, v, λ, ν, ν 0, J ν 0, K ν \ν 0, t, s is a feasible solution of DI and ϕx = λ. If w were not optimal, then there would exist a feasible solution x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s of DI such that ϕx = λ < λ, which contradicts Theorem 3.. Therefore, we conclude that w is an optimal solution of DI. We also have the following converse duality result for P and DI. Theorem 3.3 Strict Converse Duality. Let x be a normal optimal solution of P, let x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s be an optimal solution of DI, and assume that the function z Lz, ṽ, λ, t, s is strictly α, β, γ, η, 0, θ- pseudoinvex at x and γx, x > 0. Then x = x, that is, x is an optimal solution of P and ϕx = λ. Proof. a: Suppose to the contrary that x x. By Theorem 2., there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is a feasible solution of DI and ϕx = λ. Now proceeding as in the proof of Theorem 3. with x replaced by x and y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s by x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s and using the condition set forth above, we arrive at the strict inequality fx λgx > 0, and so ϕx > λ, which contradicts the fact that ϕx = λ λ. Therefore, we conclude that x = x and ϕx = λ.

7 7 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 4 4. DUALITY MODEL II In this section, we consider two duality models with special constraint structures that allow for a greater variety of generalized α, β, γ, η, ρ, θ-invexity conditions under which duality can be established. Consider the following two problems: DII subject to sup y,v,λ,ν,ν 0,J ν0,k ν\ν0, t, s H 4. fy λ gy + ν 0 m= λ v m G jm y, t m fy λgy 0, ν m=ν G jm y, t m 0, m ν 0, 4.4 v m H km y, s m 0, m ν\ν 0 ; DII sup λ y,v,λ,ν,ν 0,J ν0,k ν\ν0, t, s H v m H km y, s m = 0, subject to 3.3 and The remarks and observations made earlier about the relationships between DI and DI are, of course, also valid for DII and DII. The next two theorems show that DII is a dual problem for P. Theorem 4. Weak Duality. Let x and w y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s be arbitrary feasible solutions of P and DII, respectively, and assume that any one of the following five sets of hypotheses is satisfied: a i z fz λgz is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y > 0; ii for each m ν 0, z G jm z, t m is α, β, ˆγ m, η, ˆρ m, θ-quasiinvex at y; iii for each m ν\ν 0, z v m H km z, s m is α, β, γ m, η, ρ m, θ-quasiinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ν m=ν 0 + ρ mx, y 0; b i z fz λgz is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y; iii for each m ν\ν 0, z v m H km z, s m is α, β, γ m, η, ρ m, θ-quasiinvex at y; iv ρx, y + ˆρx, y + ν m=ν 0 + ρ mx, y 0; c i z fz λgz is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y > 0; ii for each m ν 0, z G jm z, t m is α, β, ˆγ, η, ˆρ m, θ-quasiinvex at y;

8 42 G.J. Zalmai 8 iii z ν m=ν 0 + v mh km z, s m is α, β, γ, η, ρ, θ-quasiinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ρx, y 0; d i z fz λgz is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y; iii z ν m=ν 0 + v mh km z, s m is α, β, γ, η, ρ, θ-quasiinvex at y; iv ρx, y + ˆρx, y + ρx, y 0; e i z fz λgz is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m + ν m=ν 0 + v mh km z, s m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y; iii ρx, y + ˆρx, y 0. Then ϕx λ. Proof. a: From the primal feasibility of x and 4.3 we see that G jm x, t m 0 G jm y, t m for each m ν 0, and hence αx, y ˆγ mx, y e αx,y[g jm x,tm G jm y,t m 0, which in view of ii implies that Gjm y, t m, e βx,yηx,y ˆρ m x, y θx, y 2. As v m > 0 for each m ν 0, the above inequalities yield 4.5 ν 0 ν 0 v m G jm y, t m, e βx,yηx,y v m ˆρ m x, y θx, y 2. m= m= Similarly, from the primal feasibility of x, 4.4, and iii we deduce that 4.6 ν ν v m H km y, s m, e βx,yηx,y ρ m x, y θx, y 2. m=ν 0 + m=ν 0 + Combining 4. with 4.5 and 4.6, and using iv, we obtain fy λ gy, e βx,yηx,y [ ν 0 ν v m ˆρ m x, y + ρ m x, y θx, y 2 ρx, y θx, y 2, m= m=ν 0 + which in view of i implies that αx, y γx, y e αx,y{fx λgx [fy λgy} 0.

