Necessary and sufficient conditions for achieving global optimal solutions in multiobjective quadratic fractional optimization problems

Journal of Global Optimization (2019) 74:233–253 https://doi.org/10.1007/s10898-019-00766-1 Necessary and sufficient conditions for achieving global optimal solutions in multiobjective quadratic fractional optimization problems Washington Alves de Oliveira1·Marko Antonio Rojas-Medar2· Antonio Beato-Moreno3·Maria Beatriz Hernández-Jiménez4 Received: 9 November 2017 / Accepted: 21 March 2019 / Published online: 1 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract If x∗is a local minimum solution, then there exists a ball of radius r>0 such that f(x)≥ f(x∗)for all x∈B(x∗,r). The purpose of the current study is to identify the suitable B(x∗,r)of the local optimal solution x∗for a particular multiobjective optimization problem. We provide a way to calculate the largest radius of the ball centered at local Pareto solution in which this solution is optimal. In this process, we present the necessary and sufficient conditions for achieving a global Pareto optimal solution. The results of this investigation might be useful to determine stopping criteria in the algorithms development. Keywords Pareto optimality conditions ·Multiobjective optimization ·Quadratic fractional optimization problems BWashington Alves de Oliveira washington.oli[email protected] Marko Antonio Rojas-Medar [email protected] Antonio Beato-Moreno [email protected] Maria Beatriz Hernández-Jiménez [email protected] 1School of Applied Sciences, University of Campinas, R. Pedro Zaccaria, 1300, Limeira, São Paulo 13484-350, Brazil 2Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile 3Department of Statistics and Operations Research, College of Mathematics, University of Sevilla, 41012 Sevilla, Spain 4Department of Economics, Quantitative Methods and H. Economic, Universidad Pablo de Olavide, 41013 Sevilla, Spain 123 234 Journal of Global Optimization (2019) 74:233–253 1 Introduction We consider the following multiobjective optimization problem (MOP): (MOP) Minimize u(x)=(u1(x),...,um(x)) subject to hj(x)0,j∈J, x∈Ω, where Ω⊆Rnis an open set, uiand hjare real-valued functions defined on Ωfor each i∈I≡{1,...,m}and j∈J≡{1,...,p}.LetSdenotes the set of feasible solutions defined by S={x∈Ω|hj(x)0,j∈J}. Let R+be the set of nonnegative real numbers and xTbe the transpose of x∈Rn.The inequality among vectors is to be understood in a component-wise sense: if x,y∈Rn,then x<yif and only if xi<yi;xyif and only if xiyi;andx≤yif and only if xy and x= y. By considering Euclidean distance, let B(x,r)be the open ball and ¯ B(x,r)be the closed ball of radius r>0 centered at x.Wedenoteby∂B(x,r)the boundary of B(x,r). Similar to Chankong and Haimes [8], we present two optimality definitions for MOP. Definition 1 A feasible solution x∗is said to be a (locally) Pareto optimal solution of MOP, if there does not exist another solution (x∈B(x∗,r), r>0) x∈Ssuch that u(x)≤u(x∗). Definition 2 A feasible solution x∗is said to be a (locally) weakly Pareto optimal solution of MOP, if there does not exist another solution (x∈B(x∗,r), r>0) x∈Ssuch that u(x)<u(x∗). We observe that x∈B(x∗,r)is not a restrictive hypothesis, since for an arbitrary layout neighborhood N(x∗)of x∗, it is possible that B(x∗,r)⊂N(x∗). The literature presents different ways to characterize Pareto optimality conditions (see [8, 21] and references therein). This paper extends the results of the