Pareto optimality conditions and duality for vector quadratic fractional optimization problems

Research Article Pareto Optimality Conditions and Duality for Vector Quadratic Fractional Optimization Problems W. A. Oliveira,1A. Beato-Moreno,2A. C. Moretti,1andL.L.SallesNeto 3 1School of Applied Sciences, State University of Campinas, 13484-350 Limeira, SP, Brazil 2Department of Statistics and Operations Research, College of Mathematics, University of Sevilla, 41012 Sevilla, Spain 3Department of Science and Technology, Federal University of S˜ ao Paulo, 12247-014 S˜ ao Jos´ e dos Campos, SP, Brazil Correspondence should be addressed to W. A. Oliveira; washington.o[email protected]p.br Received 12 March 2014; Revised 25 July 2014; Accepted 6 August 2014; Published 25 November 2014 AcademicEditor:ChaoYan Copyright © 2014 W. A. Oliveira et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. One of the most important optimality conditions to aid in solving a vector optimization problem is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. However, to obtain the sufficient optimality conditions, it is necessary to impose additional assumptions on the objective functions and on the constraint set. The present work is concerned with the constrained vector quadratic fractional optimization problem. It shows that sufficient Pareto optimality conditions and the main duality theorems can be established without the assumption of generalized convexity in the objective functions, by considering some assumptions on a linear combination of Hessian matrices instead. The main aspect of this contribution is the development of Pareto optimality conditions based on a similar second-order sufficient condition for problems with convex constraints, without convexity assumptions on the objective functions. These conditions might be useful to determine termination criteria in the development of algorithms. 1. Introduction There are many contributions, concepts, and definitions that characterize and give the Pareto optimality conditions for solutions of a vector optimization problem (see, for instance, [1,2]). One of the most important condition is the firstorder necessary optimality condition that generalizes the Karush-Kuhn-Tucker (KKT) condition. However, to obtain the sufficient optimality conditions, it is necessary to impose additional assumptions (like convexity and its generalizations) on the objective functions and on the constraint set. In this paper, we deal with a particular case of vector optimization problem (VOP), where each objective function consists of a ratio of two quadratic functions. Without generalized convexity assumptions on the objective functions, but by imposing some additional assumptions on a linear combination of Hessian matrices, Pareto optimality conditions are obtained and duality theorems are established. Let us consider the following vector quadratic fractional optimization problem (VQFP): Minimize 𝑓(𝑥) 𝑔(𝑥)=(𝑓1(𝑥) 𝑔1(𝑥),...,𝑓𝑚(𝑥) 𝑔𝑚(𝑥)) subject to ℎ𝑗(𝑥)≦0 𝑗∈𝐽,𝑥∈Ω, (VQFP) where Ω⊆R𝑛is an open set, 𝑓𝑖,𝑔𝑖,𝑖∈𝐼≡{1,...,𝑚},and ℎ𝑗,𝑗∈𝐽≡{1,...,ℓ}, are continuously differentiable realvalued functions defined on Ω.Inaddition,weassumethat 𝑓𝑖,𝑔𝑖,𝑖∈𝐼, are quadratic functions and 𝑔𝑖(𝑥)>0for 𝑥∈Ω and 𝑖∈𝐼.Wedenoteby𝑆the feasible set of elements 𝑥∈Ω satisfying ℎ𝑗(𝑥)≦0.Wesaythat𝑥is a feasible point if 𝑥∈𝑆. The value 𝑓𝑖(𝑥)/𝑔𝑖(𝑥)is the result of the 𝑖th objective function if the decision maker chooses the action 𝑥∈𝑆. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 983643, 13 pages http://dx.doi.org/10.1155/2014/983643 2Journal of Applied Mathematics Fractional optimization problems arise frequently in decision making applications, including science management, portfolio selection, cutting and stock, and game theory, in the optimization of the ratio performance/cost, or profit/investment, or cost/time. There are many contributions dealing with the scalar (single-objective) fractional optimization problem (FP) and vector fractional optimization problem (VFP). In most of them, using convexity or generalized convexity, optimality conditions in the KKT sense and the main duality theorems for optimal points are obtained. With a parametric approach, which transforms the original problem in a simpler associated