An efficient method for generalized linear multiplicative programming problem with multiplicative constraints

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1 DOI 0.86/s RESEARCH An efficient method for generalized linear multilicative rogramming roblem with multilicative constraints Yingfeng Zhao,2* and Sanyang Liu Oen Access *Corresondence: School of Mathematics and Statistics, Xidian University, Xi an 7007, China Full list of author information is available at the end of the article Abstract We resent a ractical branch and bound algorithm for globally solving generalized linear multilicative rogramming roblem with multilicative constraints. To solve the roblem, a relaxation rogramming roblem which is equivalent to a linear rogramming is roosed by utilizing a new two-hase relaxation technique. In the algorithm, lower and uer bounds are simultaneously obtained by solving some linear relaxation rogramming roblems. Global convergence has been roved and results of some samle examles and a small random exeriment show that the roosed algorithm is feasible and efficient. Keywords: Generalized linear multilicative rogramming, Global otimization, Branch and bound Mathematics Subject Classification: 90C26, 90C30 Background Multilicative rogramming refers to a class of otimization roblems which contains roducts of real functions in objective and (or) constraint functions. In this study, we consider the following generalized linear multilicative rogramming roblem: s.t. (GLMP) : min f 0 (x) = 0 j= f 0j(x) f i (x) = i j= f ij(x) 0, i =, 2,..., M, f i (x) = i j= f ij(x) 0, i = M +, M + 2,..., N, x D ={x R n Ax b, x 0, where f ij (x) = φ ij (x)ψ ij (x), φ ij (x) = n k= a ijk x k + b ij, ψ ij (x) = n k= c ijk x k + d ij, while the coefficients a ijk, c ijk and the constant terms b ij, d ij are all arbitrary real numbers, i = 0,, 2,..., N, j =, 2,..., i, k =, 2,..., n; A R m n is a matrix, b R m is a vector, set D is nonemty and bounded. Generalized linear multilicative rogramming (GLMP) with multilicative objective and constraint functions is a secial case of multilicative rogramming. It has attracted considerable attention of researchers and ractitioners for many years. This is mainly because, from the ractical oint of view, it ossesses imortant alication foreground 206 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (htt://creativecommons.org/licenses/by/4.0/), which ermits unrestricted use, distribution, and reroduction in any medium, rovided you give aroriate credit to the original author(s) and the source, rovide a link to the Creative Commons license, and indicate if changes were made.

2 Page 2 of 4 in various fields, including microeconomics (Henderson and Quandt 96), multileobjective decision (Benson 979; Keeney and Raiffa 993; Geoffrion 967), lant layout design (Quesada and Grossmann 996), data mining\attern recognition (Bennett and Mangasarian 994), marketing and service lanning (Samadi et al. 203), robust otimization (Mulvey et al. 995), and so on. And from the algorithmic design oint of view, a roduct of two affine functions, as noted in Avriel et al. (200), need not be convex(even not be quasi-convex), and hence roblem (GLMP) is a global otimization roblem which may have multile non-global local solutions, global solution methods for roblem (GLMP) have great difficulties and challenges. Due to the facts above, design efficient solution methods for globally solving the (GLMP) has imortant theoretical and the ractical significance. In the ast few decades, many solution methods have been devised for solving the roblem (GLMP). These methods are mainly classified as arameter-based methods (Konno et al. 994; Thoai 99), outer-aroximation methods (Gao et al. 2006; Kuno et al. 993), outcome-sace cutting lane methods (Benson and Boger 2000), branchand-bound methods (Ryoo and Sahinidis 2003; Shen and Jiao 2006; Konno and Fukaishi 2000; Kuno 200) and various heuristic methods (Benson and Boger 997; Liu et al. 999; Fan et al. 206). Recently, Wang rooses a global otimization algorithm for a kind of generalized linear multilicative rogramming by using simlicial artition techniques (Wang et al. 202), but his method is only valid for roblems in which the constraint functions are all linear. Jiao and Liu (205) resent an effective algorithm for solving the generalized linear multilicative roblem with generalized olynomial constraints by converting it into an equivalent generalized geometric rogramming roblem, the roblem they considered is more general but only valid under the assumtion φ ij (x) >0, ψ ij (x) >0, x X. There are many other solution methods not mentioned for (GLMP) and its secial case, nevertheless, most of these methods are either develoed for secial circumstances or can only obtain a local solution of roblem (GLMP). In this aer, we ut forward a fast global otimization algorithm for generalized linear multilicative rogramming roblem (GLMP). Our research can be divided into three stes. First, a well erformed linear relaxation rogramming roblem for the (GLMP) is established by using a new two-hase relaxation technique. Second, two key oerations for develoing a branch and bound algorithm for the (GLMP) are described. Finally, global convergence roerty is roved and some numerical exeriments are executed to illustrate the feasibility and robustness of the roosed algorithm. Comared with some existing methods, the new two-hase relaxation technique we used in the algorithm has a very good aroximation effect, and it doesn t require the condition φ ij (x) >0, ψ ij (x) >0, x X. Further more, relative to the algorithm in Jiao (2009), Quesada and Grossmann (996), the roosed algorithm can be alied to a more general case of linear multilicative rogramming roblem. The reminder of this article is arranged as follows. Section "Two-hase relaxation technique" exlains how the two-hase relaxation method is realized, section "Algorithm and its convergence" introduces the branch and bound oeration for deriving the resented algorithm. The algorithm statement as well as the convergence roerty are described in section "Numerical exeriments". In section "Concluding remarks", the results of some