literature by proposing necessary and sufficient conditions for achieving a global Pareto optimal solution from a local solution. We calculate the largest radius of the ball centered at a local Pareto solution, in which this solution is optimal; then we verify the conditions for this solution to be globally optimal. This paper is organized as follows: Sect. 2presents the multiobjective quadratic fractional optimization problem. Our methodology is introduced in Sect. 3. Section 4provides some optimality conditions. Conclusions are given in Sect. 5. 2 Quadratic fractional problem In this paper, we consider the multiobjective quadratic fractional optimization problem (MQFP), where for each i∈I,ui(x)≡fi(x) gi(x)in MOP, and fi,gi, are quadratic functions of nreal variables, gi(x)>0forallx∈Ω. Fractional optimization problems frequently arise in the decision-making applications, including portfolio selection, cutting and stock problems, and game theory, in the optimization of the ratio performance/cost, or profit/investment, or cost/time, and so on. Convexity and generalized convexity are used in the literature to achieve optimality conditions and main duality theorems for optimal solutions, including the scalar (single-objective) fractional optimization problem (FP) and multiobjective fractional optimization problem (MFP). 123 Journal of Global Optimization (2019) 74:233–253 235 Jagannathan [11] has established optimality conditions by transforming the original problem FP to an associated non fractional problem. Liang et al. [19] obtained results that relate the primal-dual pair of FP. Craven [9] has presented several results for FP, and Stancu- Minasian [28] has provided a book containing various practical and theoretical aspects for FP. Emam [10] has studied a multiobjective integer quadratic programming problem under uncertainty. Liang et al. [20] have extended their approach [19] to MFP and examined the dual problem reported in [22]. Osuna-Gómez et al. [24] have used the parametric approaches of Jagannathan [11] and Bector et al. [6], and two classes of generalized convexity to establish weak Pareto optimality conditions and the main duality theorems for the differentiable MFP. Santos et al. [26] have expanded Osuna-Gómez’s results to a more general, non-differentiable case of MFP. We find few studies that involve quadratic functions at both the numerator and denominator in the ratio objective function. In most cases, the mixing of linear and quadratic functions is included. Schaible and Shi [27] and Benson [7] have considered a similar case of the scalar quadratic fractional optimization problem (QFP). Benson [7] has developed some theoretical properties and optimality conditions for a QFP consisting of the convex functions, and also presented an algorithm and its convergence properties. Jiao and Liu [12]havesolvedaQFP consisting of the quadratically constrained sum of quadratic ratios problem by using a range division and compression algorithm for global optimization. Beato-Moreno et al. [4,5], Arévalo and Zapata [3], Konno and Inori [13], Rhode and Weber [25], Kornbluth and Steuer [16], and Korhonen and Yu [14,15] have addressed more similar approaches to MQFP. From an iterative computational test, Beato-Moreno et al. [4,5] have characterized the Pareto optimal solution for MQFP with linear and quadratic functions, and achieved some theoretical results by using the function linearization approach of Bector et al. [6]. Arévalo and Zapata [3], Konno and Inori [13], and Rhode and Weber [25]have analyzed the portfolio selection problem. Kornbluth and Steuer [16] have used a suitable Simplex method for MFP consisting of linear functions. Korhonen and Yu [14,15]have proposed an iterative computational method based on search directions and weighted sums for MQFP using linear and quadratic functions. In their previous work, Oliveira et al. [23] have developed sufficient Pareto optimality conditions and duality results for a legitimate MQFP, where they have avoided the generalized convexities. Ammar [1,2] has studied the solutions of MQFP from a fuzzy random technique. Lachhwani [17,18] has provided a fuzzy goal programming approach for MQFP. Stancu- Minasian and Stancu [31] have presented a new class of generalized type-I univex functions and derived the weak, strong and converse duality theorems for MQFP. Stancu-Minasian [29, 30] has listed a complete updated bibliography of fractional programming problems. The approach taken in this paper is different from previous ones. Without generalized convexity assumptions on the objective functions, some necessary and sufficient conditions are given for a local Pareto optimal solution to be a global Pareto optimal solution. The central aspect of this contribution is to show how to calculate the largest radius of the ball centered at local Pareto optimal solution where this solution is globally optimal. These conditions might be useful to determine stopping criteria in the development of algorithms, including those based on quadratic approximations in more general problems. 123 236 Journal of Global Optimization (2019) 74:233–253 3 Radius of efficiency If u(x)≤u(z), we say that x∈S dominates z ∈Sin MOP. The Pareto (or globally efficient) optimal solution set is denoted by Eff (MOP), and the set of locally Pareto (or locally efficient) optimal solutions is denoted by Leff (MOP). We introduce the new concept of radius of efficiency for MOP, and then we extend this concept to MQFP. 3.1 Radius of efficiency for MOP Definition 3 A feasible solution x∗∈Sis said to be λ-efficient or has radius of efficiency λ in MOP if x∗∈Leff (MOP) and there does not exist another solution x∈B(x∗,λ)∩Sthat dominates x∗. We note that if x∗is λ-efficient, then it is β-efficient, for all β<λ.Ifx∗is a globally optimal solution in S, we say that it is ∞-efficient. If x∗∈Leff (MOP), there exists r>0 such that x∗is not dominated in B(x∗,r)∩S. However, our goal is to calculate the largest radius λ∗>0 of the ball B(x∗,λ ∗)such that x∗is not dominated in B(x∗,λ ∗)∩S. In particular, for λ∗sufficiently large we obtain a globally optimal solution in S. Thus, we consider a local Pareto optimal solution on a fixed and most appropriate ball. The value λ∗can be useful for solving MOP because if the radius of efficiency is known, the cost to find a new solution using a suitable search procedure can be estimated. Auxiliary techniques induced by the radius of efficiency can be used to conclude if a solution is globally optimal. 