problem, Dinkelbach [3], Jagannathan [4], and Antczak [5] established optimality conditions, presented algorithms, and applied their approaches in an example (FP) consisting of quadratic functions. Using some known generalized convexity, Antczak [5], Khan and Hanson [6], Reddy and Mukherjee [7], Jeyakumar [8], and Liang et al. [9] established optimality conditions and theorems that relate the pair primal-dual of problem (FP). In Craven [10]andWeir[11], other results for the scalar optimization (FP) can be found. Further, Liang et al. [12] extended their approach to the vector optimization case (VFP) considering the type duals of Mond and Weir [13], Schaible [14], and Bector [15]. Considering the parametric approach of Dinkelbach [3], Jagannathan [4], and Bector et al. [16]andtwoclassesof generalized convexity, Osuna-G´ omez et al. [17]established weak Pareto optimality conditions and the main duality theorems for the differentiable vector optimization case (VFP). Santos et al. [18] deepened these results to the more general nondifferentiable case (VFP). Jeyakumar and Mond [19]used generalized convexity to study the problem (VFP). Few studies are found involving quadratic functions at both the numerator and denominator of the ratio objective function. Most of them involve the mixing of linear and quadratic functions. The most similar approaches to the scalar quadratic fractional optimization problem (QFP) were considered in [20–24]. On the other hand, Benson [25]considered a pure (QFP) consisting of convex functions, where some theoretical properties and optimality conditions are developed, and an algorithm and its convergence properties are presented. The closest approaches to the vector optimization case (VQFP)were considered in [26–33]. Using an iterative computational test, Beato et al. [27,28] characterized the Pareto optimal point for the problem (VQFP),consistingof linear and quadratic functions, and some theoretical results were obtained by using the function linearization technique of Bector et al. [16]. Ar´ evalo and Zapata [26], Konno and Inori [29], and Rhode and Weber [33]analyzedtheportfolio selection problem. Kornbluth and Steuer [32]usedan adapted Simplex method in the problem (VFP) consisting of linear functions. Korhonen and Yu [30,31]proposed an iterative computational method for solving the problem (VQFP), consisting of linear and quadratic functions, based on search directions and weighted sums. The approach taken in this work is different from the previous ones. The main aspect of this contribution is the development of Pareto optimality conditions for a particular vector optimization problem based on a similar second-order sufficient condition for Pareto optimality for problems with convex constraints without the hypothesis of convexity on the objective functions. These conditions might be useful to determine termination criteria in the development of algorithms, and new extensions can be established from these, where more general vector optimization problems in which algorithms are based on quadratic approximations are used locally. This paper is organized as follows. We start by defining some notations and basic properties in Section 2.In Section 3, the sufficient Pareto optimality conditions are established. In Section 4, the relationship among the associated problems is presented and duality theorems are established. Finally, comments and concluding remarks are presented in Section 5. 2. Preliminaries Let R+denote the nonnegative real numbers and let 𝑥𝑇 denote the transpose of the vector 𝑥∈R𝑛.Furthermore,we will adopt the following conventions for inequalities among vectors. If 𝑥=(𝑥1,...,𝑥𝑚)𝑇∈R𝑚and 𝑦=(𝑦1,...,𝑦𝑚)𝑇∈ R𝑚,then 𝑥=𝑦if and only if 𝑥𝑖=𝑦𝑖,∀𝑖∈𝐼; 𝑥<𝑦if and only if 𝑥𝑖<𝑦𝑖,∀𝑖∈𝐼; 𝑥≦𝑦if and only if 𝑥𝑖≦𝑦𝑖,∀𝑖∈𝐼; 𝑥≤𝑦if and only if 𝑥≦𝑦and 𝑥 =𝑦. Similarly we consider the equivalent convention for inequalities >,≧,and≥. Different optimality definitions for the problem (VQFP) are referred to as Pareto optimal solutions [34], two of which are defined as follows. Definition 1. Feasible point 𝑥∗is said to be a Pareto optimal solution of (VQFP), if there does not exist another 𝑥∈𝑆such that 𝑓(𝑥)/𝑔(𝑥)≤𝑓(𝑥∗)/𝑔(𝑥∗). Definition 2. Feasible point 𝑥∗is said to be a weakly Pareto optimal solution of (VQFP), if there does not exist another 𝑥∈𝑆such that 𝑓(𝑥)/𝑔(𝑥)<𝑓(𝑥∗)/𝑔(𝑥∗). Hypotheses of convexity or generalized convexity on the objective functions will be avoided in this work, but we will use such hypotheses on the constraint set. We recall the definition of convexity, where ∇𝑓(𝑥)denotes the gradient of the function 𝑓:R𝑛→Rat the point 𝑥. Definition 3. Let 𝑓:Ω⊆R𝑛→Rbe a function defined on an open convex set Ωand differentiable at 𝑥∗∈Ω.