OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX MULTIPLE OBJECTIVE PROBLEM TWO SPECIAL CASES1

Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 26 OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX MULTIPLE OBJECTIVE PROBLEM: TWO SPECIAL CASES 1 Assoc. Prof. Nguyen Thi Bach Kim 1 and M.Sc.Tran Ngoc Thang 2 , 1,2 School of Applied Mathematics and Informatics Hanoi University of Science and Technology 1 Email: kim.nguyenthibach@hust.edu.vn 2 Email: thang.tranngoc@hust.edu.vn Abstract. Optimizing over the efficient set is a very hard and interesting task in multiple objective optimization. Besides, this problem has some important applications in finance, economics, engineering, and other fields. In this article, we propose convex programming procedures for solving the problem of minimizing a real function over the efficient set of a convex multiple objective programming problem in the two special cases. Preliminary computational experiments show that these procedures can work well. AMS Subject Classification: 2000 Mathematics Subject Classification. Primary: 90 C29; Secondary: 90 C26 Key words: Global optimization, Optimization over the efficient set, Outcome set, Convex programming. 1. Introduction Let X be a nonempty, compact and convex set in n R . Let ( ), =1, , i f x i k , 2k  , be convex functions defined on a suitable open set containing X . Then the convex multiple objective programming problem may be written as 1 Min ( ) = ( ( ), , ( )) s.t. . T k f x f x f x x X (CMP) When = 2,k problem ()CMP is called a bicriteria convex programming problem. Such problem are not uncommon and have received special attention in the literature (see, for instance, H.P.Benson and D.Lee [2], J.Fulop and L.D.Muu [3], N.T.B.Kim and T.N.Thang [6], H.X.Phu [10], B.Schandle, K.Klamroth and M.M.Wiecek [11], Y.Yamamoto [14] and references therein). Let h be a real-valued function on n R . The central problem of interest in this paper is the following problem min ( ) s.t. , X h x x E (OP 0 ) where X E is the efficient decision set for problem ()CMP and defined as follows: 0 0 0 ={ | such that ( ) ( ) and ( ) ( )}. X E x X x X f x f x f x f x     1 THIS RESEARCH IS FUNDED BY VIETNAM NATIONAL FOUNDATION FOR SCIENCE AND TECHNOLOGY DEVELOPMENT (NAFOSTED) UNDER GRANT NUMBER "101.01-2013.19" Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 27 As usual, we use the notation 12 yy , where 12 , k yyR , to indicate 12 ii yy for all =1, ,ik . It is well-known that, the set X E is generally neither convex set nor given explicitly as the form of a standard mathematical programming problems, even in the case of linear multiple objective programming problem when the component functions 1 ,, k ff of f are linear and X is a polyhedral convex set. Therefore, problem 0 ()OP is one of hard global programming problems. This problem has applications in finance, economics, engineering, and other fields. Recently this problem has been attracted a great deal of attention from researcher (see e.g. [1, 2, 3, 4, 5, 6, 8, 9, 12, 14] and references therein). In this article, simple convex programming procedures are proposed for solving two special cases of problem 0 ()OP . These special-case procedures require quite little computational effort in comparison to ones required by algorithms for the general problem 0 ()OP . 