3.2 Radius of efficiency for MQFP If x∗∈Leff (MQFP), then the smallest value λ>0, such that the inequality f(x∗+λd) g(x∗+λd)≤ f(x∗) g(x∗)is valid for every unitary direction d, provides the maximum radius of efficiency of x∗. We use the following problem closely associated with MQFP, namely MQFPx∗,where ui(x)≡fi(x)−fi(x∗) gi(x∗)gi(x),x∗∈S,and fi,gi,i∈I, are the quadratic functions of MQFP. Similar to Lemma 1.1 from [24], we consider Pareto optimal solutions to achieve next result. Theorem 1 x∗∈Leff (MQFP) if and only if x∗∈Leff (MQFPx∗). In particular, there exists r>0such that x∗is locally Pareto optimal solution for MQFP in B(x∗,r)∩S if and only if it is locally Pareto optimal solution for MQFPx∗in B(x∗,r)∩S. Proof Let x∗be locally Pareto optimal solution for MQFP in B(x∗,r)∩S. Suppose that x∗/∈Leff (MQFPx∗), then there exists another solution x∈B(x∗,r), satisfying f(x)− f(x∗) g(x∗)g(x)≤f(x∗)−f(x∗) g(x∗)g(x∗)=0⇒ f(x) g(x)≤f(x∗) g(x∗), which contradicts that x∗∈ Leff (MQFP). Therefore, x∗∈Leff (MQFPx∗). Now, let x∗be the locally Pareto optimal solution for MQFPx∗in B(x∗,r)∩S. Suppose that x∗/∈Leff (MQFP), then there exists another solution x∈B(x∗,r), satisfying f(x) g(x)≤f(x∗) g(x∗)⇒ f(x)−f(x∗) g(x∗)g(x)≤0= f(x∗)−f(x∗) g(x∗)g(x∗), which contradicts that x∗∈Leff (MQFPx∗). Therefore, x∗∈Leff (MQFP).  From Theorem 1, we must choose which is more appropriate between MQFP and MQFPx∗ to calculate the maximum radius of B(x∗,r), and analyze the dominance of a local Pareto optimal solution in the set of feasible solutions. 123 Journal of Global Optimization (2019) 74:233–253 237 4 Optimality conditions First, we establish the optimality conditions for the unconstrained problem, and then we extend the results to the constrained problem with linear constraints. 4.1 Radius of efficiency in the unconstrained case For each i∈Iand all x∈Rn, we address fi(x)=xTAix+aT ix+¯aiand gi(x)= xTBix+bT ix+¯ bi,where Ai,Bi∈Rn×nare matrices, and we assume that Aiis symmetric; Bi is symmetric and positive semidefinite; ai,bi∈Rnand ¯ai,¯ bi∈R, with ¯ bi>−(wiTBiwi+ bT iwi),wherewiis the solution of the system 2Bix+bi=0; that is, wiis the solution where the function xTBix+bT ixreaches its minimum and this choice ensures that gi(x)>0, ∀x∈Rn. We do not consider cases where 2Bix+bi=0 has no solution. Let ∇u(x)and ∇2u(x)be the gradient and the Hessian matrix of the function u:Rn→R at x, respectively. We denote by UMQFP and UMQFPx∗the unconstrained MQFP and MQFPx∗, respectively. Further, we define some important sets and parameters for this work. Definition 4 Let d∈Rnbe a fixed and unitary direction, and x∗∈Rn.Let pi(x)=xTAi−fi(x∗) gi(x∗)Bix+aT i−fi(x∗) gi(x∗)bT ix,i∈I,(1) be a quadratic function, and I0=i∈I|dT∇2pi(x∗)d,∇pi(x∗)TdT≥[0,0]T, I+={i∈I|dT∇2pi(x∗)d>0and∇pi(x∗)Td<0}, I−={i∈I|dT∇2pi(x∗)d<0and∇pi(x∗)Td>0}, λd −=max i∈I−−2∇pi(x∗)Td dT∇2pi(x∗)d,λ d +=min i∈I+−2∇pi(x∗)Td dT∇2pi(x∗)d,if I+=∅ +∞,if I+=∅, Λd −=[λd −,∞), Λd +=(0,λ d +],if I+=∅ (0,λ d +), if I+=∅,Λd=Λd −∩Λd +. Each function pidefined in (1) is an objective function of MQFPx∗without the constant term ¯ai−fi(x∗) gi(x∗)¯ bi, thus pi(x∗)=−[¯ai−fi(x∗) gi(x∗)¯ bi],i∈I. Taylor expansion around zero of the function ri(λ) =pi(x∗+λd),i∈I,λ∈R,λ0, is the quadratic function ri(λ) = pi(x∗)+λ∇pi(x∗)Td+λ2 2dT∇2pi(x∗)d,wherepi(x∗),∇pi(x∗)Tdand 1 2dT∇2pi(x∗)d are the constant, linear and quadratic terms, respectively. For a fixed and unitary direction d,thesetI+includes the index i∈Iwhose function ri decreases in value around λ=0 but is convex; i.e., dT∇2pi(x∗)d>0. The set I−includes the index i∈Iwhose function riincreases in value around λ=0 but is concave, i.e., dT∇2pi(x∗)d<0(seeFig.5); and the set I0includes the index i∈Iwhose function ri does not decrease around λ=0(seeFig.6). Figures 6,7and 10 illustrate the positive roots λd −and λd +of the equation ri(λ)−pi(x∗)= 0. For the same direction d,thesetsΛd −,Λd +and Λdare intervals of R+\{0}.