𝑓is called convex at 𝑥∗if, for all 𝑥∈Ω,𝑓(𝑥)−𝑓(𝑥∗)≧∇𝑓(𝑥∗)𝑇(𝑥−𝑥∗). When 𝑓is convex on the set Ω, we simply say that 𝑓is convex. Maeda [35] used the generalized Guignard constraint qualification (GGCQ) [36], to derive the following necessary Pareto optimality conditions for the problem (VOP) in the KKT sense. Assuming differentiability of the objective and the constraint functions, Maeda guarantees the existence of Journal of Applied Mathematics 3 Lagrange multipliers, all strictly positive, associated with the objective functions. Lemma 4 (Maeda [35]). Let 𝑥∗be a Pareto optimal solution of (VQFP). Suppose that (GGCQ) holds at 𝑥∗; then there exist vectors 𝜏∈R𝑚,𝜆∈Rℓsuch that 𝑚 ∑ 𝑖=1𝜏𝑖∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)+ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)=0, ℓ ∑ 𝑗=1𝜆𝑗ℎ𝑗(𝑥∗)=0, 𝜏>0, 𝜆≧0. (1) For each 𝑖∈𝐼and 𝑥∈R𝑛, we consider the objective functions defined as 𝑓𝑖(𝑥)=𝑥𝑇𝐴𝑖𝑥+𝑎𝑇 𝑖𝑥+𝑎𝑖and 𝑔𝑖(𝑥)= 𝑥𝑇𝐵𝑖𝑥+𝑏𝑇 𝑖𝑥+𝑏𝑖,where𝐴𝑖,𝐵𝑖∈R𝑛×𝑛,𝐴𝑖is symmetric, 𝐵𝑖is symmetric and positive semidefinite, and 𝑎𝑖,𝑏𝑖∈R𝑛and 𝑎𝑖, 𝑏𝑖∈R,with𝑏𝑖>−(𝑤𝑖𝑇𝐵𝑖𝑤𝑖+𝑏𝑇 𝑖𝑤𝑖),where𝑤𝑖is the solution of the system 2𝐵𝑖𝑥+𝑏𝑖=0;thatis,𝑤𝑖is the point in which the function 𝑥𝑇𝐵𝑖𝑥+𝑏𝑇 𝑖𝑥reaches its minimum and this ensures that 𝑔𝑖(𝑥)>0,∀𝑥∈R𝑛. We cannot consider the cases where 2𝐵𝑖𝑥+𝑏𝑖=0has no solution. 3. Sufficient Optimality Conditions Without assumptions of generalized convexity, but imposing some additional assumptions on a linear combination of Hessian matrices of the objective functions 𝑓𝑖and 𝑔𝑖,𝑖∈ 𝐼, we provide in the next theorem a sufficient condition that guarantees that a feasible point of (VQFP)is Pareto optimal point. Similar to a second-order sufficient condition for Pareto optimality, this condition explores the intrinsic characteristics of the problem (VQFP). We assume, unlike the objective functions, that each ℎ𝑗is convex. Also, given 𝑥∗∈𝑆,foreach𝑖∈𝐼, we define the scalar functions 𝑢𝑖:𝑆×𝑆→R+\{0}and 𝑠𝑖:𝑆×𝑆→Rby 𝑢𝑖(𝑥,𝑥∗) ≡𝑔𝑖(𝑥∗) 𝑔𝑖(𝑥), 𝑠𝑖(𝑥,𝑥∗) ≡1 𝑔𝑖(𝑥){(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)}. (2) Theorem 5. Let 𝑥∗be a feasible point of (VQFP).Supposethat theconstraintfunctionℎ𝑗is convex for each 𝑗∈𝐽and there exist vectors 𝜏∈R𝑚,𝜆∈Rℓ,suchthat 𝑚 ∑ 𝑖=1𝜏𝑖∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)+ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)=0, (3) ℓ ∑ 𝑗=1𝜆𝑗ℎ𝑗(𝑥∗)=0, (4) 𝜏>0, 𝜆≧0. (5) If, for any 𝑥∈𝑆,weobtain 𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)≧0, (6) then 𝑥∗is a Pareto optimal solution for (VQFP). Proof. Given 𝑥∈𝑆,weobtainforeach𝑖∈𝐼 𝑓𝑖(𝑥)−𝑓𝑖(𝑥∗) =(𝑥−𝑥∗)𝑇𝐴𝑖(𝑥−𝑥∗)+∇𝑓𝑖(𝑥∗)𝑇(𝑥−𝑥∗), 𝑔𝑖(𝑥)−𝑔𝑖(𝑥∗) =(𝑥−𝑥∗)𝑇𝐵𝑖(𝑥−𝑥∗)+∇𝑔𝑖(𝑥∗)𝑇(𝑥−𝑥∗), 𝑓𝑖(𝑥) 𝑔𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗) =𝑓𝑖(𝑥)𝑔𝑖(𝑥∗)−𝑔𝑖(𝑥)𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥)𝑔𝑖(𝑥∗) =(𝑓𝑖(𝑥)𝑔𝑖(𝑥∗)−𝑓𝑖(𝑥∗)𝑔𝑖(𝑥∗)+𝑓𝑖(𝑥∗)𝑔𝑖(𝑥∗) −𝑔𝑖(𝑥)𝑓𝑖(𝑥∗)) ⋅(𝑔𝑖(𝑥)𝑔𝑖(𝑥∗))−1 =𝑔𝑖(𝑥∗){𝑓𝑖(𝑥)−𝑓𝑖(𝑥∗)}−𝑓𝑖(𝑥∗){𝑔𝑖(𝑥)−𝑔𝑖(𝑥∗)} 𝑔𝑖(𝑥)𝑔𝑖(𝑥∗) =1 𝑔𝑖(𝑥){(𝑥−𝑥∗)𝑇𝐴𝑖(𝑥−𝑥∗)+∇𝑓𝑖(𝑥∗)𝑇(𝑥−𝑥∗)} −𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥)𝑔𝑖(𝑥∗) ⋅{(𝑥−𝑥∗)𝑇𝐵𝑖(𝑥−𝑥∗)+∇𝑔𝑖(𝑥∗)𝑇(𝑥−𝑥∗)} =1 𝑔𝑖(𝑥){(𝑥−𝑥∗)𝑇𝐴𝑖(𝑥−𝑥∗)} −𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥)𝑔𝑖(𝑥∗){(𝑥−𝑥∗)𝑇𝐵𝑖(𝑥−𝑥∗)} +1 𝑔𝑖(𝑥){∇𝑓𝑖(𝑥∗)𝑇(𝑥−𝑥∗)} −𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥)𝑔𝑖(𝑥∗){∇𝑔𝑖(𝑥∗)𝑇(𝑥−𝑥∗)} =1 𝑔𝑖(𝑥){(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)} +𝑔𝑖(𝑥∗) 𝑔𝑖(𝑥) ⋅{ { {[∇𝑓𝑖(𝑥∗)𝑔𝑖(𝑥∗)−∇𝑔𝑖(𝑥∗)𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2]𝑇(𝑥−𝑥∗)} } } 4Journal of Applied Mathematics =1 𝑔𝑖(𝑥){(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)} +𝑔𝑖(𝑥∗) 𝑔𝑖(𝑥){[∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)]𝑇(𝑥−𝑥∗)}. (7) Thus, each function 𝑓𝑖/𝑔𝑖satisfies 𝑓𝑖(𝑥) 𝑔𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)=𝑢𝑖(𝑥,𝑥∗)[∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)]𝑇(𝑥−𝑥∗) +𝑠𝑖(𝑥,𝑥∗),∀𝑥∈𝑆. (8) Suppose that 𝑥∗is not a Pareto optimal solution of (VQFP). Then there exists another point 𝑥∈𝑆such that 𝑓(𝑥) 𝑔(𝑥)≤𝑓(𝑥∗) 𝑔(𝑥∗).