On Stability of Fuzzy Multi-Objective. Constrained Optimization Problem Using. a Trust-Region Algorithm

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1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 28, On Stability of Fuzzy Multi-Objective Constrained Optimization Problem Using a Trust-Region Algorithm Bothina El-Sobky Department of Mathematics, Faculty of Science Alexandria University, Alexandria, Egypt bothinaelsobky@yahoo.com Yusria Abo-Elnaga Department of Basic Science, Higher Technological Institute Tenth of Ramadan City, Egypt Abstract In this paper, a fuzzy multi-objective constrained optimization problem (FMCOP) is converted to a single-objective constrained optimization problem with equality and inequality constraints (SCOP) by using α level set of the fuzzy vector and weighting approach. A trust-region algorithm for solving problem (SCOP) is introduced to obtain α pareto optimal solutions of problem (FMCOP). An ε α stability set for problem (FMCOP) which represent a range of α pareto optimal solutions is obtained. A numerical example to clarify the proposed of the paper is introduced in the end. Keywords: Multi-objective, Fuzzy environment, Single-objective, Trustregion, Stability, Active set

2 1368 B. El-Sobky and Y. Abo-Elnaga 1 Introduction A fuzzy multi-objective optimization problem is minimize subject to : c(x) 0, f(x, p) =(f 1 (x, p 1 ),..., f m (x, p m )) T (1.1) where x Ω={x R n c(x) 0} and p =( p 1,..., p m ) T represented a vector of fuzzy parameters that can be characterized as a fuzzy number Sakawa [15]. The functions f i (x, p i ), i =1, 2,..., m, and c(x) R me are twice continuously differentiable functions. In this paper, we convert the above problem to a single-objective constrained optimization problem with equality and inequality constraints (SCOP) by using α level set of the fuzzy vector and weighting approach. To obtain an α pareto optimal solutions of problem (FMCOP), we solve the singleobjective constrained optimization problem by using a trust-region algorithm. A trust-region algorithms have proved to be robust techniques for solving unconstrained or constrained single-objective optimization problems, see([3], [5], [6], [7], [8], [9], [13], and [19]). In an earlier work, Orloviski[12] formulate multi-objective nonlinear programming problem with fuzzy parameters. Many authors such as [[10], [11], [14], [15], [16], [17], [18]] introduced the stability of fuzzy multi-objective optimization problem. But in many applications of fuzzy multi-objective optimization problem, the decision maker not only needs an α pareto optimal solution but also he prefers to get a set of α pareto optimal solutions which denoted by ε α stability set. Our proposed approach is allowing a decision maker to control the α pareto optimal set by choosing an appropriate ε value according to his needs. In this paper, the ε α stability set for problem (FMCOP) is obtained. The ε α stability set is the set of all α pareto optimal solutions of problem (FM- COP) that lie in the range of α pareto optimal solutions. In the next section, some basic fuzzy concepts and how the problem (FM- COP) is converted to problem (SCOP) are discussed. In section 3, the trustregion algorithm for solving the single-objective optimization problem is introduced to obtain α pareto optimal solutions of problem (FMCOP). In section 4, an ε α stability set is introduced.

3 Fuzzy multi-objective constrained optimization problem 1369 In section 5, a numerical example for the multi-objective optimization problem is introduced in the end to clarify the proposed of the paper. Finally Section 6, contains concluding remarks. The following notations are used throughout the rest of the paper. Subscripted functions are function values at particular points; for example, f k = f(x k ), g k = g(x k ), h k = h(x k ), f k = x f(x k ), g k = x g(x k ), h k = x h(x k ), l k = l(x k,λ k,ν k ), l k = x l(x k,λ k,ν k ), U k = U(x k ), and so on. However, the arguments of the functions are not abbreviated when emphasizing the dependence of the functions on their arguments. The i th component of a vector ν k is denoted by (ν k ) i. Finally, all norms used in this paper are l 2 norms. 2 Theoretical Fuzzy Foundations Fuzzy set theory has developed for solving problems in which descriptions of activities and observation are imprecise, vague and uncertain. The term fuzzy refers to the situation in which here are no well-defined boundaries of the set of activities or observations to which the descriptions apply. A fuzzy set is a class of objects with membership grades. A membership function, which assigns to each object a grade of membership, is associated with each fuzzy set. Usually the membership grades are in [0, 1]. When the grade of the membership for an object in a set is one, this object is absolutely in that set when the grade of membership function is zero, the object is absolutely not in that set. Borderline cases are assigned numbers between zero and one. A fuzzy number is defined differently by many authors such as [[14], [18]]. That most frequently used definition belongs to a trapezoidal fuzzy type as follows: Definition 2.1 (A fuzzy number) A fuzzy number is appropriate to recall that a real fuzzy number p is a continuous fuzzy subset from real line R whose membership function μ p (β) defined by: 1. μ β(β) :R [0, 1], is continuous function, 2. μ β(β) =0 β (,β 1 ], 3. μ β(β) is strictly increasing on [β 1,β 2 ],