9 9 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 43 Since γx, y > 0, this inequality implies that fx λgx fy λgy, and therefore using 4.2 we have the desired inequality ϕx λ. b e: The proofs are similar to that of part a. Theorem 4.2 Strong Duality. Let x be a normal optimal solution of P and assume that any one of the five sets of conditions set forth in Theorem 4. is satisfied for all feasible solutions of DII. Then there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is an optimal solution of DII and ϕx = λ. Proof. The proof is similar to that of Theorem 3.2. Theorem 4.3 Strict Converse Duality. Let x be a normal optimal solution of P, let w x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s be an optimal solution of DII, and assume that any one of the five sets of conditions specified in Theorem 4. is satisfied and that the function z fz λgz is strictly α, β, γ, η, ρ, θ-pseudoinvex at x. Then x = x, that is, x is an optimal solution of P, and ϕx = λ. Proof. a: Suppose to the contrary that x x. By Theorem 2., there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is a feasible solution of DII and ϕx = λ. Now proceeding as in the proof of Theorem 4. with x replaced by x and w by w and using our strict α, β, γ, η, ρ, θ-pseudoinvexity assumption, we arrive at the strict inequality fx λgx > 0 and so ϕx > λ, which contradicts the fact that ϕx = λ λ. Therefore, we conclude that x = x and ϕx = λ. b e: The proofs are similar to that of part a. Theorem 4.4 Weak Duality. Let x and y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s be arbitrary feasible solutions of P and DII, respectively, and assume that any one of the following five sets of hypotheses is satisfied: a b i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each m ν 0, z G jm z, t m is α, β, ˆγ m, η, ˆρ m, θ-quasiinvex at y; iii for each m ν\ν 0, z v m H km z, s m is α, β, γ m, η, ρ m, θ-quasiinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ν m=ν 0 + ρ mx, y > 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y;

10 44 G.J. Zalmai 0 c d e iii for each m ν\ν 0, z v m H km z, s m is α, β, γ m, η, ρ m, θ-quasiinvex at y; iv ρx, y + ˆρx, y + ν m=ν 0 + ρ mx, y > 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each m ν 0, z G jm z, t m is α, β, ˆγ m, η, ˆρ m, θ-quasiinvex at y; iii z ν m=ν 0 + v mh km z, s m is α, β, γ, η, ρ, θ-quasiinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ρx, y > 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y; iii z ν m=ν 0 + v mh km z, s m is α, β, γ, η, ρ, θ-quasiinvex at y; iv ρx, y + ˆρx, y + ρx, y > 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m + ν m=ν 0 + v mh km z, s m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y; iii ρx, y + ˆρx, y > 0; Then ϕx λ. Proof. a : Because of our assumptions specified in ii and iii, 4.5 and 4.6 remain valid for the present case. From 4., 4.5, 4.6, and iv we deduce that fy λ gy, e βx,yηx,y [ ν 0 v m ˆρ m x, y + m= ν m=ν 0 + ρ m x, y θx, y 2 > ρx, y θx, y 2, which in view of i implies that αx, y γx, y e αx,y{fx λgx [fy λgy} 0. Since γx, y > 0, it follows from this inequality that fx λgx fy λgy 0, where the second inequality follows from 4.2. Therefore, we have the desired inequality ϕx λ. b e: The proofs are similar to that of part a. Theorem 4.5 Strong Duality. Let x be a normal optimal solution of P and assume that any one of the five sets of conditions set forth in

11 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 45 Theorem 4.4 is satisfied for all feasible solutions of DII. Then there exist v, λ, ν, ν 0, J ν 0, K ν \ν 0, t, and s such that x, v, λ, ν, ν 0, J ν 0, K ν \ν 0, t, s is an optimal solution of DII and ϕx = λ. Proof. The proof is similar to that of Theorem 3.2. Theorem 4.6 Strict Converse Duality. Let x be a normal optimal solution of P, let w x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s be an optimal solution of DII, and assume that any one of the five sets of conditions set forth in Theorem 4.4 is satisfied for the feasible solution w of DII and that the function z fz λgz is α, β, γ, η, ρ, θ-quasiinvex at x. Then x = x and ϕx = λ. Proof. The proof is similar to that of Theorem 4.3. Theorem 4.7 Weak Duality. Let x and y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s be arbitrary feasible solutions of P and DII, respectively, and assume that any one of the following seven sets of hypotheses is satisfied: a b c d i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each m ν 0, z G jm z, t m is strictly α, β, ˆγ m, η, ˆρ m, θ- pseudoinvex at y; iii for each m ν\ν 0, z v m H km z, s m is α, β, γ m, η, ρ m, θ-quasiinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ν m=ν 0 + ρ mx, y 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is strictly α, β, ˆγ, η, ˆρ, θ-pseudoinvex at y; iii for each m ν\ν 0, z v m H km z, s m is α, β, γ m, η, ρ m, θ-quasiinvex at y; iv ρx, y + ˆρx, y + ν m=ν 0 + ρ mx, y 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each m ν 0, z G jm z, t m is α, β, ˆγ m, η, ˆρ m, θ-quasiinvex at y; iii for each m ν\ν 0, z v m H km z, s m is strictly α, β, γ m, η, ρ m, θ- pseudoinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ν m=ν 0 + ρ mx, y 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0;