λd −is the left extreme of Λd −,andλd +is the right extreme of Λd +.Ifλd −λd +they are the end points of Λd,andifλd −>λ d +we have Λd=∅(see Fig. 8). Note that if I+=∅it is appropriate to 123 238 Journal of Global Optimization (2019) 74:233–253 choose λd +=+∞, because if I−=∅we have Λd=[λd −,∞)∩(0,∞)=[λd −,∞)=∅ (see Fig. 9). Corollary 2 Let d ∈∂B(0,1).IfΛd=∅,thenp i(x)pi(x∗)for all x =x∗+λd, λ∈Λd, and i ∈I+∪I−. Proof By considering the Taylor expansion around zero of the functions ¯ri(λ) =fi(x∗+λd) and ˜ri(λ) =gi(x∗+λd),λ∈Λd,i∈I+∪I−, ¯ri(λ) =fi(x∗)+λ∇fi(x∗)Td+λ2dTAid, ˜ri(λ) =gi(x∗)+λ∇gi(x∗)Td+λ2dTBid, we obtain fi(x∗+λd) gi(x∗+λd)fi(x∗) gi(x∗)⇐⇒ λdTλAi−λfi(x∗) gi(x∗)Bid λfi(x∗) gi(x∗)∇gi(x∗)−λ∇fi(x∗)T d. By using pi(x)from Eq. (1), we have λdTλAi−λfi(x∗) gi(x∗)Bidλfi(x∗) gi(x∗)∇gi(x∗)−λ∇fi(x∗)T d ⇐⇒ λ∇pi(x∗)Td+λ 2dT∇2pi(x∗)d0,(2) where 1 2∇2pi(x∗)=Ai−fi(x∗) gi(x∗)Biand ∇pi(x∗)=∇fi(x∗)−fi(x∗) gi(x∗)∇gi(x∗). Therefore, from (2) pi(x∗+λd)−pi(x∗)=λ∇pi(x∗)Td+λ 2dT∇2pi(x∗)d0 ⇐⇒ ri(λ) =pi(x∗+λd)pi(x∗).  Figures 5,9and 10 illustrate the conclusion of Corollary 2. Condition 1 Let d ∈∂B(0,1), and suppose that I0=∅,I +=∅and I−=∅.Ifλd −=λd += −2∇pi(x∗)Td dT∇2pi(x∗)d, for all i ∈I+∪I−, then we define Λd:= ∅. Condition 2 Let x∗∈Leff (UMQFPx∗) and d ∈∂B(0,1). Then it does not simultaneously occur ∇pi(x∗)Td=dT∇2pi(x∗)d=0for all i ∈I. By imposing Conditions 1and 2, we eliminate some rare cases that might occur with respect to the size of sets I0,I+and I−. For instance, if |I+|=|I−|=1andλd −=λd +it is more suitable to use Λd=∅instead of using Λd={λd −}. Note that these conditions are useless for weakly Pareto optimal solutions. Lemma 1 Let x∗∈Leff (UMQFP). Then x dominates x∗if and only if there exists d ∈ ∂B(0,1)such that I0=∅,I −=∅and Λd=∅,whereλ∗∈Λdand x =x∗+λ∗d. 123 Journal of Global Optimization (2019) 74:233–253 239 Proof (⇒)See Appendix A. (⇐)Suppose that I0=∅,I−=∅and Λd=∅in the direction d.IfI+=∅then there exists λ∗∈Λd(see Fig. 9), in which x=x∗+λ∗dand p(x)≤p(x∗) have a solution. If I+=∅then either there exists a λ∗∈Λd,inwhichx=x∗+λ∗dand p(x)≤p(x∗)have a solution, orλd −<λ d +(see Fig. 10) and there exists λ∗∈Λd=[λd −,λ d +], in which x=x∗+λ∗dand p(x)≤p(x∗)have a solution. Thus, xdominates x∗. Theorem 3 Let x∗∈Leff (UMQFP). Then x∗∈Eff (UMQFP) if and only if for all d ∈ ∂B(0,1),I 0=∅or Λd=∅whenever I+=∅. Proof (⇒)Suppose there exists d∈∂B(0,1)where I+=∅,I0=∅and Λd=∅.Asx∗∈ Leff (UMQFP), from Theorem 1,x∗∈Leff (UMQFPx∗)andI−=∅in this direction. By Lemma 1,thereexistλ∗∈Λdand x=x∗+λ∗dsuch that p(x)≤p(x∗)has a solution. By Eq. (1), f(x) g(x)≤f(x∗) g(x∗)has a solution. Which contradicts that x∗∈Eff (UMQFP). Therefore, for all d∈∂B(0,1),I0=∅or Λd=∅whenever I+=∅. (⇐)By Theorem 1,x∗∈Leff (UMQFPx∗). Suppose that I0=∅for an arbitrary d∈∂B(0,1). Then, by definition, there exists i∈Ifor which dT∇2pi(x∗)d>0and ∇pi(x∗)Td0, or dT∇2pi(x∗)d0and∇pi(x∗)Td>0, and then ri(λ) grows indefinitely for λ>0, and p(x)≤p(x∗)does not have a solution for each x=x∗+λdnear to x∗(see Fig. 11). Therefore, x=x∗+λddoes not dominate x∗. Now, suppose that I+=∅and I0=∅,andΛd=∅in the direction d,thenI−=∅.By Lemma 1,ifI0=∅,I+=∅,I−=∅and Λd=∅,p(x∗+λd)≤p(x∗)does not have a solution for λ>0(seeFig.8). Therefore, x=x∗+λddoes not dominate x∗. We conclude that if d∈∂B(0,1)and I0=∅,x∗is non-dominated; or whenever I+=∅and Λd=∅,we again have x∗non-dominated. Thus x∗∈Eff (UMQFP).  