(9) Since 𝑢𝑖(𝑥,𝑥∗)>0,𝑖∈𝐼,from(8) we obtain 1 𝑢𝑖(𝑥,𝑥∗)(𝑓𝑖(𝑥) 𝑔𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)) =[∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)]𝑇(𝑥−𝑥∗)+𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗),𝑖∈𝐼. (10) From (9),wehave 𝑓(𝑥) 𝑔(𝑥)−𝑓(𝑥∗) 𝑔(𝑥∗)≤0, (11) and we obtain 𝑚inequalities [∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)]𝑇(𝑥−𝑥∗)+𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)≦0, 𝑖∈𝐼, (12) with at least one strict inequality. Multiplying the 𝑚inequalities above by their respective 𝜏𝑖>0,𝑖∈𝐼, and summing all the products, we obtain 𝑚 ∑ 𝑖=1𝜏𝑖[∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)]𝑇(𝑥−𝑥∗)+𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)<0. (13) Then, we have [𝑚 ∑ 𝑖=1𝜏𝑖∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)]𝑇(𝑥−𝑥∗)+𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)<0. (14) Substituting (3) into (14),weget [ [−ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)] ] 𝑇(𝑥−𝑥∗)+𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)<0. (15) Using (6) and (15),weobtain 0≦𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)<[ [ ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)] ] 𝑇(𝑥−𝑥∗).(16) That is, [ [ ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)] ] 𝑇(𝑥−𝑥∗)>0. (17) On the other hand, by convexity of ℎ𝑗, we have, for each 𝑗∈𝐽, ℎ𝑗(𝑥)−ℎ𝑗(𝑥∗)≧∇ℎ𝑗(𝑥∗)𝑇(𝑥−𝑥∗).(18) Since 𝜆𝑗≧0,𝑗∈𝐽,wehave ℓ ∑ 𝑗=1𝜆𝑗(ℎ𝑗(𝑥)−ℎ𝑗(𝑥∗))≧ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)𝑇(𝑥−𝑥∗) =[ [ ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)] ] 𝑇(𝑥−𝑥∗). (19) However, since 𝑥is feasible point, condition (4) and 𝜆𝑗≧0, 𝑗∈𝐽, imply that ℓ ∑ 𝑗=1𝜆𝑗(ℎ𝑗(𝑥)−ℎ𝑗(𝑥∗))≦0. (20) We conclude that [ [ ℓ ∑ 𝑗=1𝜆𝑗∇ℎ𝑗(𝑥∗)] ] 𝑇(𝑥−𝑥∗)≦0, (21) which contradicts (17). Therefore 𝑥∗is a Pareto optimal solution for (VQFP). The expression 𝑓𝑖(𝑥)/𝑔𝑖(𝑥)−𝑓𝑖(𝑥∗)/𝑔𝑖(𝑥∗)in Theorem 5 is manipulated in a similar manner in [6,7,9,12,19]; however some generalized convexity on the functions 𝑓𝑖and 𝑔𝑖is imposed. In most of them, for each 𝑖∈𝐼and 𝑥∈𝑆, the hypotheses 𝑓𝑖(𝑥) ≧ 0,𝑔𝑖(𝑥) > 0,and𝑓𝑖,−𝑔𝑖satisfy some generalized convexity. This is not the purpose of this work,buttheconstraintfunctionscanbeassumedina more general class of convex functions; for example, the generalized convexity of Liang et al. [9]canbeused. In the following, the Pareto optimal solution set is denoted by Eff (VQFP). Corollary 6. Let 𝑥∗be a feasible point of (VQFP).Suppose that the constraint function ℎ𝑗is convex for each 𝑗∈𝐽and there exist vectors 𝜏∈R𝑚,𝜆∈Rℓ,suchthat(3),(4),and(5) are valid. If [𝐴𝑖−(𝑓𝑖(𝑥∗)/𝑔𝑖(𝑥∗))𝐵𝑖]are positive semidefinite matrices for each 𝑖∈𝐼,then𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Journal of Applied Mathematics 5 Proof. By hypothesis, given 𝑥∈𝑆and 𝑖∈𝐼,weobtain (𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)≧0 󳨐⇒ 𝜏𝑖1 𝑔𝑖(𝑥∗){(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)}≧0 󳨐⇒ 𝑚 ∑ 𝑖=1𝜏𝑖𝑔𝑖(𝑥) 𝑔𝑖(𝑥∗)1 𝑔𝑖(𝑥) ⋅{(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)}≧0 󳨐⇒ 𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗)≧0. (22) Therefore, inequality (6) is valid and the result follows from Theorem 5. To ensure that inequality (6) is valid, we start exploring the features of the Hessian matrices of the objective functions of (VQFP). Negative values can occur in each term 𝜏𝑖(𝑠𝑖(𝑥,𝑥∗)/𝑢𝑖(𝑥,𝑥∗))of the sum ∑𝑚 𝑖=1 𝜏𝑖(𝑠𝑖(𝑥,𝑥∗)/𝑢𝑖(𝑥,𝑥∗)), which depends on each matrix [𝐴𝑖−(𝑓 𝑖(𝑥∗)/𝑔𝑖(𝑥∗))𝐵𝑖], 𝑖∈𝐼,andthevector(𝑥−𝑥∗).Letuschecknewconditions for which (6) is satisfied; that is, we want to ensure the result of Theorem 5 by analysing the function 𝑍(𝑥,𝑥∗)≡𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗) =𝑚 ∑ 𝑖=1𝜏𝑖𝑔𝑖(𝑥) 𝑔𝑖(𝑥∗)1 𝑔𝑖(𝑥) ⋅{(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)} =(𝑥−𝑥∗)𝑇{𝑚 ∑ 𝑖=1 [𝜏𝑖 𝑔𝑖(𝑥∗)𝐴𝑖−𝜏𝑖𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2𝐵𝑖]}(𝑥−𝑥∗). (23) Note that 𝑍(⋅,𝑥∗)is a quadratic function without the linear part; thus in (VQFP)we obtain 𝑍(𝑥,𝑥∗)≧0on 𝑆if and only if min𝑥∈𝑆 𝑍(𝑥,𝑥∗)≧0;thatis,wecanuse the classical results on quadratic optimization to check if min𝑥∈𝑆 𝑍(𝑥,𝑥∗)≧0. The next corollary follows immediately from Theorem 5. Corollary 7. Let 𝑥∗be a feasible point of (VQFP).Supposethat