12 46 G.J. Zalmai 2 e f g ii for each m ν 0, z G jm z, t m is α, β, ˆγ m, η, ˆρ m, θ-quasiinvex at y; iii z ν m=ν 0 + v mh km z, s m is strictly α, β, γ, η, ρ, θ-pseudoinvex at y; iv ρx, y + ν 0 m= v m ˆρ m x, y + ρx, y 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is strictly α, β, ˆγ, η, ˆρ, θ-pseudoinvex at y; iii z ν m=ν 0 + v mh km z, s m is α, β, γ, η, ρ, θ-quasiinvex at y; iv ρx, y + ˆρx, y + ρx, y 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m is α, β, ˆγ, η, ˆρ, θ-quasiinvex at y; iii z ν m=ν 0 + v mh km z, s m is strictly α, β, γ, η, ρ, θ-pseudoinvex at y; iv ρx, y + ˆρx, y + ρx, y 0; i z fz λgz is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii z ν 0 m= v mg jm z, t m + ν m=ν 0 + v mh km z, s m is strictly α, β, ˆγ, η, ˆρ, θ-pseudoinvex at y; iii ρx, y + ˆρx, y 0. Then ϕx λ. Proof. a: Suppose to the contrary that ϕx < λ. This means that fx λgx < 0, and hence αx, y γx, y e αx,y{fx λgx [fy λgy} < 0, which in view of i implies 4.7 fy λ gy, e βx,yηx,y ρx, y θx, y 2. From the primal feasibility of x and 4.3 we see that G jm x, t m 0 G jm y, t m for each m ν 0, and hence αx, y ˆγ mx, y e αx,y[g jm x,tm G jm y,t m 0, which in view of ii implies that Gjm y, t m, e βx,yηx,y < ˆρ m x, y θx, y 2.

13 3 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 47 As v m > 0 for each m ν 0, the above inequalities yield 4.8 ν 0 ν 0 v m G jm y, t m, e βx,yηx,y < v m ˆρ m x, y θx, y 2. m= m= Now combining 4., 4.6 which is valid for the present case because of our assumptions set forth in iii, 4.8, and iv, we get fy λ gy, e βx,yηx,y > ρx, y θx, y 2, which contradicts 4.7. Therefore, we conclude that ϕx λ. b g: The proofs are similar to that of part a. Theorem 4.8 Strong Duality. Let x be a normal optimal solution of P and assume that any one of the seven sets of conditions set forth in Theorem 4.7 is satisfied for all feasible solutions of DII. Then there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is an optimal solution of DII and ϕx = λ. Proof. The proof is similar to that of Theorem 3.2. Theorem 4.9 Strict Converse Duality. Let x be a normal optimal solution of P, let w x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s be an optimal solution of DII, and assume that any one of the seven sets of conditions set forth in Theorem 4.7 is satisfied for the feasible solution w of DII, and the function z fz λgz is α, β, γ, η, ρ, θ-quasiinvex at x. Then x = x and ϕx = λ. Proof. a: By Theorem 2., there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is a feasible solution of DII and ϕx = λ. Suppose to the contrary that x x. From the primal feasibility of x and 4.3 we see that G jm x, t m 0 G jm x, t m for each m ν 0, and hence αx, x ˆγ mx, x e αx, x[g jm x,t m G jm x, t m 0, which in view of ii implies that βx Gjm x, t m, e βx, xηx, x < ˆρ m x, x θx, x 2., x As ṽ m > 0 for each m ν 0, the above inequalities yield 4.9 ν 0 ν 0 βx ṽ m G jm x, t m, e βx, xηx, x < ṽ m ˆρ m x, x θx, x 2., x m= m=

14 48 G.J. Zalmai 4 Now, combining 4., 4.6 which is valid for the present case because of our assumptions set forth in iii, 4.9, and iv, we get f x λ g x, e βx, xηx, x βx > ρx, x θx, x 2,, x which by virtue of the α, β, γ, η, ρ, θ-quasiinvexity of the function z fz λgz at x implies that αx, x γx, x e αx, x{fx λgx [f x λg x} > 0. Since γx, x > 0 and f x λg x 0, this inequality implies that fx λgx > 0, and so ϕx > λ, which contradicts the fact that ϕx = λ λ. Therefore, we conclude that x = x and ϕx = λ. b g: The proofs are similar to that of part a. 5. DUALITY MODEL III In this section, we discuss several families of duality results under various generalized α, β, γ, η, ρ, θ-invexity hypotheses imposed on certain combinations of the problem functions. This is accomplished by employing a certain partitioning scheme which was originally proposed in [ for the purpose of constructing generalized dual problems for nonlinear programming problems. For this we need some additional notation. Let ν 0 and ν be integers, with ν 0 ν n+, and let {J 0, J,..., J M } and {K 0, K,..., K M } be partitions of the sets ν 0 and ν\ν 0, respectively; thus, J i ν 0 for each i M {0}, J i J j = for each i, j M {0} with i j, and M i=0 J i = ν 0. Obviously, similar properties hold for {K 0, K,..., K M }. Moreover, if m and m 2 are the numbers of the partitioning sets of ν 0 and ν\ν 0, respectively, then M = max{m, m 2 } and J i = or K i = for i > min{m, m 2 }. In addition, we use the real-valued functions z Φz, v, λ, t, s, and z Λ τ z, v, t, s defined, for fixed v, λ, ν, ν 0, J ν0, K ν\ν0, t, and s, on X as follows: Φz, v, λ, t, s = fz λgz + v m G jm z, t m + v m H km z, s m, m J 0 Λ τ z, v, t, s = m J τ v m G jm z, t m + m K τ v m H km z, s m, τ M {0}. Making use of the sets and functions defined above, we can state our general duality models as follows:

15 5 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 49 DIII sup y,v,λ,ν,ν 0,J ν0,k ν\ν0, t, s H λ subject to 5. fy λ gy + ν 0 m= v m G jm y, t m + ν m=ν fy λgy + m J 0 v m G jm y, t m m J τ v m G jm y, t m + DIII sup λ y,v,λ,ν,ν 0,J ν0,k ν\ν0, t, s H v m H km y, s m = 0, v m H km y, s m 0, m K τ v m H km y, s m 0, τ M; subject to 3.3, 5.2, and 5.3. The remarks and observations made earlier about the relationships between DI and DI are, of course, also valid for DIII and DIII. The next two theorems show that DIII is a dual problem for P. Theorem 5. Weak Duality. Let x and w y, v, λ, ν, ν 0, J ν0, K ν\ν0, t, s be arbitrary feasible solutions of P and DIII, respectively, and assume that any one of the following four sets of hypotheses is satisfied: a b c i z Φz, v, λ, t, s is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y>0; ii for each τ M, z Λ τ z, v, t, s is α, β, γ τ, η, ρ τ, θ-quasiinvex at y; iii ρx, y + M τ= ρ τ x, y 0; i z Φz, v, λ, t, s is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each τ M, z Λ τ z, v, t, s is α, β, γ τ, η, ρ τ, θ-quasiinvex at y; iii ρx, y + M τ= ρ τ x, y > 0; i z Φz, v, λ, t, s is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each τ M, z Λ τ z, v, t, s is strictly α, β, γ τ, η, ρ τ, θ- pseudoinvex at y; iii ρx, y + M τ= ρ τ x, y 0; d i z Φz, v, λ, t, s is α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each τ M, z Λ τ z, v, t, s is α, β, γ τ, η, ρ τ, θ-quasiinvex at y, and for each τ M 2, z Λ τ z, v, t, s is strictly α, β, γ τ, η, ρ τ, θ-pseudoinvex at y, where {M, M 2 } is a partition of M;

16 50 G.J. Zalmai 6 iii ρx, y + M τ= ρ τ x, y 0. Then ϕx λ. Proof. a: It is clear that 5. can be expressed as follows: 5.4 fy λ gy + v m G jm y, t m + v m H km y, s m m J 0 M [ + v m G jm y, t m + v m H km y, s m = 0. τ= m J τ m K τ Since for each τ M, Λ τ x, v, t, s = v m G jm x, t m + v m H km x, s m m J τ m K τ 0 by the primal feasibility of x and positivity of v m, m ν 0 m J τ v m G jm y, t m + m K τ v m H km y, s m by 5.3 = Λ τ y, v, t, s, it follows that αx, y γ τ x, y e αx,y[λτ x,v, t, s Λ τ y,v, t, s 0, which in view of ii implies that v m G jm y, t m + v m H km y, s m, e βx,yηx,y m J τ m K τ Summing over τ, we obtain ρ τ x, y θx, y M [ v m G jm y, t m + v m H km y, s m, e βx,yηx,y τ= m J τ m K τ M ρ τ x, y θx, y 2. Combining 5.4 and 5.5, and using iii we get fy λ gy + v m G jm y, t m m J 0 + M v m H km y, s m, e βx,yηx,y ρ τ x, y θx, y 2 τ= τ= ρx, y θx, y 2,

17 7 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 5 which by virtue of i implies that αx, y γx, y e αx,y[φx,v,λ, t, s Φy,v,λ, t, s 0, Since γx, y > 0, we can deduce from this inequality that Φx, v, λ, t, s Φy, v, λ, t, s 0, where the second inequality follows from 5.2. Therefore, we have 0 Φx, v, λ, t, s = fx λgx + m J 0 v m G jm x, t m + fx λgx by the primal feasibility of x. v m H km x, s m Hence, ϕx λ. b: The proof is similar to that of part a. c : Suppose to the contrary that ϕx < λ. This gives fx λgx < 0. Using this inequality and keeping in mind that v m > 0 for each m ν 0, we have Φx, v, λ, t, s fx λgx by the primal feasibility of x < 0 Φy, v, λ, t, s by 5.2, and hence αx, y γx, y e αx,y[φx,v,λ, t, s Φy,v,λ, t, s < 0, which in view of i implies that 5.6 fy λ gy + v m G jm y, t m m J 0 + v m H km y, s m, e βx,yηx,y ρx, y θx, y 2. As shown in the proof of part a, for each τ M, we have Λ τ x, v, t, s Λ τ y, v, t, s, and hence αx, y γ τ x, y e αx,y[λτ x,v, t, s Λ τ y,v, t, s 0, which in view of ii implies that v m G jm y, t m + v m H km y, s m, e βx,yηx,y m J τ m K τ < ρ τ x, y θx, y 2.