We do not require any kind of generalized convexity to obtain the optimality conditions of Theorem 3. In the results that follow, we define D={d∈∂B(0,1)|Λd=∅}. Corollary 4 Let x∗∈Leff (UMQFP) and β=inf d∈Dλd −. Then there does not exist another solution x ∈B(x∗,β)such that f(x) g(x)≤f(x∗) g(x∗). Proof It follows immediately from Theorem 3,ifd∈D, then there exists λ>0inΛdsuch that another solution x∈Rndominates x∗in this direction. However, the first solution xthat dominates x∗along the direction dis x=x∗+λd −d. By checking all directions in the set D, we conclude that the first solution xwhich dominates x∗is x=x∗+βd. Therefore x∗is β-efficient and there does not exist another solution x∈B(x∗,β)such that f(x) g(x)≤f(x∗) g(x∗). Figure 10 illustrates a direction d∈D. There, we observe that I0=∅,I+={2,3}, I−={1,4},Λd=∅,andifx=x∗+λd −d,p(x)≤p(x∗)has a solution. It is possible to rewrite Corollary 4replacing the set Dwith the set D={d∈∂Bn(0,1)|I0=∅}(see the charts in Figs. 8and 10). Next, we present a desirable result which gives a lower bound for the radius of efficiency of a local Pareto optimality solution. Corollary 5 Let x∗∈Leff (UMQFP) and F(d)=max i∈I−2∇pi(x∗)Td. Suppose there exists ρ∈R, such that for all d ∈∂B(0,1),F(d)ρ. Then there does not exist another solution x∈B(x∗,ρ −γ)such that f(x) g(x)≤f(x∗) g(x∗),whereγ<0,γ=min i∈I{γi}and γiis the smallest negative eigenvalue of the matrix ∇2pi(x∗),i∈I. 123 240 Journal of Global Optimization (2019) 74:233–253 Proof By Theorem 3and the value of βin Corollary 4, if we search a solution x=x∗+λd that dominates x∗,wehavetofindavalueλ>0 that satisfies λβ=inf d∈Dmax i∈I−−2∇pi(x∗)Td dT∇2pi(x∗)d. From the symmetric and diagonalizable matrix ∇2pi(x∗),i∈I−,weobtain0 > dT∇2pi(x∗)d≥γi,whereγiis the smallest negative eigenvalue of the matrix ∇2pi(x∗), thus λβ=inf d∈Dmax i∈I−−2∇pi(x∗)Td dT∇2pi(x∗)d inf d∈Dmax i∈I−−2∇pi(x∗)Td γi inf d∈Dmax i∈I−−2∇pi(x∗)Td γ=inf d∈D F(d) −γ, where γ=min i∈I{γi}.SinceF(d)ρ,βρ −γ.IfΛd=∅for all d∈∂B(0,1),thenx∗is ∞-efficient; but if there exists dsuch that Λd=∅,thenx∗is β-efficient. As ∞>βρ −γ, the theorem holds.  Because i∈I−, the changeover from the first to the second inequality of the Corollary 5 demonstration is valid, and it also provides ρ>0. Therefore ρ −γ>0. In order to limit the search space, we present another beneficial result which gives an upper bound for the radius of efficiency of a local Pareto optimality solution. Let ·be the Euclidean norm. Theorem 6 Let x∗∈Leff (UMQFP) and M =min i∈I+2∇pi(x∗) α. Suppose that for some d∈D and for all i ∈I+=∅,d T∇2pi(x∗)dα>0. If there does not exist another solution x ∈B(x∗,M)such that f(x) g(x)≤f(x∗) g(x∗),thenx ∗∈Eff (UMQFP). Proof Lemma 1and Corollary 4provide that if there exists a solution xthat dominates x∗ in the direction d,thenI−=∅and λd −x−x∗λd +.LetP=supd∈D{λd +}.Bythe hypotheses and x−x∗P, we also obtain P=sup d∈Dmin i∈I+−2∇pi(x∗)Td dT∇2pi(x∗)d sup d∈Dmin i∈I+−2∇pi(x∗)Td α min i∈I+2∇pi(x∗) α=M. Therefore, if there does not exist a solution that dominates x∗in B(x∗,M), then there does not exist another solution in Rnwith this property, and x∗∈Eff (UMQFP).  Because i∈I+, the changeover from the first to the second inequality of the Theorem 6 demonstration is valid. It only has usefulness if M<∞. Our results can be use to determine whether a locally Pareto optimal solution is dominated by some point within a limited subset 123 Journal of Global Optimization (2019) 74:233–253 241 x2 x1 x∗ x B(x∗,ρ −γ) B(x∗,β) B(x∗,P) B(x∗,M) Fig. 1 Some interesting spherical regions of the solution x∗∈Leff (UMQFP) of Rn. Note that Corollary 4provides the maximum radius β=inf d∈Dλd −of B(x∗,β) where