theconstraintfunctionℎ𝑗is convex for each 𝑗∈𝐽and there exist vectors 𝜏∈R𝑚,𝜆∈Rℓ,suchthat(3),(4),and(5) are valid. If min𝑥∈𝑆 𝑍(𝑥,𝑥∗)≧0,then𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Using the previous results to check whether a feasible point is a Pareto optimal solution of (VQFP),weproposethe following computational test method. Pareto Optimality Test Step 1. Given 𝑥∗∈𝑆, find the vectors 𝜏>0and 𝜆≧0such that (3) and (4) are valid. If the vectors 𝜏and 𝜆do not exist, then 𝑥∗∉Eff (VQFP). Step 2. Otherwise, solve 𝑍(𝑥,𝑥∗)=min𝑥∈𝑆𝑍(𝑥,𝑥∗).If 𝑍(𝑥,𝑥∗)≧0,wesaythat𝑥∗has passed the Pareto optimality test and 𝑥∗∈Eff (VQFP). Pareto optimality test starts with a feasible point; then it seeks to solve a system of linear equations containing 𝑚+ℓ unknowns, 𝜏and 𝜆,theinequalities𝜏>0,𝜆≧0,andtwo equalities (3) and (4). If this system has no solution, then the point 𝑥∗does not satisfy the first-order necessary condition for Pareto optimality, so the method terminates concluding that 𝑥∗∉Eff (VQFP).Otherwise,inStep 2, a quadratic optimization problem on 𝑆should be solved. If the minimum of the quadratic problem is nonnegative, then the procedure ends, concluding that 𝑥∗∈Eff (VQFP).Otherwise,we say that 𝑥∗has not passed the Pareto optimality test. Its complexity lies in solving a system of linear inequalities plus a quadratic optimization problem. The next results, which address a linear combination of the Hessian matrices, can be used to develop a computational search method. Looking at the previous Pareto optimality test, if the fixed point 𝑥∗is assumed to be a variable 𝑦, then the linear system in Step 1 becomes a nonlinear system for the variables 𝜏>0, 𝜆≧0,𝑦∈𝑆. And the quadratic optimization problem in Step 2 becomes a quadratic optimization problem of the type min𝑥,𝑦∈𝑆 𝑍(𝑥,𝑦). This raises considerable difficulties. In order to reduce these difficulties, we further explore the characteristics of the matrix  𝐹(𝑦∗)≡𝑚 ∑ 𝑖=1 [𝜏𝑖 𝑔𝑖(𝑦∗)𝐴𝑖−𝜏𝑖𝑓𝑖(𝑦∗) [𝑔𝑖(𝑦∗)]2𝐵𝑖]. (24) One possibility is to search for points 𝑦∗such that  𝐹(𝑦∗) becomes positive semidefinite. In this case 𝑍(𝑥,𝑦∗)=(𝑥− 𝑦∗)𝑇 𝐹(𝑦∗)(𝑥−𝑦∗)≧0depends only on 𝑦∗∈𝑆. Consider a fixed point 𝑥∗; the next theorem takes advantage of the symmetry and diagonalizations of the matrices 𝐴𝑖 and 𝐵𝑖,𝑖∈𝐼, to give sufficient Pareto optimality conditions for a feasible point of (VQFP). Consider the usual inner product ⟨⋅,⋅⟩in R𝑛. Theorem 8. Let 𝑥∗be a feasible point of (VQFP).Supposethat theconstraintfunctionℎ𝑗is convex for each 𝑗∈𝐽and there exist vectors 𝜏∈R𝑚,𝜆∈Rℓ,suchthat(3),(4),and(5) are 6Journal of Applied Mathematics valid. Consider also, for each 𝑖∈𝐼and 𝑘∈𝐾≡{1,...,𝑛},the following functions: 𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)≡ 𝜏𝑖 𝑔𝑖(𝑥∗)⟨𝑥−𝑥∗,𝑝𝑘 𝑖⟩2, 𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏)≡𝜏𝑖𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2⟨𝑥−𝑥∗,𝑞𝑘 𝑖⟩2,(25) where 𝑝𝑘 𝑖and 𝑞𝑘 𝑖are the columns of orthogonal matrices 𝑃𝑖 and 𝑄𝑖, constructed from the normalized eigenvectors of the matrices 𝐴𝑖and 𝐵𝑖,respectively.Ifforall𝑥∈𝑆the following inequality 𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)≧𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏), ∀𝑖∈𝐼,∀𝑘∈𝐾, (26) is valid, where 𝜇𝐴𝑖 𝑘and 𝜇𝐵𝑖 𝑘are the eigenvalues of the matrices 𝐴𝑖and 𝐵𝑖associated with the eigenvectors 𝑝𝑘 𝑖and 𝑞𝑘 𝑖,respectively, then 𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Proof. The matrices 𝐴𝑖and 𝐵𝑖,𝑖∈𝐼,arediagonalizableand can be rewritten as 𝐴𝑖=𝑃𝑖𝐷𝐴𝑖𝑃𝑇 𝑖=∑𝑛 𝑘=1 𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖𝑝𝑘 𝑖𝑇and 𝐵𝑖= 𝑄𝑖𝐷𝐵𝑖𝑄𝑇 𝑖=∑𝑛 𝑘=1 𝜇𝐵𝑖 𝑘𝑞𝑘 𝑖𝑞𝑘 𝑖𝑇,where𝐷𝐴𝑖and 𝐷𝐵𝑖are diagonal matrices, with their diagonal formed by the eigenvalues 𝜇𝐴𝑖 𝑘 and 𝜇𝐵𝑖 𝑘,𝑘∈𝐾,ofthematrices𝐴𝑖and 𝐵𝑖,respectively.Thus, we obtain 𝑚 ∑ 𝑖=1𝜏𝑖𝑠𝑖(𝑥,𝑥∗) 𝑢𝑖(𝑥,𝑥∗) =𝑚 ∑ 𝑖=1𝜏𝑖𝑔𝑖(𝑥) 𝑔𝑖(𝑥∗)1 𝑔𝑖(𝑥) ⋅{(𝑥−𝑥∗)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥−𝑥∗)} =𝑚 ∑ 𝑖=1 {(𝑥−𝑥∗)𝑇[𝜏𝑖 𝑔𝑖(𝑥∗)𝐴𝑖−𝜏𝑖𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2𝐵𝑖](𝑥−𝑥∗)} =𝑚 ∑ 𝑖=1 {(𝑥−𝑥∗)𝑇[𝜏𝑖 𝑔𝑖(𝑥∗)(𝑛 ∑ 𝑘=1𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖𝑝𝑘 𝑖𝑇) −𝜏𝑖𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2(𝑛 ∑ 𝑘=1𝜇𝐵𝑖 𝑘𝑞𝑘 𝑖𝑞𝑘 𝑖𝑇)](𝑥−𝑥∗)} =𝑚 ∑ 𝑖=1 [𝑛 ∑ 𝑘=1𝜇𝐴𝑖 𝑘(𝜏𝑖 𝑔𝑖(𝑥∗)⟨𝑥−𝑥∗,𝑝𝑘 𝑖⟩2) −𝑛 ∑ 𝑘=1𝜇𝐵𝑖 𝑘(𝜏𝑖𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2⟨𝑥−𝑥∗,𝑞𝑘 