3 Page 3 of 4 numerical exeriments aeared in recent literatures are listed and some concluding remarks are reorted in the last section. Two hase relaxation technique As is known to all, construct a well erformed relaxation roblem can bing great convenience for designing branch and bound algorithm of global otimization roblems. In this section, equivalent transformation technique and a new two-hase relaxation skill will be used to establish a linear relaxation rogramming roblem for underestimating the otimal value of roblem (GLMP). First, we comute the initial variable bounds by solving the following linear rogramming roblems: x l i = min x D x i, then an initial rectangle X 0 = x u i = max x D x i, i =, 2,..., n, {x R n x l i x i x u i, i =, 2,..., n will be obtained. To construct the first-hase relaxation rogramming roblem of the (GLMP) over subrectangle X X 0, we further solve some linear rogramming roblems as follows: l ij = L ij = min x D φ ij(x), u ij = max X x D φ ij(x), X min x D ψ ij(x), U ij = max X x D ψ ij(x). X Uon criteria (), it is clear that ( )( ) ( )( ) φij (x) l ij ψij (x) L ij 0, φij (x) u ij ψij (x) U ij 0, () (2) and ( φij (x) l ij )( ψij (x) U ij ) 0, ( φij (x) u ij )( ψij (x) L ij ) 0, (3) by taking (2) and (3) together, we have φ ij (x)ψ ij (x) max { u ij ψ ij (x) + U ij φ ij (x) U ij u ij, l ij ψ ij (x) + L ij φ ij (x) L ij l ij, (4) and φ ij (x)ψ ij (x) min { u ij ψ ij (x) + L ij φ ij (x) u ij L ij, l ij ψ ij (x) + U ij φ ij (x) U ij l ij. (5) For each i = 0,, 2,..., N, by denoting g ij (x) u ijψ ij (x) + U ij φ ij (x) U ij u ij, g 2 ij (x) l ijψ ij (x) + L ij φ ij (x) L ij l ij, and g ij (x) u ijψ ij (x) + L ij φ ij (x) u ij L ij, g 2 ij (x) l ijψ ij (x) + U ij φ ij (x) U ij l ij, conclusion (4) and (5) can be exressed as φ ij (x)ψ ij (x) min {g ij (x), g2 ij (x) g ij (x), (6)

4 Page 4 of 4 and φ ij (x)ψ ij (x) max {g ij (x), g2ij (x) g ij (x), (7) resectively. Then we can obtain a lower bound function g i (x) and uer bound function g i (x) for f i (x), which satisfy g i (x) f i (x) g i (x), i = 0,, 2,..., N, where i g i (x) = g ij (x), g i (x) = g ij (x), i = 0,, 2,..., N. i j= j= (8) So far, based on the above discussion, we can get the first-hase relaxation rogramming roblem for the (GLMP) which we formulated as follows: s.t. (RMP0) : min g 0 (x) = 0 j= g 0j (x) g i (x) = i j= g (x) 0, i =, 2,..., M, ij g i (x) = i j= g ij (x) 0, i = M +, M + 2,..., N, x D X ={x X Ax b, x 0, To get the second-hase linear relaxation rogramming roblem, we will once again relax each nonlinear function aeared in roblem (RMP0) according the following conclusion: i g i (x) = max {g ij (x), g2ij (x) j= i i max g ij (x), g 2 ij (x) g i(x), i = 0,, 2,..., M, j= j= (9) and i g i (x) = min {g ij (x), g2 ij (x) j= i i min g ij (x), g 2 ij (x) g i(x), i = M +, M + 2,..., N. j= j= (0) With conclusion (9) and (0), the second-hase relaxation rogramming roblem (RMP) of the (GLMP) can be exressed as follows:

5 Page 5 of 4 s.t. (RMP) : { 0 min g 0 (x) = max j= g 0j (x), 0 j= g2 0j (x) { i g i (x) = max j= g ij (x), i j= g2 ij (x) 0, i =, 2,..., M, { i g i (x) = min j= g ij (x), i j= g2 ij (x) 0, i = M +, M + 2,..., N, x D X ={x X Ax b, x 0, which is roved equivalent to the following linear rogramming roblem: min t 0 s.t. (x) t 0, (ERMP) : j= g 0j 0 j= g2 0j i j= g ij i j= g2 ij i j= g ij (x) t 0, (x) 0, i =, 2,..., M, (x) 0, i =, 2,..., M, (x) 0, i = M +, M + 2,..., N, i j= g2 ij (x) 0, i = M +, M + 2,..., N, x D X ={x X Ax b, x 0. Theorem If (x, t) R n+ is a global otimal solution for the (ERMP), then x R n is a global otimal solution for the (RMP). Conversely, if x R n is a global otimal solution for the (RMP), then (x, t) R n+ is a global otimal solution for the (ERMP), where t = g 0 (x ). Proof The roof of this theorem can be easily followed according to the definition of roblems (RMP) and (ERMP), therefore, it is omitted here. Theorem 2 () For any x X, we have g i (x) f i (x), i = 0,, 2,..., M, and g i (x) f i (x), i = M +, M + 2,..., N. (2) gi (x) f i (x) 0, as Ui L i 0, ui l i 0, where U i = (U i, U i2,..., U ii ), L i = (L i, L i2,..., L ii ) and u i = (u i, u i2,..., u ii ), l i = (l i, l i2,..., l ii ). Proof () can be easily verified from conclusion (4), (5), (9) and (0), thus the detailed roof is omitted here. For (2), according to the Cauchy Schwarz inequality, we know that for i = 0,,..., M,

6 Page 6 of 4 gi (x) f i (x) i = max i g ij (x), i g 2 ij (x) φ ij (x)ψ ij (x) j= j= j= i = max i g ij (x) i φ ij (x)ψ ij (x), g 2 ij (x) i φ ij (x)ψ ij (x) j= j= j= j= = max i ( lij ψ ij (x) + L ij φ ij (x) l ij L ij φ ij (x)ψ ij (x) ), j= i ( uij ψ ij (x) + U ij φ ij (x) U ij u ij φ ij (x)ψ ij (x) ) j= i = max ) i ) (ψ ij (x) L ij )(φ ij (x) l ij, (ψ ij (x) U ij )(φ ij u ij j= max { (U L) (u l), for the case i = M +, M + 2,..., N, it can be roved with the similar method, so omitted here, and thus the Proof of Theorem 2 is comleted. Remark From Theorems and 2, we only need to solve roblem (ERMP) instead of solving the (RMP) to obtain the lower and uer bounds of the otimal value in roblem (GLMP). Remark 2 Based on the continuity of linear function, U i L i 0 and ui l i 0 will hold when the diameter of X aroximate to zero, this indicated that we can erform the branching oeration in variable sace X with the convergence roerty is guaranteed. Remark 3 Theorem 2 ensures that roblem (ERMP) can infinitely aroximate the roblem (GLMP), as X 0, this will guarantee the global convergence of the roosed algorithm. j= Algorithm and its convergence In this section, we will describe two key oeration for designing an efficient branch and bound algorithm for roblem (GLMP), that is, branching and bounding. Then the algorithm stes will be summarized with roof rocess of global convergence roerty followed. Branching and bounding The branching oeration iteratively subdivides the rectangle X into subregions according to an exhaustive artition rule, such that any infinite iterative sequence of artition sets shrinks to a singleton. For this, we shall adot an standard range bisection aroach, which is adequate to insure global convergence of the roosed algorithm. Detailed rocess is described as follows.