2 Preliminaries Let Q be a nonempty subset in k R . The set of all efficient points of Q is denoted by ep Q and given by 0 0 0 ={ | such that and }. ep Q q Q q Q q q q q     It is clear that a point 0 ep qQ if 00 ( ) ={ } k Q q q  R , where ={ | 0, =1, , } kk i y y i k  RR . Let ={ | = ( ) for some }. k Y y y f x x XR As usual, the set Y is said to be the outcome set for problem ()CMP ; see, for instance, [2, 6, 12, 15]. Since the functions , =1, , i f i k are continuous and n X  R is a nonempty, compact set, the outcome set Y is also compact set in k R . Therefore the efficient set ep Y is nonempty [7]. The relationship between the efficient decision set X E and the efficient set ep Y of the outcome set Y is described as follows. Proposition 2.1 i) For any 0 ep yY , if 0 xX satisfies 00 ()f x y , then 0 X xE . ii) For any 0 X xE , if 00 = ( )y f x , then 0 . ep yY Proof. This fact follows from the definitions.  Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 28 Let = ={ | for some }. kk T Y z z y y Y     RR It is clear that T is a nonempty, full-dimension closed convex set. The following interesting property of T (Theorem 3.2 in [15]) will be used in the sequel . Proposition 2.2 =. ep ep TY For each =1,2, , ,ik let I i y be the optimal value of the following convex programming problem min s.t. . i y y T It is clear that I i y is also the optimal value of the following convex programming problem min ( ) s.t. . i f x y X (P i ) The point 1 = ( , , ) I I I k y y y is called the ideal point of the set T . Notice that the ideal point I y need not belong to T . Proposition 2.3 If I yT then ={ } I ep Yy . Proof. This fact follows from the definitions and Proposition 2.2.  It is clear that the case of ={ } I ep Yy is a special case of problem ()CMP . Let ={ | ( ) , =1, , }. id I ii X x X f x y i k By the definition, id X is a convex set. The following corollary is immediate from Proposition 2.1 and Proposition 2.3. Proposition 2.4 If id X is not empty then I yT and = id X EX . Otherwise, I y does not belong to T . The next discuss concerns with the case that ()CMP is a bicriteria convex programming problem, i.e. =2k , and the objective function ()hx of the problem ()OP is defined by 1 1 2 2 ( ) = ( ) ( ) = , ( )h x f x f x f x       (1) where 2 12 = ( , ) T    R . The case could happen in certain common situations. For instance, problem 0 ()OP represents the minimization of a criterion function () i fx , {1,2}i , of problem ()CMP over the efficient decision set X E . Let Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 29 ={ ( )| }. YX E f x x E The set Y E is called the efficient outcome set for problem ()CMP . The outcome-space reformulation of problem 0 ()OP can be given by min , s.t. . Y y y E     (OP 1 ) By the definition, it is easy to see that = Y ep EY . Combining this fact and Proposition 2.2, problem 1 ()OP can be rewritten as follows min , s.t. . ep y y T     (OP 2 ) Here, instead of solving problem 1 ()OP , we solve problem 2 ()OP . Since T is a nonempty convex subset in 2 R , it is well know [9] that the efficient set ep T is homeomorphic to a nonempty closed interval of 1 R . By geometry, it is easily seen that the problem 2 1 1 min{ : , = } I y y T y y (P S ) has an unique optimal solution S y and the problem 1 2 2 min{ : , = } I y y T y y (P E ) has an unique optimal solution E y . If I yT then SE yy and the efficient set ep T is a curve on the boundary of T with starting point S y and the end point E y . Direct computation shows that the equation of the line through S y and E y is ,=ay   , where 12 1 1 2 2 1 1 2 2 11 = ( , ) and = . EE E S S E E S S E yy a y y y y y y y y       (2) It is clear that the vector a is strictly positive. Now, let * ={ | , } and = \ ( , ), SE M y T a y M M y y       where ( , ) ={ = (1 ) |0 < <1} S E S E y y y ty t y t and M is the boundary of the set M . It is clear that M is a compact convex set. By the definition and geometry, we can see that * M contains the set