18 52 G.J. Zalmai 8 Summing over τ, we obtain M [ v m G jm y, t m + v m H km y, s m, e βx,yηx,y τ= m J τ m K τ M < ρ τ x, y θx, y 2. Combining this inequality with 5.4 and using iii, we get fy λ gy + v m G jm y, t m m J 0 + M v m H km y, s m, e βx,yηx,y > ρ τ x, y θx, y 2 τ= τ= which contradicts 5.6. Therefore, we conclude that ϕx λ. d: The proof is similar to that of part c. ρx, y θx, y 2, Theorem 5.2 Strong Duality. Let x be a normal optimal solution of P and assume that any one of the four sets of conditions set forth in Theorem 5. is satisfied for all feasible solutions of DIII. Then there exist v, λ, ν, ν0, J ν0, K ν \ν0, t, and s such that x, v, λ, ν, ν0, J ν0, K ν \ν0, t, s is an optimal solution of DIII and ϕx = λ. Proof. The proof is similar to that of Theorem 3.2. Theorem 5.3 Strict Converse Duality. Let x be a normal optimal solution of P, let w x, ṽ, λ, ν, ν 0, J ν0, K ν\ ν0, t, s be an optimal solution of DIII, and assume that any one of the following four sets of conditions holds: a The assumptions specified in part a of Theorem 5. are satisfied for the feasible solution w of DIII, and the function z Φz, ṽ, λ, t, s is strictly α, β, γ, η, ρ, θ-pseudoinvex at x. b The assumptions specified in part b of Theorem 5. are satisfied for the feasible solution w of DIII, and the function z Φz, ṽ, λ, t, s is α, β, γ, η, ρ, θ-quasiinvex at x. c The assumptions specified in part c of Theorem 5. are satisfied for the feasible solution w of DIII, and the function z Φz, ṽ, λ, t, s is α, β, γ, η, ρ, θ-quasiinvex at x. d The assumptions specified in part d of Theorem 5. are satisfied for the feasible solution w of DIII, and the function z Φz, ṽ, λ, t, s is α, β, γ, η, ρ, θ-quasiinvex at x. Then x = x and ϕx = λ.

19 9 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 53 Proof. a : Since x is a normal optimal solution of P, by Theorem 2., there exist v, λ, ν, ν 0, J ν 0, K ν \ν 0, t, and s such that x, v, λ, ν, ν 0, J ν 0, K ν \ν 0, t, s is a feasible solution of DIII and ϕx = λ. Suppose to the contrary that x x. Now proceeding as in the proof of part a of Theorem 5. with x replaced by x and w by w, we arrive at the inequality βx f x, x λ g x + ṽ m G jm x, t m m J 0 + ṽ m H km x, s m, e βx, xηx, x ρx, x θx, x 2, which by virtue of our strict HAα, β, γ, η, ρ, θ-pseudoinvexity assumption implies that αx, x γx, x e αx, x[φx,ṽ, λ, t, s Φ x,ṽ, λ, t, s > 0. Since γx, x > 0, this inequality implies that Φx, ṽ, λ, t, s > Φ x, ṽ, λ, t, s 0, where the second inequality follows from the dual feasibility of w and 5.2. In view of the primal feasibility of x and dual feasibility of w, the above inequality reduces to fx λgx > 0, and so we get ϕx > λ, which contradicts the fact that ϕx = λ λ. Therefore, we conclude that x = x and ϕx = λ. b d: The proofs are similar to that of part a. As pointed out earlier, the duality models DIII and DIII can be viewed as two families of dual problems for P whose members can easily be identified by appropriate choices of the partitioning sets J µ and K µ, µ M {0}. The special cases and variants of DIII and DIII provide a multitude of dual problems for various classes of semiinfinite as well as conventional nonlinear programming problems. The remainder of this paper will be devoted to a brief discussion of some nonparametric duality models for P. It is of course possible to formulate the nonparametric analogues of the duality models DI, DI, DII and DII, and state and prove the counterparts of Theorems and Theorems However, for the sake of avoiding excessive repetition, we shall not treat these nonparametric duality models separately. Instead, in the next section we shall consider two general nonparametric duality models which subsume the semiinfinite versions of several well-known nonparametric dual problems that have been studied previously in the literature of fractional programming.