7 Page 7 of 4 For any region X =[x l, x u ] X 0, let r argmin{xi u xi l i =, 2,..., n and mid = (xr l + xu r )/2, then the current region X can be divided into the following two sub-regions: { X = x R n xi l x i xi u, i =r, xl r x r mid, and X = { x R n xi l x i xi u, i =r, mid x r xr u. For each artition subset X generated by the above branching oeration, the bounding oeration is mainly concentrate on estimating a lower bound LB(X) and a uer bound UB(X) for the otimal value of roblem (GLMP). This oeration is realized by solving the linear relaxation rogramming roblem (ERMP) over all artition sets in the k th iteration, and the one with the smallest otimal value will rovide the lower bound for otimal value of roblem (GLMP) over the initial region X 0. Moreover, since any feasible solution of the relaxation rogramming roblem will also be feasible to the (GLMP), so we can evaluate the initial objective value and make the one with smallest value as a new uer bound if ossible. Algorithm and its convergence Based on the former discussion, the algorithm stes can be summarized as follows: Ste 0 (Initialization) Choose convergence tolerance ǫ = 0 8, set iteration counter k := 0 and the initial artition set as 0 = X 0. Solve the initial linear relaxation roblem (ERMP) over region X 0, if the (ERMP) is not feasible then there is no feasible solution for the initial roblem. Otherwise, denote the otimal value and solution as f bar and xot 0, resectively. Then we can obtain the initial uer and lower bound of the otimal value for roblem (GLMP), that is, UB := f 0 (xot 0 ), and LB := f bar. And then, if UB LB <ǫ, the algorithm can sto, and xot 0 is the otimal solution of the (GLMP), otherwise roceed to ste. Ste (Branching) Partition X k into two new sub-rectangles according to the artition rule described in section Branching and bounding. Deleting X k and add the new nods into the active nods set X k, still denote the set of new artitioned sets as X k. Ste 2 (Bounding) For each subregion still of interest X kµ X 0, µ =, 2, obtain the otimal solution and value for roblem (RMFP) by solving the relaxation linear rogramming roblem over X kµ, if LB(X k,µ )>UB, delete X kµ from X k. Otherwise, we can udate the lower and uer bounds: LB = min{lb(x k,µ ) µ =, 2 and UB = min{ub, f (x k,µ ) µ =, 2. Ste 3 (Termination) If UB LB ǫ, the algorithm can be stoed, UB is the global otimal value for (GLMP). Otherwise, set k := k +, and select the node with the smallest otimal value as the current active node, and return to Ste. Theorem 3 The roosed algorithm either terminates within finite iterations with an otimal solution for (GLMP) be found, or generates an infinite sequence of iterations such that along any infinite branches of the branch-and-bound tree, any accumulation oint of the sequence {x k will be the global otimal solution of the (GLMP).

8 Page 8 of 4 Proof () If the roosed algorithm is finite, assume it stos at the kth iteration, k 0. From the termination criteria, we know that UB LB ǫ. Based on the uer bounding technique described in Ste 3, it imlies that f (x k ) LB ǫ. Let v ot be the otimal value of roblem (GLMP), then by section Branching and bounding and Ste 3 above, we known that UB = f (x k ) v ot LB. Hence, taken together, it imlies that v ot + ǫ LB + ǫ f (x k ) v ot, and thus the roof of art () is comleted. (2) If the algorithm doesn t terminate within finite iterations and generates an infinite feasible solution sequence {x k for the (GLMP) via solving the (RMP). According to the structure of the roosed algorithm, we have LB k min x X f 0(x), () assume that: X k arg min LB(X), x k = x(x k ) X k X 0. X k (2) Horst (998) has roved that LB k is non-decrease and bounded above by min x X f 0 (x), thus the existence of the limit LB := lim k LB k min x X f 0 (x) can be guaranteed. Further more, since x k is a sequence on a comact set, it must have a convergent subsequence. For any accumulation oint ˆx of {x k, there exists a subsequence of {x k which, without loss of generality, we might still denote as {x k satisfied lim k x k = ˆx. With similar method in Tuy (99), we can easily follow that the subdivision of artition sets in ste is exhaustive on X 0, and the selection of elements to be artitioned is bound imroving, thus there exists a decreasing subsequence X r X k where X r r with x r X r, LB r = LB(X r ) = g 0 (x r ), lim r x r = ˆx. Based on the construction rocess of the relaxation roblem, we know that the linear relaxation functions g i (x)(i = 0,,..., N) used in roblem (RMP)(and thus for (ERMP)) are strongly consistent on X 0, hence it follows that lim k LB k = LB = g 0 (ˆx). Since ˆx is feasible to (GLMP) and combining with () we can deduce that ˆx is a global solution for the (GLMP). Numerical exeriments To verify the erformance of the roosed algorithm, we solve some test roblems in recent literatures (Thoai 99; Wang et al. 202; Jiao and Liu 205; Wang and Liang 2005; Gao et al. 200; Chen and Jiao 2009; Shen et al. 2008; Shen and Jiao 2006; Jiao 2009) and construct a