of all extreme points of M and * =. ep TM (3) Consider following convex problem min , s.t. ,y y M     (OP 3 ) that has the explicit information as follows Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 30 min , s.t. ( ) 0 , ,, y f x y xX ay         where vector 2 aR and the real number  is determined by (2) . The relationship between the optimal solutions to problem 0 ()OP and the optimal solutions to problem 3 ()OP is presented in the following proposition. Proposition 2.5 Suppose that ** ( , )xy is an optimal solution of the problem 3 ()OP . Then * x is an optimal solution of problem 0 ()OP . Proof. It is well known that a convex programming problem with the linear objective function has an optimal solution which belongs to the extreme point set of the feasible solution [13]. This fact and (3) implies that * y is an optimal solution of problem 2 ()OP . Since == ep Y ep T E Y , by definition, we have * ,,yy       for all Y yE and * ep yY . Then * , , ( ) , . X y f x x E         (4) Since ** ( , )xy is a feasible solution of problem 3 ()OP , we have ** ()f x y . By Proposition 2.1, * X xE . Furthermore, we have * ()f x Y and * ep yY . This infers that ** = ( )y f x . Combining this fact and (4) prove that * x is an optimal solution of problem 0 ()OP . 3 Solving Two Special Cases Case 1. The ideal point I y belongs to the outcome set Y and the objective function ()hx of problem 0 ()OP is convex. By Proposition 2.4, to detect whether the ideal point I y belongs to T and solve problem 0 ()OP in this case, we solve the following convex programming problem min ( ) s.t. . id h x x X (CP I ) Namely, the procedure for this case is described as follows. Procedure 1. Step 1. For each =1, ,ik , find the optimal value I i y of problem () i P . Step 2. Solve the convex programming problem () I CP . If problem () I CP is not feasible Then STOP (the Case 1 does not apply). Else Find any optimal solution * x to problem () I CP . STOP Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 31 ( * x is an optimal solution to problem 0 ()OP ). Below we present two numerical examples to illustrate Procedure 1. Example 1. Consider problem 0 ()OP , where X E is the efficient solution set to the problem 1 2 1 2 Vmin ( ( ), ( )) = ( , )f x f x x x 22 12 s.t. ( 2) ( 2) 4xx    12 2xx     12 2xx   12 2xx    and 22 1 2 1 2 ( ) = min{0.5 0.25 0.2;2 4.6 5.8}h x x x x x    . In the case =0  Step 1. Solving problem 1 ()P and 2 ()P , we obtain the ideal point = (0.6667.0.6667) I y . Step 2. Solving problem () I CP , we find that it is not feasible and the algorithm is terminated. It means that the ideal point I y does not belong to T . In the case =1  Step 1. Solving problem 1 ()P and 2 ()P , we obtain the ideal point = (1,1) I y . Step 2. Solving problem () I CP , we find an optimal solution * = (1,1)x . Then * x is the optimal solution to problem 0 ()OP and the optimal value * ( ) = 0.9500.hx Example 2. Consider problem 0 ()OP , with 22 1 1 2 ( ) =10 (1 2 3 )h x x x x   and X E is the efficient solution set to problem ()CMP where =2k and 1 1 2 2 1 2 ( ) = 3 2 3, ( ) = 1,f x x x f x x x    2 2 2 1 1 2 2 1 2 ={ |10 14 5 32 22 22 0,X x x x x x x x      R 1 1 2 1 2 4, 5 3 8, 4 3 4}x x x x x       Step 1. Solving problem 1 ()P and 2 ()P , we obtain the ideal point = (1.8000,2.6000) I y . Step 2. Solving problem () I CP , we can find an optimal solution * = (1.6000,0.0000)x . Then * x is the optimal solution to problem 0 ()OP and the optimal value * ( ) = 43.2400.hx Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 32 Case 2. Problem ()CMP is a bicriteria convex programming problem and the objective function ()hx of the problem 0 ()OP has the form as (1) . In this case, the procedure for solving problem is established basing on Proposition 2.4 and Proposition 2.5. Procedure 2. Step 1. For each =1,2i , find the optimal value I i y of problem () i P . Step 2. Solve the convex programming problem () I CP . If problem () I CP is not feasible Then Go to Step 3 (the ideal point I yT ). Else Find any optimal solution * x to the problem () I CP , where =2k . STOP ( * x is an optimal solution for problem 0 ()OP ). Step 3. Solve the convex programming problems () S P and () E P to find the efficient point S y and E y , respectively. Step 4. Solve problem 3 ()OP to find any optimal solution ** ( , )xy . STOP ( * x is an optimal solution of problem 0 ()OP ) We give below some examples to illustrate Procedure 2. Example 3. Consider the problem 0 ()OP , where X E is the efficient solution set to problem ()CMP with =2k and 22 1 1 2 2 ( ) = ( 2) 1, ( ) = ( 4) 1,f x x f x x      2 2 2 1 2 1 2 = | 25 4 100 0, 2 4 0 ,X x x x x x      R and 1 1 2 2 ( ) = ( ) ( )h x f x f x   . Case  * x * y * ()hx 1 (1.0,0.0) (1.9931,0.4128) (1.0000,13.8730) 1.0000 2 (0.8,0.2) (1.6471,1.1765) (1.1246,8.9723) 2.6941 3 (0.5,0.5) (0.8000,1.6000) (2.4400,6.7600) 4.6000 4 (0.2,0.8) ( 1.0000,2.5000) (10.0000,3.2500) 4.6000 Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 33 5 (0.0,1.0) ( 1.6502,2.8251) (14.3236,2.3804) 2.3804 6 ( 0.2,0.8) ( 1.6502,2.8251) (14.3236,2.3804) 0.9604 7 (0.8, 0.2) (1.9932,0.4121) (1.0000,13.8780) 1.9756 8 ( 0.5, 0.5) ( 1.6502,2.8251) (14.3236,2.3804) 8.3520 Table 1: Computational results of Example 3 Step 1. Solving problem 1 ()P and 2 ()P , we obtain the ideal point = (1.000,2.3804) I y . Step 2. Solving problem () I CP , we can find that it is not feasible. Then go to Step 3. Step 3. Solve two problems () S P and () E P to obtain the points = (14.3236,2.3804) S y and = (1.0000,13.8781) E y . Step 4. For each 2 12 = ( , )    R , solve problem 3 ()OP to find the optimal solution ** ( , )xy . Then * x is an optimal solution to problem 0 ()OP and * ()hx is the optimal value of 0 ()OP . The computational results are shown in Table 1. Example 4. Consider the problem 0 ()OP , where 1 1 2 2 ( ) = ( ) ( )h x f x f x   and X E is the efficient solution set to the following problem   12 Vmin ( ) = ( ), ( ) s.t. , 0, ( ) 0, f x f x f x Ax b x g x   where 1.0 2.0 1.0 1.0 1.0 1.0 = , = , 2.0 1.0 4.0 2.0 5.0 10.0 1.0 1.0 1.5 Ab                                      22 12 ( ) = 0.5( 1) 1.4( 0.5) 1.1,g x x x    22 1 1 2 1 2 ( ) = 0.4 4 ,f x x x x x   and   2 1 2 1 2 ( ) = max (0.5 0.25 0.2); 2 4.6 5.8 .f x x x x x      Case  * x * y * ()hx 1 (1.0,0.0) (0.2724,1.2724) ( 3.2875, 0.4916) 3.2875 Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 34 2 (0.8,0.2) (0.2500,1.2500) ( 3.2750, 0.5500) 2.7300 3 (0.5,0.5) (0.3829,1.2731) ( 3.1639, 0.7021) 1.9330 4 (0.2,0.8) (0.8623,1.3826) ( 2.5304, 0.9768) 1.2875 5 (0.0,1.0) (1.3170,1.3659) ( 1.3365, 1.2000) 1.2000 6 ( 0.2,0.8) (1.3170,1.3659) ( 1.3365, 1.2000) 0.6927 7 (0.8, 0.2) (0.2724,1.2724) ( 3.2875, 0.4916) 2.5316 8 ( 0.5, 0.5) (1.3170,1.3659) ( 1.3365, 1.2000) 1.2683 Table 2: Computational results of Example 4 Step 1. Solving problem 1 ()P and 2 ()P , we obtain the ideal point = ( 3.2875, 1.2000) I y  . Step 2. Solving problem () I CP , we can find that it is not feasible. Then go to Step 3. Step 3. Solve two problems () S P and () E P to obtain the points = ( 1.3365, 1.2000) S y  and = ( 3.2875, 0.4916) E y  . Step 4. For each 2 12 = ( , )    R , solve problem 3 ()OP to find the optimal solution ** ( , )xy . Then * x is an optimal solution to problem 0 ()OP and * ()hx is the optimal value of 0 ()OP . The computational results are shown in Table 2. 