𝑖⟩2)] =𝑚 ∑ 𝑖=1 [𝑛 ∑ 𝑘=1𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)−𝑛 ∑ 𝑘=1𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏)] =𝑚 ∑ 𝑖=1 𝑛 ∑ 𝑘=1 [𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)−𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏)]. (27) Since, for all 𝑥∈𝑆,wehave𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)≧𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏), for all 𝑖∈𝐼and 𝑘∈𝐾,weconcludethat𝑍(𝑥,𝑥∗)≧0. Therefore, inequality (6) is valid and the result follows from Theorem 5. Theorem 8 is not simple to use since (26) depends on all points of the feasible set; that is, it depends on the functions 𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏),𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏),∀𝑖 ∈ 𝐼,∀𝑘 ∈ 𝐾,and𝑥∈𝑆. However, even if, for some 𝑖∈𝐼and 𝑘∈𝐾,𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)< 𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏)occurs, inequality (6) can still be satisfied. In order to obtain (26),wepresentthenextcorollary,which follows immediately from the previous theorem. Corollary 9. Let 𝑥∗be a feasible point of (VQFP).Suppose that the constraint function ℎ𝑗is convex for each 𝑗∈𝐽and there exist vectors 𝜏∈R𝑚,𝜆∈Rℓ,suchthat(3),(4),and(5) arevalid.Consideralso,foreach𝑖∈𝐼and 𝑘∈𝐾, 𝑎+ 𝑖,𝑘 ≡√𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖+√𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑞𝑘 𝑖, 𝑎− 𝑖,𝑘 ≡√𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖−√𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑞𝑘 𝑖, 𝛼𝑖,𝑘 ≡⟨𝑥∗,𝑎+ 𝑖,𝑘⟩𝑎− 𝑖,𝑘 +⟨𝑥∗,𝑎− 𝑖,𝑘⟩𝑎+ 𝑖,𝑘, 𝛽𝑖,𝑘 ≡⟨𝑥∗,𝑎+ 𝑖,𝑘⟩⟨𝑥∗,𝑎− 𝑖,𝑘⟩, 𝐻𝑖,𝑘 (𝑥)≡𝑥𝑇[𝑎+ 𝑖,𝑘𝑎− 𝑖,𝑘𝑇]𝑥−𝛼𝑇 𝑖,𝑘𝑥+𝛽𝑖,𝑘, (28) where 𝑝𝑘 𝑖and 𝑞𝑘 𝑖are the columns of orthogonal matrices 𝑃𝑖and 𝑄𝑖, constructed from the normalized eigenvectors of the matrices 𝐴𝑖,𝐵𝑖and 𝜇𝐴𝑖 𝑘,𝜇𝐵𝑖 𝑘are the eigenvalues of the matrices 𝐴𝑖and 𝐵𝑖associated with the eigenvectors 𝑝𝑘 𝑖and 𝑞𝑘 𝑖, respectively. If, for all 𝑥∈𝑆,oneobtains𝐻𝑖,𝑘(𝑥)≧0for each 𝑖∈𝐼and 𝑘∈𝐾,then𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Proof. According to Theorem 8,itisenoughtoshowthat, for every feasible point and for all 𝑖∈𝐼and 𝑘∈𝐾, 𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)≧𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏)is valid. Given 𝑥∈𝑆and apair{𝑖,𝑘}∈𝐼×𝐾,weobtain 𝜇𝐴𝑖 𝑘𝛾𝑘 𝑖(𝑥,𝑥∗,𝜏)≧𝜇𝐵𝑖 𝑘𝜂𝑘 𝑖(𝑥,𝑥∗,𝜏) ⇐⇒ 𝜏𝑖𝜇𝐴𝑖 𝑘 𝑔𝑖(𝑥∗)⟨𝑥−𝑥∗,𝑝𝑘 𝑖⟩2≧𝜏𝑖𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2⟨𝑥−𝑥∗,𝑞𝑘 𝑖⟩2 ⇐⇒𝜇𝐴𝑖 𝑘⟨𝑥−𝑥∗,𝑝𝑘 𝑖⟩2≧𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)⟨𝑥−𝑥∗,𝑞𝑘 𝑖⟩2 ⇐⇒(√𝜇𝐴𝑖 𝑘⟨𝑥−𝑥∗,𝑝𝑘 𝑖⟩+√𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)⟨𝑥−𝑥∗,𝑞𝑘 𝑖⟩) ⋅(√𝜇𝐴𝑖 𝑘⟨𝑥−𝑥∗,𝑝𝑘 𝑖⟩−√𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)⟨𝑥−𝑥∗,𝑞𝑘 𝑖⟩)≧0 Journal of Applied Mathematics 7 ⇐⇒(⟨𝑥−𝑥∗,√𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖+√𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑞𝑘 𝑖⟩) ⋅(⟨𝑥−𝑥∗,√𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖−√𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑞𝑘 𝑖⟩)≧0 ⇐⇒(⟨𝑥−𝑥∗,𝑎+ 𝑖,𝑘⟩)(⟨𝑥−𝑥∗,𝑎− 𝑖,𝑘⟩)≧0 ⇐⇒(⟨𝑥,𝑎+ 𝑖,𝑘⟩−⟨𝑥∗,𝑎+ 𝑖,𝑘⟩)(⟨𝑥,𝑎− 𝑖,𝑘⟩−⟨𝑥∗,𝑎− 𝑖,𝑘⟩)≧0 ⇐⇒⟨𝑥,𝑎+ 𝑖,𝑘⟩⟨𝑥,𝑎− 𝑖,𝑘⟩−⟨𝑥∗,𝑎− 𝑖,𝑘⟩⟨𝑥,𝑎+ 𝑖,𝑘⟩ −⟨𝑥∗,𝑎+ 𝑖,𝑘⟩⟨𝑥,𝑎− 𝑖,𝑘⟩+⟨𝑥∗,𝑎+ 𝑖,𝑘⟩⟨𝑥∗,𝑎− 𝑖,𝑘⟩≧0 ⇐⇒⟨𝑥,𝑎+ 𝑖,𝑘⟩⟨𝑥,𝑎− 𝑖,𝑘⟩ −⟨𝑥,(⟨𝑥∗,𝑎+ 𝑖,𝑘⟩𝑎− 𝑖,𝑘 +⟨𝑥∗,𝑎− 𝑖,𝑘⟩𝑎+ 𝑖,𝑘)⟩ +⟨𝑥∗,𝑎+ 𝑖,𝑘⟩⟨𝑥∗,𝑎− 𝑖,𝑘⟩≧0 ⇐⇒⟨𝑥,𝑎+ 𝑖,𝑘⟩⟨𝑥,𝑎− 𝑖,𝑘⟩−⟨𝑥,𝛼𝑖,𝑘⟩+𝛽𝑖,𝑘 ≧0 ⇐⇒𝑥𝑇[𝑎+ 𝑖,𝑘𝑎− 𝑖,𝑘𝑇]𝑥−𝛼𝑇 𝑖,𝑘𝑥+𝛽𝑖,𝑘 ≧0 ⇐⇒𝐻𝑖,𝑘 (𝑥)≧0. (29) Therefore, the result follows from Theorem 8. From Corollary 9,ifeachquadraticfunction𝐻𝑖,𝑘(𝑥), {𝑖,𝑘}∈𝐼×𝐾,isnonnegativeinthefeasibleset,thenafeasible point satisfying (3),(4),and(5) is a Pareto optimal solution of (VQFP). Let [𝐻𝑖,𝑘]≡[𝑎+ 𝑖,𝑘𝑎− 𝑖,𝑘𝑇]∈R𝑛×𝑛.Then𝛽𝑖,𝑘 =𝑥∗𝑇[𝐻𝑖,𝑘]𝑥∗, and the nonnegativity of the quadratic 𝐻𝑖,𝑘(𝑥)=𝑥𝑇[𝐻𝑖,𝑘]𝑥− 𝛼𝑇 𝑖,𝑘𝑥+𝛽𝑖,𝑘 depends on each matrix [𝐻𝑖,𝑘]and each vector 𝛼𝑖,𝑘 ∈R𝑛,where{𝑖,𝑘} ∈ 𝐼×𝐾and 𝑥∗∈𝑆. For example, the unconstrained (VQFP)requires that each matrix [𝐻𝑖,𝑘] be positive semidefinite and that 𝛽𝑖,𝑘 ≧(𝛼 𝑖,𝑘 −𝑤𝑇[𝐻𝑖,𝑘]𝑤), where 𝑤is a solution of the system 2[𝐻𝑖,𝑘]𝑥=𝛼𝑖,𝑘. Corollary 10. Let 𝑥∗be a feasible point of (VQFP).Suppose that the constraint function ℎ𝑗is convex for each 𝑗∈𝐽and there exist vectors 𝜏∈R𝑚,𝜆∈Rℓ,suchthat(3),(4),and(5) are valid. If, for each pair {𝑖,𝑘}∈𝐼×𝐾,thematrix[𝐻𝑖,𝑘]is positive semidefinite and 𝛼𝑖,𝑘 =0(see (28)), then 𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Proof. By hypothesis, for all 𝑥∈𝑆,wehave𝐻𝑖,𝑘(𝑥)≧0for each pair {𝑖,𝑘}∈𝐼×𝐾. Therefore, the result follows from Corollary 9. Given a pair {𝑖,𝑘}∈𝐼×𝐾, writing each entry of the matrix [𝐻𝑖,𝑘]=(𝐻𝑖,𝑘(𝑟,𝑠))and each entry of the vector 𝛼𝑖,𝑘 =(𝛼 𝑖,𝑘(𝑟))according to the entries of the eigenvectors 𝑝𝑘 𝑖=(𝑝𝑘 𝑖(𝑟))and 𝑞𝑘 𝑖=(𝑞𝑘 𝑖(𝑟)),where𝑟,𝑠∈𝐾,weobtain,for each pair {𝑟,𝑠}∈𝐾×𝐾, 𝐻𝑖,𝑘 (𝑟,𝑠)=𝑎+ 𝑖,𝑘 (𝑟)𝑎− 𝑖,𝑘 (𝑠) =𝜇𝐴𝑖 𝑘𝑝𝑘 𝑖(𝑟)𝑝𝑘 𝑖(𝑠) +√𝜇𝐴𝑖 𝑘𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)(𝑝𝑘 𝑖(𝑠)𝑞𝑘 𝑖(𝑟)−𝑝𝑘 𝑖(𝑟)𝑞𝑘 𝑖(𝑠)) −𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑞𝑘 𝑖(𝑟)𝑞𝑘 𝑖(𝑠), 𝐻𝑖,𝑘 (𝑟,𝑟)=𝜇𝐴𝑖 𝑘(𝑝𝑘 𝑖(𝑟))2−𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)(𝑞𝑘 𝑖(𝑟))2, 𝛼𝑖,𝑘 (𝑟)=2𝜇𝐴𝑖 𝑘⟨𝑥∗,𝑝𝑘 𝑖⟩𝑝𝑘 𝑖(𝑟)−2𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)⟨𝑥∗,𝑞𝑘 𝑖⟩𝑞𝑘 𝑖(𝑟). (30) We can draw some conclusions from (30). For example, for a fixed pair {𝑖,𝑘}∈𝐼×𝐾,thevector𝛼𝑖,𝑘 is a linear combination of the eigenvectors 𝑝𝑘 𝑖and 𝑞𝑘 𝑖.If𝜇𝐴𝑖 𝑘=0,𝜇𝐵𝑖 𝑘=0, or 𝑓𝑖(𝑥∗)=0,then[𝐻𝑖,𝑘]is a symmetric matrix. Moreover, if 𝜇𝐴𝑖 𝑘𝜇𝐵𝑖 𝑘𝑓𝑖(𝑥∗)<0and there exists a pair {𝑟,𝑠}∈𝐾×𝐾such that 𝑝𝑘 𝑖(𝑠)𝑞𝑘 𝑖(𝑟) =𝑝𝑘 𝑖(𝑟)𝑞𝑘 𝑖(𝑠), then the matrix [𝐻𝑖,𝑘]∉R𝑛×𝑛. In this case, if there exists 𝑥∈𝑆such that 𝐻𝑖,𝑘(𝑥)∈C\R, 𝐻𝑖,𝑘(𝑥) ≧ 0does not make sense. However, when (26) is required, it is possible to show that 𝐻𝑖,𝑘(𝑥) ∈ C\Ris not possible. The results of Theorems 5and 8and their corollaries can be used in order to develop a method of searching for Pareto optimal solutions of (VQFP),anditmightbeuseful to determine the termination criteria in the development of algorithms. 4. Duality Matrix (24) defines a specific function, and by adding some assumptions about it, we obtain new results, such as, a relationship between the problem (VQFP)and a scalar problem associated with it, and the main duality theorems. In the scalar optimization problem case, Dinkelbach [3]andJagannathan[4] used a parametric approach that transforms the fractional optimization problem in a new scalar optimization problem. Similarly, we consider the following parameterized problem (VQFP)𝑥∗associated with the problem (VQFP): Minimize 𝑓(𝑥)−𝑓(𝑥∗) 𝑔(𝑥∗)𝑔(𝑥) =(𝑓1(𝑥)−𝑓1(𝑥∗) 𝑔1(𝑥∗)𝑔1(𝑥),...,𝑓𝑚(𝑥) −𝑓𝑚(𝑥∗) 𝑔𝑚(𝑥∗)𝑔𝑚(𝑥)) subject to ℎ𝑗(𝑥)≦0 𝑗∈𝐽,𝑥∈Ω, (VQFP)𝑥∗ 8Journal of Applied Mathematics where Ω⊆R𝑛,𝑓𝑖,𝑔𝑖,𝑖∈𝐼,andℎ𝑗,𝑗∈𝐽, are defined in (VQFP),and𝑥∗∈𝑆. Using assumptions of generalized convexity, Osuna- G´ omez et al. [17] presented the problem (VFP)𝑥∗and obtained necessary and sufficient conditions for weakly Paretooptimalityandmaindualitytheorems.Theresults presented in [3,4,17] considered each objective function as 𝑓𝑖(𝑥)−𝛼𝑖𝑔𝑖(𝑥),𝑖∈𝐼, and they studied the properties of the parameter 𝛼𝑖∈R. Following the ideas presented by Osuna- G´ omez et al. [17], we obtain new results by considering directly 𝛼𝑖≡𝑓 𝑖(𝑥∗)/𝑔𝑖(𝑥∗),𝑖∈𝐼,where𝑥∗∈Eff (VQFP). However, by imposing hypothesis on the linear combination of matrices [𝐴𝑖−(𝑓𝑖(𝑥∗)/𝑔𝑖(𝑥∗))𝐵𝑖],𝑖∈𝐼and 𝑥∗∈𝑆,we consider Pareto optimal solutions rather than weakly Pareto optimal solutions. To characterize the solutions of the problems (VOP), Geoffrion [37] used the solutions of the associated scalar problems.Similarly,weconsiderthefollowingweighted scalar problem (VQFP)𝑤 𝑥∗associated with the problem (VQFP): Minimize 𝑚 ∑ 𝑖=1𝑤𝑖(𝑓𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥)) subject to ℎ𝑗(𝑥)≦0 𝑗∈𝐽,𝑥∈Ω, (VQFP)𝑤 𝑥∗ where Ω⊆R𝑛,𝑓𝑖,𝑔𝑖,𝑖∈𝐼,andℎ𝑗,𝑗∈𝐽, are defined in (VQFP),𝑥∗∈𝑆and 𝑤=(𝑤1,...,𝑤𝑚)𝑇∈R𝑚,𝑤>0. 4.1. The Relationship between the Associated Problems. The next theorem and its proof are similar to Lemma 1.1 from [17], when Pareto optimal solutions (not necessarily weak) are considered. Theorem 11. 𝑥∗∈Eff (𝑉𝑄𝐹𝑃) if and only if 𝑥∗∈ Eff (𝑉𝑄𝐹𝑃)𝑥∗. Proof. SeeLemma1.1in[17], considering “≤” instead of “<.” In Section 3, we define the matrix  𝐹(𝑥∗)= ∑𝑚 𝑖=1[(𝜏𝑖/𝑔𝑖(𝑥∗))𝐴𝑖−(𝜏 𝑖𝑓𝑖(𝑥∗)/[𝑔𝑖(𝑥∗)]2)𝐵𝑖],where 𝑥∗∈𝑆and 𝜏∈R𝑚,𝜏>0. Let us define now the set W={𝑤∈R𝑚|𝑤>0},thefunction𝐹:W×𝑆→R𝑛×𝑛 given by 𝐹(𝑤,𝑥)≡𝑚 ∑ 𝑖=1𝑤𝑖[𝐴𝑖−𝑓𝑖(𝑥) 𝑔𝑖(𝑥)𝐵𝑖], (31) and, for each 𝑖∈𝐼,thefunctions𝐹𝑖:𝑆→R𝑛×𝑛 given by 𝐹𝑖(𝑥)≡𝐴𝑖−𝑓𝑖(𝑥) 𝑔𝑖(𝑥)𝐵𝑖.