9 Page 9 of 4 roblem to illustrate the nature that (GLMP) may have multile local otimal solutions (see Fig. ), comutational results are given in Table, where the following notations have been used in row headers: Exa.: the serial number of exeriments; Ref.: reference which we contrast with; Ot.Val.: otimal value; Ot.Sol.: otimal solution; Iter: numbers of iterations; Time: CPU time in seconds; Pre.: recision we used in the algorithm. We used the TPRM to reresent the two-hase relaxation method given in this aer. We coded the algorithms in Matlab 204a, and ran the tests in a micro comuter with Intel(R) Xeon(R) rocessor of 2.4 GHz, 4 GB of RAM memory, under the Win0 oerational system. We used linrog solver to solve all linear rogramming roblems. Table shows that our algorithm erforms more efficient than that in references Ryoo and Sahinidis (2003), Shen and Jiao (2006), Thoai (99), Tuy (99), Wang et al. (202) and Wang and Liang (2005). Esecially for Examles, 2, 5, 6, 8 and 0, our algorithm only need one iteration to determine the global otimal solutions, this indicates that our new relaxation technique is so efficient that the global otimal solution can be founded in the initialization ste. Further more, we constructed an examle (Examle and Fig. ) with multile local otimum to test our algorithm. Examle (Refs. Wang and Liang 2005; Jiao 2009). min x 2 + x2 2 s.t. 0.3x x 2, 2 x 5, x 2 3, Fig. 3-D surface and contour lot over [ 5, 5; 5, 5] of the objective function in Examle. From this figure we can see that the objective function in Examle may have multile local otimal solutions over the feasible region

10 Page 0 of 4 Table Results of the numerical contrast exeriments Exa. Ref. Ot. val. Ot. sol. Iter Time Pre. Wang and Liang (2005) ( ,.66665) Jiao (2009) (2.0, ) 58 < 0 8 TPRM (2.0000,.6667) Jiao (2009) 4.0 (2.0,.0, 3.0) TPRM 4.0 (2.0000,.0000, ) Gao et al. (200) (2.0003, ) Chen and Jiao (2009) ( , ) TPRM 0.0 (2.0000,8.0000) Gao et al. (200) (0.0002, 0.000, 0, 0, 0, 0, 0, 0, 0, 0, 0) TPRM (0.00, , , 6.42,.9434, 0.00, , , , 5.800, 5.340) Thoai (99) (.34792,.39555, 0, ) TPRM (.348, 0.396, , ) Shen et al. (2008).475 (0.6824, ) TPRM (0.0000, ) Chen and Jiao (2009) (2.0, ) TPRM (2.0000,.0000) Shen and Jiao (2006) (2.00,.00) Jiao and Liu (205) ( ,.0) TPRM (2.0000,.0000) Shen and Jiao (2006).77 (.7709, 2.772) Jiao and Liu (205).7708 (.7709, 2.772) TPRM.770 (.77088, 2.778) Jiao and Liu (205) (0.0000,4.0000) TPRM (0.0000,4.0000) TPRM (0.0000, ) Examle 2 (Refs. Jiao 2009). min x 2 + x2 2 x2 3 s.t. 0.3x x x 2 x x x 3 4, 2 x 5, x 2 3, x 3 3, Examle 3 (Refs. Gao et al. 200; Chen and Jiao 2009). min (x + x 2 )(x x 2 + 7) s.t. 2x + x 2 4, x + x 2 0, 4x + x 2 0, 2x + x 2 6, x + x 2 6, x 5, x + x 2 0, x x