x∗is not dominated. Corollary 5provides a lower bound for β, i.e., a new B(x∗,ρ −γ). On the other hand, Theorem 6provides two balls centered at x∗, one of them with radius P=supd∈D{λd +}, and another with radius M=min i∈I+2∇pi(x∗) α(upper bound of P). Therefore, we have four balls such that B(x∗,ρ −γ)⊆B(x∗,β)⊆B(x∗,P)⊆B(x∗,M). The lower and upper bounds are more attractive computationally to calculate. A suitable search can be made in the subset ¯ B(x∗,P)\B(x∗,β), or alternatively in the subset ¯ B(x∗,M)\B(x∗,ρ −γ). Figure 1illustrates a particular case in R2. It shows four spheres centered at the solution x∗: in dashed lines and nearer to the center is the radius of B(x∗,ρ −γ); in continuous lines and near the center is the radius of B(x∗,β)and B(x∗,P), respectively; the closed subset ¯ B(x∗,P)\B(x∗,β) is the shaded area; finally, the radius of the biggest ball B(x∗,M)is represented by the outer dashed lines. If there exists a solution xthat dominates x∗,itmust belong to subset ¯ B(x∗,P)\B(x∗,β). Auxiliary computational methods can be designed to further reduce the size of subset ¯ B(x∗,P)\B(x∗,β). It is enough to observe that x∈¯ B(x∗,P)\B(x∗,β)if and only if there exists a unitary direction dsuch that λd −λλd +and x=x∗+λd. Therefore, all those directions participating in subset ¯ B(x∗,P)\B(x∗,β), but with Λd=∅, can be excluded. Figure 2illustrates this possibility, where there exist two shaded subsets of ¯ B(x∗,P)\B(x∗,β), each containing a possible solution x. The shaded subsets represent only the directions dsuch that λd −λλd +. 123 248 Journal of Global Optimization (2019) 74:233–253 ri(λ) λ p4(x∗) p3(x∗) λd + p2(x∗) λd − p1(x∗) d/∈D Λd=∅ 1,3∈I+ 2,4∈I− Fig. 8 The search for xthat dominates x∗∈Leff (UMQFPx∗) in the search direction d/∈D, in which I0=∅, I+={1,3},I−={2,4}and Λd=∅ ri(λ) λ p4(x∗) p3(x∗) p2(x∗) λd − p1(x∗) d∈D Λd=∅ I+=∅ 1,2,3,4∈I− Fig. 9 The search for xthat dominates x∗∈Leff (UMQFPx∗) in the search direction d∈D, in which I0=∅, I+=∅,I−={1,2,3,4}and Λd=∅ 123 Journal of Global Optimization (2019) 74:233–253 249 ri(λ) λ p4(x∗) λd − p3(x∗) λd + p2(x∗) p1(x∗) d∈D Λd=∅ 2,3∈I+ 1,4∈I− Fig. 10 The search for xthat dominates x∗∈Leff (UMQFPx∗) in the search direction d∈D, in which I0=∅,I+={2,3},I−={1,4}and Λd=∅ ri(λ) λ p4(x∗) p3(x∗) p2(x∗) p1(x∗) d/∈D I0=∅ 1∈I0 3∈I+ 2,4∈I− Fig. 11 The search for xthat dominates x∗∈Leff (UMQFPx∗) in the search direction d/∈D, in which I0={1},I+={3}and I−={2,4} 123 250 Journal of Global Optimization (2019) 74:233–253 ri(λ) λ p4(x∗) p3(x∗) λd + p2(x∗) λd − p1(x∗) Λd=∅ λd 1,3∈I+ 2,4∈I− Fig. 12 The search for xthat dominates x∗∈Leff (MQFPx∗) in the direction d, in which I0=∅,I+={1,3}, I−={2,4},λd +<λ d −λd and Λd=∅ ri(λ) λ p4(x∗) λd − p3(x∗) p2(x∗) p1(x∗) Λd=∅ λd λd +←λd 2,3∈I+ 1,4∈I− Fig. 13 The search for xthat dominates x∗∈Leff (MQFPx∗) in the direction d, in which I0=∅,I+={2,3}, I−={1,4},λd <λ d −,λd +←λd and Λd=∅ 123 Journal of Global Optimization (2019) 74:233–253 251 ri(λ) λ p4(x∗) λd − p3(x∗) λd + p2(x∗) p1(x∗) Λd=∅ λd 2,3∈I+ 1,4∈I− Fig. 14 The search for xthat dominates x∗∈Leff (MQFPx∗) in the direction d, in which I0=∅,I+={2,3}, I−={1,4},λd −λd +<λ d and Λd=∅ ri(λ) λ p4(x∗) p3(x∗) p2(x∗) λ d − p1(x∗) Λd=∅ I+=∅ λ d λ d + ←λ d 1,2,3,4∈I− Fig. 15 The search for xthat dominates x∗∈Leff (MQFPx∗) in the direction d, in which I0=∅,I+=∅, I−={1,2,3,4},λd <λ d −,λd +←λd and Λd=∅ 123 252 Journal of Global Optimization (2019) 74:233–253 References 1. 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