(32) Then, we have  𝐹(𝑥∗) = 𝐹(𝜏/𝑔(𝑥∗),𝑥∗),where𝜏/𝑔(𝑥∗)= (𝜏1/𝑔1(𝑥∗),...,𝜏𝑚/𝑔𝑚(𝑥∗))𝑇∈W,𝐹(𝑤,𝑥) = ∑𝑚 𝑖=1 𝑤𝑖𝐹𝑖(𝑥), and we can establish some relations among the associated problems (VQFP),(VQFP)𝑥∗,and(VQFP)𝑤 𝑥∗. Theorem 12. If 𝑥∗is an optimal solution of the weighted scalar problem (VQFP)𝑤 𝑥∗,then𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Proof. Suppose that 𝑥∗∉Eff (VQFP); then there exists another point 𝑥∈𝑆such that 𝑓𝑖(𝑥) 𝑔𝑖(𝑥)≤𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗) 󳨐⇒ 𝑓𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥)≤0 󳨐⇒ 𝑚 ∑ 𝑖=1𝑤𝑖(𝑓𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥))<0 󳨐⇒ 𝑚 ∑ 𝑖=1𝑤𝑖(𝑓𝑖(𝑥)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥)) <𝑚 ∑ 𝑖=1𝑤𝑖(𝑓𝑖(𝑥∗)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥∗)). (33) This contradicts the minimality of 𝑥∗in (VQFP)𝑤 𝑥∗. Lemma 13. Let 𝑥∗∈Eff (𝑉𝑄𝐹𝑃). Suppose that the constraint qualification (GGCQ) is satisfied at 𝑥∗; then there exist vectors 𝜏∗>0and 𝜆∗≧0such that 𝑚 ∑ 𝑖=1𝜏∗ 𝑖(∇𝑓𝑖(𝑥∗)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)∇𝑔𝑖(𝑥∗))+ℓ ∑ 𝑗=1𝜆∗ 𝑗∇ℎ𝑗(𝑥∗)=0, ℓ ∑ 𝑗=1𝜆∗ 𝑗ℎ𝑗(𝑥∗)=0. (34) Proof. Let 𝑥∗∈𝑆,𝜇∈R𝑚,𝜇>0,and𝜏𝑖=𝜇𝑖/𝑔𝑖(𝑥∗)>0, 𝑖∈𝐼.Then 𝑚 ∑ 𝑖=1𝜇𝑖∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)=𝑚 ∑ 𝑖=1𝜏𝑖𝑔𝑖(𝑥∗)∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗) =𝑚 ∑ 𝑖=1𝜏𝑖𝑔𝑖(𝑥∗)(∇𝑓𝑖(𝑥∗)𝑔𝑖(𝑥∗)−∇𝑔𝑖(𝑥∗)𝑓𝑖(𝑥∗) [𝑔𝑖(𝑥∗)]2) =𝑚 ∑ 𝑖=1𝜏𝑖(∇𝑓𝑖(𝑥∗)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)∇𝑔𝑖(𝑥∗)), (35) and if 𝑥∗∈Eff (VQFP),byLemma 4, there exist 𝜇∗>0and 𝜆∗≧0such that (𝑥∗,𝜇∗,𝜆∗)is a critical point, in the KKT sense, of the problem (VQFP).Thatis, 𝑚 ∑ 𝑖=1𝜇∗ 𝑖∇𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)+ℓ ∑ 𝑗=1𝜆∗ 𝑗∇ℎ𝑗(𝑥∗)=0, ℓ ∑ 𝑗=1𝜆∗ 𝑗ℎ𝑗(𝑥∗)=0. (36) Journal of Applied Mathematics 9 From (35), there exist 𝜏∗>0,𝜏∗ 𝑖=𝜇∗ 𝑖/𝑔𝑖(𝑥∗)>0,𝑖∈𝐼,and 𝜆∗≧0such that 𝑚 ∑ 𝑖=1𝜏∗ 𝑖(∇𝑓𝑖(𝑥∗)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)∇𝑔𝑖(𝑥∗))+ℓ ∑ 𝑗=1𝜆∗ 𝑗∇ℎ𝑗(𝑥∗)=0, ℓ ∑ 𝑗=1𝜆∗ 𝑗ℎ𝑗(𝑥∗)=0. (37) Therefore, the result is valid. Lemma 14. Let 𝑥∗∈𝑆. If there exists 𝑤∈W,suchthat the matrix 𝐹(𝑤,𝑥∗)is positive semidefinite, then the objective function of (VQFP)𝑤 𝑥∗is convex. Proof. Given 𝑥1,𝑥2∈𝑆, we have, for each 𝑖∈𝐼, 𝑓𝑖(𝑥1)−𝑓𝑖(𝑥2)=(𝑥1−𝑥2)𝑇𝐴𝑖(𝑥1−𝑥2) +∇𝑓𝑖(𝑥2)𝑇(𝑥1−𝑥2), 𝑔𝑖(𝑥1)−𝑔𝑖(𝑥2)=(𝑥1−𝑥2)𝑇𝐵𝑖(𝑥1−𝑥2) +∇𝑔𝑖(𝑥2)𝑇(𝑥1−𝑥2). (38) Hence, for each objective function of (VQFP)𝑥∗,wehave (𝑓𝑖(𝑥1)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥1)) −(𝑓𝑖(𝑥2)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥2)) =(𝑓𝑖(𝑥1)−𝑓𝑖(𝑥2))−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)(𝑔𝑖(𝑥1)−𝑔𝑖(𝑥2)) ={(𝑥1−𝑥2)𝑇𝐴𝑖(𝑥1−𝑥2)+∇𝑓𝑖(𝑥2)𝑇(𝑥1−𝑥2)} −𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗) ⋅{(𝑥1−𝑥2)𝑇𝐵𝑖(𝑥1−𝑥2)+∇𝑔𝑖(𝑥2)𝑇(𝑥1−𝑥2)} =(𝑥1−𝑥2)𝑇[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥1−𝑥2) +(∇𝑓𝑖(𝑥2)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)∇𝑔𝑖(𝑥2))𝑇(𝑥1−𝑥2). (39) If there exists 𝑤∈Wsuch that the matrix 𝐹(𝑤,𝑥∗)is positive semidefinite, then 𝑚 ∑ 𝑖=1𝑤𝑖(𝑓𝑖(𝑥1)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥1)) −𝑚 ∑ 𝑖=1𝑤𝑖(𝑓𝑖(𝑥2)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝑔𝑖(𝑥2)) =(𝑥1−𝑥2)𝑇𝑚 ∑ 𝑖=1𝑤𝑖[𝐴𝑖−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)𝐵𝑖](𝑥1−𝑥2) +𝑚 ∑ 𝑖=1𝑤𝑖(∇𝑓𝑖(𝑥2)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)∇𝑔𝑖(𝑥2))𝑇(𝑥1−𝑥2) ≧(𝑚 ∑ 𝑖=1𝑤𝑖(∇𝑓𝑖(𝑥2)−𝑓𝑖(𝑥∗) 𝑔𝑖(𝑥∗)∇𝑔𝑖(𝑥2)))𝑇(𝑥1−𝑥2). (40) Therefore, the objective function of (VQFP)𝑤 𝑥∗is convex. Note that the hypothesis of semidefiniteness on the matrix 𝐹(𝑤,𝑥∗)or on the matrices 𝐹𝑖(𝑥∗),𝑖∈𝐼,𝑥∗∈𝑆,is punctual. However, in the next example, we draw a situation in which, for all 𝑥∈𝑆and 𝑖∈𝐼,wehave𝑦𝑇𝐹𝑖(𝑥)𝑦≧0,forall 𝑦∈𝑆,andthen𝑦𝑇𝐹(𝑤,𝑥)𝑦≧0,forall𝑦∈𝑆. Example. Consider the problem (VQFP),where𝑆=[−2,2] and for all 𝑥∈𝑆 𝑓1(𝑥) 𝑔1(𝑥)=𝑥−2 𝑥2+2, 𝑓2(𝑥) 𝑔2(𝑥)=2𝑥2−𝑥−1 𝑥2+1 , 𝑓3(𝑥) 𝑔3(𝑥)=−2𝑥2−2𝑥−5 𝑥2+𝑥+1 . (41) Forallthesefunctions,weobtainforall𝑦∈𝑆 𝑦𝑇[𝐴1−𝑓1(𝑥) 𝑔1(𝑥)𝐵1]𝑦=𝑦2(0− 𝑥−2 𝑥2+21) =𝑦2(2−𝑥) 𝑥2+2 ≧0, 𝑦𝑇[𝐴2−𝑓2(𝑥) 𝑔2(𝑥)𝐵2]𝑦=𝑦2(2−2𝑥2−𝑥−1 𝑥2+1 1) =𝑦2(𝑥+3) 𝑥2+1 ≧0, 𝑦𝑇[𝐴3−𝑓3(𝑥) 𝑔3(𝑥)𝐵3]𝑦=𝑦2(−2−−2𝑥2−2𝑥−5 𝑥2+𝑥+1 1) =3𝑦2 𝑥2+𝑥+1≧0. (42) Therefore, for this example, each point 𝑥∗satisfying (3),(4), and (5) is Pareto optimal. For example, for (𝜏1,𝜏2,𝜏3)𝑇= (0.5,1,0.25)𝑇and (𝜆1,𝜆2)𝑇= (1,1),wehavethat𝑥∗= 0is Pareto optimal solution. Likewise, for (𝜏1,𝜏2,𝜏3)𝑇≈ (0.62,1,0.89)𝑇and (𝜆1,𝜆2)𝑇= (0,0)𝑇,wehavethat𝑥∗= −0.25is Pareto optimal solution. Theorem 11 shows an equivalence between the associated problems (VQFP)and (VQFP)𝑥∗. The next theorem shows a relation between the problems (VQFP)𝑥∗and (VQFP)𝑤 𝑥∗;then it provides a converse to the Theorem 12.