11 Page of 4 Examle 4 (Refs. Gao et al. 200; Chen and Jiao 2009). { min (c T x + d )(c2 T x + d 2) where s.t. Ax b, b = (8, 72, 72, 9, 9, 9, 8, 8) T, d = 0, d 2 = 0, ( c =, 0, ) T ( 9,0,0,0,0,0,0,0,0, c 2 = 0,, ) T 9,0,0,0,0,0,0,0, A = , Examle 5 (Refs. Wang et al. 202; Thoai 99). min ( x x x x ) ( x x x x ) s.t x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x 0, x 2 0, x 3 0, x 4 0. Examle 6 (Refs. Chen and Jiao 2009). min (6x + x 2 + )(x + 2x 2 + ) + ( x + 3)(x + x 2 + ) s.t. 2x + x 2 0, x + x 2 8, 0 x 2.5, x 2 0. Examle 7 (Refs. Shen et al. 2008). min 4x 2 5x2 2 + x x 2 + 2x s.t. x x 2 0, 3 x2 3 x2 2, 2 x x 2, 0 x 3, x 2 0.

12 Page 2 of 4 Examle 8 (Refs. Shen and Jiao 2006; Jiao and Liu 205). min x x 2 2x + x 2 + s.t. 8x2 2 6x 6x 2, x x + 2x 2 7, x 2.5, x Examle 9 (Refs. Shen and Jiao 2006; Jiao and Liu 205). min x s.t. Examle 0 (Refs. Jiao and Liu 205). min x + (2x 3x 2 + 3)(x + x 2 ) Examle Figure. Examle 2 4 x + 2 x 2 6 x2 6 2 x2, 4 x2 + 4 x x 3 7 x 2, x 5.5, x s.t. x + 2x 2 8, x 2 3, x + 2x 2 2, x 2x 2 5, x 0, x 2 0. min 6x 2 x2 2 s.t. 2x + x 2 0, x + x 2 8, 64 x2 64 x x 2, 4 x x 2 8 x2 2, 5 x 5, 5 x 2 5. min f (x) = i= (at 0i x + d 0i)(c0i T x + e 0i) s.t. i= (at i x + b i)(ci T x + d i) 0, i= (at 2i x + b 2i)(c2i T x + d 2i) 0, x D ={x R n Ax b.

13 Page 3 of 4 Table 2 Numerical results of Examle 2 m n Iter Time where the real numbers a ij, c ij, d ij and e ij are randomly generated in the range [, ], the real elements of A and b are randomly generated in the range [0, ]. For this roblem, we tested twenty different random instances and listed the comutational results in Table 2, where the notations used in the head line have the following means: Iter:average numbers of iterations in the algorithm; Time: average CPU time in seconds; m and n denote the number of linear constraints and variables, resectively. Concluding remarks In this study, a new global otimization algorithm is resented for solving generalized linear multilicative rogramming roblem with multilicative constraints. This method has three main features. First, the relaxation roblem erforms well in aroximation effect. Second, to obtain the lower and uer bounds of the otimal value, we only need to solve some linear rogramming roblems. Finally, the roblem we investigated is more general than those in many other literatures and results of numerical contrast exeriments show that our method erforms better than those methods. Authors contributions Both authors contributed equally to the manuscrit. Both authors read and aroved the final manuscrit. Author details School of Mathematics and Statistics, Xidian University, Xi an 7007, China. 2 School of Mathematical Science, Henan Institute of Science and Technology, Xinxiang , China. Acknowledgments This aer is suorted by the National Natural Science Foundation of China (637374); the Science and Technology Key Project of Education Deartment of Henan Province (4A0024), (5A0023) and (6A0030); the natural science foundation of Henan Province ( ); the Major Scientific Research Projects of Henan Institute of Science and Technology (205ZD07). Cometing interests The authors declare that they have no cometing interests. Received: 23 Aril 206 Acceted: 2 August 206 References Avriel M, Diewert WE, Schaible S, Zhang I (200) Generalized concavity. Plenum Press, New York Bennett KF, Mangasarian OL (994) Bilinear searation of two sets in n-sace. Comut Otim Al 2: Benson HP (979) Vector maximization with two objective functions. J Otim Theory Al 28: Benson HP, Boger GM (997) Multilicative rogramming roblems: analysis and efficient oint search heuristic. J Otim Theory Al 94(2): Benson HP, Boger GM (2000) Outcome-sace cutting-lane algorithm for linear multilicative rogramming. J Otim Theory Al 04(2):30 22 Chen YQ, Jiao HW (2009) A nonisolated otimal solution of general linear multilicative rogramming roblems. Comut Oer Res 36:

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