4. Conclusion Problem 0 ()OP is a very hard and interesting task in multiple objective optimization and has some important applications in finance, economics, engineering, and other fields. In this paper, we propose simple convex programming procedures for solving problem 0 ()OP in two special cases: i) The ideal point I y belongs to the outcome set Y and the objective function ()hx of problem 0 ()OP is convex; ii) Problem ()CMP is a bicriteria convex programming problem and the objective function ()hx of the problem 0 ()OP has the form as (1) . These procedures require quite little computational effort in comparison to that required to solve the general problem 0 ()OP . Therefore, when solving problem 0 ()OP , they can be used as screening devices to detect and solve this two special cases. Acknowledgements. The authors wish to thank Prof. Tran Vu Thieu for his help. 5. References [1] An, L.T.H., Tao, P. D., Muu, L. D. (1996), “Numerical solution for optimization over the efficient set by d.c. optimization algorithm”, Oper. Res. Lett., 19, 117-128. Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 35 [2] Benson, H. P., Lee, D. (1996), “Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem”, J. Optim. Theory Appl., 88, 77-105. [3] Fulop, J., Muu, L.D. (2000), “Branch-and-bound variant of an outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem”, J. Optim. Theory Appl., 105, 37-54. [4] Horst, R., Thoai, N.V. (2007), “Yamamoto, Y., Zenke, D.: On optimization over the efficient set in linear multicriteria programming”, J. Optim. Theory Appl., 134, 433-443. [5] Kim, N.T.B., Muu, L.D. (2002), “On the projection of the efficient set and potential application”, Optim., 51, 401-421. [6] Kim, N.T.B., Thang, T.N. (2013) “Optimization over the Efficient Set of a Bicriteria Convex Programming Problem”, Pacific J. Optim., 9, 103-115. [7] Luc, D.T. (1989), Theory of Vector Optimization, Springer-Verlag, Berlin, Germany. [8] Luc L. T., Muu L. D. (1997), “Global optimization approach to optimization over the efficient set”, The Proceedings of the 8th French-German Conference on Optimiza- tion, Lecture Notes in Economics and Mathematical System, 452 , Springer-Verlag, Berlin, 213-221. [9] Philip, J. (1972), “Algorithms for the vector maximization problem”, Math. Program., 2, 207-229. [10] Phu, H. X. (2005), “On efficient sets in 2 R ”, Vietnam J. Math., 33, 463-468. [11] Schandle B., Klamroth K., Wiecek M. M. (2001), “Norm-Based Approximation in Bicriteria Programming”, Computational Optimization and Applications, 20, pp 23-42. [12] Thoai, N. V. (2010), “Reverse convex programming approach in the space of extreme criteria for optimization over efficient sets”, J. Optim. Theory and Appl., 147, 263- 277. [13] Tuy H. (1998), Convex Analysis and Global Optimization, Academic Kluwer Publisher. [14] Yamamoto, Y. (2002), “Optimization over the efficient set: overview”, J. Global Optim., 22, 285-317. [15] Yu, P. L. (1985), Multiple-Criteria Decision Making, Plenum Press, New York and London. . is generally neither convex set nor given explicitly as the form of a standard mathematical programming problems, even in the case of linear multiple objective programming problem when the component. programming problem in the two special cases. Preliminary computational experiments show that these procedures can work well. AMS Subject Classification: 2000 Mathematics Subject Classification khoa học 2014 – Phần I Trường Đại học Thăng Long 26 OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX MULTIPLE OBJECTIVE PROBLEM: TWO SPECIAL CASES 1 Assoc. Prof. Nguyen Thi Bach Kim 1 and