20 54 G.J. Zalmai DUALITY MODEL IV In this section, we formulate two generalized nonparametric duality models for P and prove appropriate duality theorems under various generalized α, β, γ, η, ρ, θ-invexity conditions. In addition to the notation used in Section 5, we also need the function z Πz, y, v, t, s defined, for fixed y, v, t, and s, on X by Πz, y, v, t, s = gy [fz + m J0 v m G jm z, t m + v m H km z, s m Let K = DIV [fy + Λ 0 y, v, t, sgz. { y, v, ν, ν 0, J ν0, K ν\ν0, t, s : y X; 0 ν 0 ν n+; v R ν, v i > 0, for i ν 0 ; J ν0 = j, j 2,..., j ν0, j i q; K ν\ν0 = k ν0 +,..., k ν, k i r; t = t, t 2,..., t ν 0, t i T ji ; s = s ν 0+,..., s ν, s i S ki }. Consider the following two nonparametric duality models for P : sup y,v,ν,ν 0,J ν0,k ν\ν0, t, s K fy + Λ 0 y, v, t, s gy subject to 6. gy [ fy + m J0 v m G jm y, t m + v m H km y, s m [fy+ Λ 0 y, v, t, s gy+ v m G jm y, t m + v m H km y, s m = 0, m ν 0 \J 0 m ν\ν 0 \K v m G jm y, t m + v m H km y, s m 0, τ M; m J τ m K τ DIV sup y,v,ν,ν 0,J ν0,k ν\ν0, t, s K subject to 6.2 and fy + Λ 0 y, v, t, s gy gy [ fy+ m J0 v m G jm y, t m + v m H km y, s m [fy+ Λ 0 y, v, t, s gy + v m G jm y, t m + v m H km y, s m, m ν 0 \J 0 m ν\ν 0 \K 0 e βx,yηx,y 0 for all x F, where β : X X R and η : X X R n are given functions.

21 2 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 55 The remarks and observations made earlier about the relationships between DI and DI are, of course, also valid for DIV and DIV. The next two theorems show that DIV is a dual problem for P. Theorem 6. Weak Duality. Let x and w y, v, ν, ν 0, J ν0, K ν\ν0, t, s be arbitrary feasible solutions of P and DIV, respectively, and assume that fy + Λ 0 y, v, t, s 0, gy > 0, and any one of the following four sets of hypotheses is satisfied: a b c d i z Πz, y, v, t, s is α, β, γ, η, ρ, θ-pseudoinvex at y and γx, y>0; ii for each τ M, z Λ τ z, v, t, s is α, β, γ τ, η, ρ τ, θ-quasiinvex at y; iii ρx, y + m τ= ρ τ x, y 0; i z Πz, y, v, t, s is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each τ M, z Λ τ z, v, τ, s is α, β, γ τ, η, ρ τ, θ-quasiinvex at y; iii ρx, y + m τ= ρ τ x, y > 0; i z Πz, y, v, t, s is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each τ M, z Λ τ z, v, t, s is strictly α, β, γ τ, η, ρ τ, θ- pseudoinvex at y; iii ρx, y + m τ= ρ τ x, y 0; i z Πz, y, v, t, s is prestrictly α, β, γ, η, ρ, θ-quasiinvex at y and γx, y > 0; ii for each τ M, z Λ τ z, v, τ, s is α, β, γ τ, η, ρ τ, θ-quasiinvex at y, and for each τ M 2, z Λ τ z, v, τ, s is strictly α, β, γ τ, η, ρ τ, θ-pseudoinvex at y, where {M, M 2 } is a partition of M; iii ρx, y + m τ= ρ τ x, y 0. Then ϕx ψw, where ψ is the objective function of DIV. Proof. a: It is clear that 6. can be expressed as follows: 6.3 gy [ fy + m J0 v m G jm y, t m + v m H km y, s m [fy + Λ 0 y, v, t, s gy + Since for each τ M, M [ τ= v m G jm y, t m m J τ + v m H km y, s m = 0. m K τ

22 56 G.J. Zalmai 22 Λ τ x, v, t, s = m J τ v m G jm x, t m + m K τ v m H km x, s m 0 by the primal feasibility of x and positivity of v m, m ν 0 m J τ v m G jm y, t m + m K τ v m H km y, s m by 6.2 = Λ τ y, v, t, s, it follows that αx, y γ τ x, y e αx,y[λτ x,v, t, s Λ τ y,v, t, s 0, which in view of ii implies that v m G jm y, t m + v m H km y, s m, e βx,yηx,y m J τ m K τ ρ τ x, y θx, y 2. Summing over τ, we obtain M [ 6.4 v m G jm y, t m + v m H km y, s m, e βx,yηx,y τ= m J τ m K τ M ρ τ x, y θx, y 2. Combining 6.3 and 6.4, and using iii we get gy [ fy + m J0 v m G jm y, t m + v m H km y, s m M [fy + Λ 0 y, v, t, s gy, e βx,yηx,y ρ τ x, y θx, y 2 which by virtue of i implies that αx, y γx, y e αx,y[πx,y,v, t, s Πy,y,v, t, s 0. τ= τ= ρx, y θx, y 2, Since γx, y > 0, we can deduce from this inequality that Πx, y, v, t, s Πy, y, v, t, s = 0, where the equality follows from the definitions of Π and Λ 0. Therefore, we have 0 Πx, y, v, t, s = gy [fx+ m J0 v m G jm x, t m + v m H km x, s m [fy+λ 0 y, v, t, sgx gyfx [fy + Λ 0 y, v, t, sgx by the primal feasibility of x.

23 23 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 57 Hence, we get the desired inequality ϕx = fx gx fy + Λ 0y, v, t, s = ψw. gy b: The proof is similar to that of part a. c: Suppose to the contrary that ϕx < ψw. This means that gyfx [fy + Λ 0 y, v, t, sgx < 0. Using this inequality and keeping in mind that v m > 0 for each m ν 0, we have Πx, y,v, t, s = gy [fx + m J0 v m G jm x, t m + v m H km x, s m [fy + Λ 0 y, v, t, sgx gyfx [fy + Λ 0 y, v, t, sgx by the primal feasibility of x < 0 = Πy, y, u, v, t, s by the definitions of Π and Λ 0, and hence αx, y γx, y e αx,y[πx,y,v, t, s Πy,y,v, t, s < 0, which in view of i implies that 6.5 gy [ fy + m J0 v m G jm y, t m + v m H km y, s m [fy + Λ 0 y, v, t, s gy, e βx,yηx,y ρx, y θx, y 2. As shown in the proof of part a, for each τ M, we have Λ τ x, v, t, s Λ τ y, v, t, s, and hence αx, y γ τ x, y e αx,y[λτ x,v, t, s Λ τ y,v, t, s 0, which in view of ii implies that v m G jm y, t m + v m H km y, s m, e βx,yηx,y m J τ m K τ Summing over τ, we obtain < ρ τ x, y θx, y 2. M [ v m G jm y, t m + v m H km y, s m, e βx,yηx,y τ= m J τ m K τ M < ρ τ x, y θx, y 2. τ= Combining this inequality with 6.3 and using iii, we get

24 58 G.J. Zalmai 24 gy [ fy + m J0 v m G jm y, t m + v m H km y, s m M [fy + Λ 0 y, v, t, s gy, e βx,yηx,y > ρ τ x, y θx, y 2 τ= ρx, y θx, y 2, which contradicts 6.5. Therefore, we conclude that ϕx ψw. d: The proof is similar to that of part c. Theorem 6.2 Strong Duality. Let x be a normal optimal solution of P and assume that any one of the four sets of conditions set forth in Theorem 6. is satisfied for all feasible solutions of DIV. Then there exist v, ν, ν0, J ν0, K ν \ν0, t, and s such that w x, v, ν, ν0, J ν0, K ν \ν0, t, s is an optimal solution of DIV and ϕx = ψw. Proof. a: Since x is a normal optimal solution of P, by Theorem 2.2, there exist v, ν, ν0, J ν0, K ν \ν0, t, and s such that gx fx fx gx + ν0 ν v m G jm x, t m + + v m H km x, s m =0. m=ν0 m= From this equation and the fact that x F and t m ˆT jm x for each m ν 0, it is easily seen that 6.6 gx [ fx + m J0 vm G jm x, t m + vm H km x, s m [fx + Λ 0 x, v, t, s gx + vm G jm x, t m m ν 0 \J m J τ v mg jm x, t m + + m ν\ν 0 \K 0 v m H km x, s m = 0, m K τ v mh km x, s m = 0, τ M {0}, 6.8 ϕx = fx + Λ 0 x, v, t, s gx, where vm = v m /gx for each m J 0 K 0, and vm = v m for each m ν\j 0 K 0. From 6.6 and 6.7 it is clear that w is a feasible solution of DIV. From 6.8 we see that ϕx = ψw and hence the optimality of w for DIV follows from Theorem 6.. b d: The proofs are similar to that of part a.

25 25 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 59 Theorem 6.3 Strict Converse Duality. Let x be a normal optimal solution of P, let w = x, ṽ, ν, ν 0, J ν0, K ν\ ν0, t, s be an optimal solution of DIV, and assume that any one of the following four sets of conditions holds: a The assumptions specified in part a of Theorem 6. are satisfied for the feasible solution w of DIV, and z Πz, x, ṽ, t, s is strictly α, β, γ, η, ρ, θ-pseudoinvex at x; b The assumptions specified in part b of Theorem 6. are satisfied for the feasible solution w of DIV, and z Πz, x, ṽ, t, s is α, β, γ, η, ρ, θ- quasiinvex at x; c The assumptions specified in part c of Theorem 6. are satisfied for the feasible solution w of DIV, and z Πz, x, ṽ, t, s is α, β, γ, η, ρ, θ- quasiinvex at x; d The assumptions specified in part d of Theorem 6. are satisfied for the feasible solution w of DIV, and z Πz, x, ṽ, t, s is α, β, γ, η, ρ, θ- quasiinvex at x. Then x = x and ϕx = ψ w. Proof. a: Suppose to the contrary that x x. Since x is a normal optimal solution of P, by Theorem 6.2, there exist v, ν, ν0, J ν0, K ν \ν0, t, and s such that w x, v, ν, ν0, J ν0, K ν \ν0, t, s is a feasible solution of DIV and ϕx = ψw. Now, proceeding as in the proof of part a of Theorem 6. with x replaced by x and w by w, we arrive at the inequality g x [ f i x + m J0 ṽ m G jm x, t m + ṽ m H km x, s m [f x + Λ 0 x, ṽ, t, s g x, e βx, xηx, x ρx, x θx, x 2, βx, x which by virtue of our strict α, β, γ, η, ρ, θ-pseudoinvexity assumption implies that αx, x γx, x e Πx, x,ṽ, t, s Π x, x,ṽ, t, s > 0. Since γx, x > 0, this inequality implies that Πx, x, ṽ, t, s > Π x, x, ṽ, t, s = 0. In view of the primal feasibility of x, this inequality reduces to and so we get g xfx [f x + Λ 0 x, ṽ, t, sgx > 0, ϕx = fx gx > f x + Λ 0 x, ṽ, t, s = ψ w, g x

26 60 G.J. Zalmai 26 which contradicts the fact that ϕx = ψw ψ w. Hence x = x and ϕx = ψ w. b d: The proofs are similar to that of part a. As pointed out earlier, the duality models DIV and DIV can be viewed as two families of nonparametric dual problems for P whose members can easily be identified by appropriate choices of the partitioning sets J µ and K µ, µ M {0}. In particular, if we let J 0 = ν 0 and K 0 = ν\ν 0, then we see that DIV and DIV reduce to the following dual problems for P : DIV a subject to [ gy fy + where sup y,v,ν,ν 0,J ν0,k ν\ν0, t, s K ν 0 m= Λy, v, t, s = v m G jm y, t m + ν 0 m= DIV a sup y,v,ν,ν 0,J ν0,k ν\ν0, t, s K subject to [ gy fy + ν 0 m= fy + Λy, v, t, s gy ν m=ν 0 + v m G jm y, t m + v m H km y, s m [fy + Λy, v, t, s gy = 0, ν m=ν 0 + fy + Λy, v, t, s gy v m G jm y, t m + ν m=ν 0 + v m H km y, s m ; v m H km y, s m [fy + Λy, v, t, s gy, e βx,yηx,y 0 for all x F, where β is a function from X X to R and η is a function from X X to R n. Similarly, if we choose J 0 = and K 0 =, then we obtain the following dual problems for P : DIV b subject to sup y,v,ν,ν 0,J ν0,k ν\ν0, t, s K gy fy fy gy ν 0 m= m J τ v m G jm y, t m + fy gy v m G jm y, t m + ν m=ν 0 + m K τ v m H km y, s m 0, τ M; v m H km y, s m = 0;

27 27 Generalized α, β, γ, η, ρ, θ-invex functions. Part II 6 DIV b sup y,v,ν,ν 0,J ν0,k ν\ν0, t, s K fy gy subject to 6.9 and gy fy fy gy + + ν m=ν 0 + ν 0 m= v m G jm y, t m v m H km y, s m, e βx,yηx,y 0 for all x F, where β is a function from X X to R and η is a function from X X to R n. In this manner, one can easily identify many other special cases of DIV and DIV. The dual problems DIV a, DIV a, DIV b, and DIV b are the semiinfinite versions of some popular dual problems that have been investigated previously in the area of fractional programming. Evidently, one can easily specialize and restate Theorems for these special cases of DIV and DIV. 7. CONCLUDING REMARKS In this study, we have constructed six first-order parametric and two firstorder nonparametric duality models and established a fairly large number of duality results under various generalized α, β, γ, η, ρ, θ-invexity hypotheses for a semiinfinite fractional programming problem. It appears that all these results are new in the area of semiinfinite programming. Furthermore, the style and techniques employed in this paper can be utilized to establish similar duality results for some other classes of related optimization problems. For example, it seems reasonable to expect that a similar approach can be used to investigate the optimality and duality aspects of the following classes of semiinfinite continuous minmax and multiobjective fractional programming problems: Minimize x F Minimize x F max y Y fx, y gx, y, f x g x,..., f px g p x We shall investigate these classes of semiinfinite programming problems in subsequent papers..

28 62 G.J. Zalmai 28 REFERENCES [ B. Mond and T. Weir, Generalized concavity and duality. In: Generalized Concavity in Optimization and Economics, S. Schaible and W.T. Ziemba Eds.. Academic Press, New York, 98, pp [2 G.J. Zalmai, Generalized α, β, γ, η, ρ, θ-invex functions in semiinfinite fractional programming. Part I: Optimality conditions. Submitted to Rev. Roumaine Math. Pures Appl. [3 G.J. Zalmai, Generalized α, β, γ, η, ρ, θ-invex functions in semiinfinite fractional programming. Part III: Second-order duality models. Submitted to Rev. Roumaine Math. Pures Appl. [4 G.J. Zalmai, Generalized α, β, γ, η, ρ, θ-invex functions in semiinfinite fractional programming. Part IV: Higher-order duality models. unpublished manuscript. [5 G.J. Zalmai and Qinghong Zhang, Generalized F, β, φ, ρ, θ-univex functions and optimality conditions in semiinfinite fractional programming, J. Interdiscip. Math , Received 4 May 202 Northern Michigan University Department of Mathematics and Computer Science Marquette, MI 49855, USA gzalmai@nmu.edu