4 1370 B. El-Sobky and Y. Abo-Elnaga 4. μ β(β) =1 β [β 2,β 3 ], 5. μ β(β) is strictly decreasing on [β 3,β 4 ], 6. μ β(β) =0 β [β 4, ]. For more details see [14]. Here, the vector of fuzzy parameters p involved in problem(1.1) is a vector of fuzzy numbers whose membership functions is μ p (p). Throughout this paper, a membership function in the following form will be elicited: 0 if <β β 1, β β 1 β 2 β 1 if β 1 β β 2, μ β(β) = 1 if β 2 β β 3, β 4 β β 3 β 4 if β 3 β β 4, (2.1) 0 if β 4 β<, Definition 2.2 (α level set): α level set of vector of fuzzy parameter p in problem(1.1) is defined as the ordinary set L α ( p) for which the degree of its membership function exceeds that level α [0, 1], where: L α ( p) ={p R μ p (p) α}. (2.2) For more details see [14]. For a certain degree of α [0, 1], estimated by the decision maker, the problem (1.1) can be written as the following α multi-objective constrained optimization problem minimize f(x, p) =(f 1 (x, p 1 ),..., f m (x, p m )), subject to : c(x) 0, p L α ( p), (2.3) where p =(p 1,..., p m ) T. It should be emphasized here in problem(2.3) that the vector of parameters p i, i =1,..., m is treated as a vectors of decision variables

5 Fuzzy multi-objective constrained optimization problem 1371 rather than the constraints. On the basis of the α level sets of the fuzzy numbers, the compact of α pareto optimal solution to the problem (2.3) is given by the following definition. Definition 2.3 (α pareto optimal solutions): A point (x,p ) where x Ω and p L α ( p) is said to be an α pareto optimal solution to the problem (2.3), if and only if there does not exists x Ω and p L α ( p), such that f(x, p) <f(x,p ), where the corresponding value of parameter p is called α optimal parameter of p. By using a weighting approach, we transform problem (2.3) to the following single-objective constrained optimization problem (SCOP): minimize f(x, p, w) = m i=1 w if i (x, p i ), subject to c j (x) 0, j=1,...,me p i1 p i p i2, i=1,...,m, w i 0, i=1,...,m, m i=1 w i =1, (2.4) where [p i1,p i2 ]=L α ( p). In the following section, we introduce the trust-region algorithm for solving (SCOP) problem (2.4) to obtain α pareto optimal solution (x,p ) of problem (FMCOP). In this algorithm, an active set strategy is used together with a reduced Hessian technique to convert the computation of the trial step to two easy trust-region subproblem similar to those for the unconstrained case. A dogleg method is used to compute a trial step in the following algorithm. 3 A Trust-region algorithm for solving (SCOP) problem In this section, we consider the solution of (SCOP) problem (2.4)which can be rewritten as follows minimize f( x) subject to h( x) = 0, (3.1) g( x) 0, where x =(x, p, w) T R n+2m, f( x) =f(x, p, w), h( x) = m i=1 w i 1, and g( x) = (c j (x),p i1 p i,p i p i2, w i ) T. The functions f( x) :R n+2m R,

6 1372 B. El-Sobky and Y. Abo-Elnaga h( x) : R n+2m R, and g( x) : R n+2m R 3m+me are twice continuously differentiable. The Lagrangian function associated with problem (3.1) is the function l( x, λ, ν) =f( x)+λ T h( x)+ν T g( x), (3.2) where λ Rand ν R 3m+me are the Lagrange multiplier vectors associated with equality and inequality constraint respectively. By using active set method, we transform the problem (3.1) to the following problem minimize f( x)+ν T g( x)+ ρ 2 U( x)g( x) 2 2, (3.3) subject to h( x) =0, where U( x) R 3m+me 3m+me is a 0-1 diagonal indicator matrix, whose diagonal entries are { 1 if g i ( x) 0, u i ( x) = (3.4) 0 if g i ( x) < 0. and ρ is a positive parameter. This matrix is similar to the one used by Dennis, El-Alem, and Williamson [1]. The Lagrangian function associated with Problem (3.3) is given by L( x, λ, ν; ρ) =l( x, λ, ν)+ ρ 2 U( x)g( x) 2 2, (3.5) and the augmented Lagrangian is the function Φ( x, λ, ν; ρ; r) =l( x, λ, ν)+ ρ 2 U( x)g( x) r h( x) 2 2, (3.6) where r>0 is a penalty parameter. In the following section, we introduce a trust-region algorithm outline for solving problem (3.1) 3.1 Algorithm Outline This section is devoted to presenting the detailed description of the trustregion algorithm for solving problem (3.1). A global convergence theory of this algorithm is proved in El-Sobky [4]. In this algorithm a reduced Hessian approach is used to compute a trial step s k. In this approach, the trial step s k is decomposed into two orthogonal components; the normal component s n k and the tangential component st k. The

7 Fuzzy multi-objective constrained optimization problem 1373 trial step s k has the form s k = s n k + Z k s t k, where Z k is a matrix whose columns form an orthonormal basis for the null space of h T k. We obtain the normal component s n k by solving the following trust-region subproblem 1 minimize 2 ht k sn + h k 2 (3.7) subject to s n ζδ k, for some ζ (0, 1), where δ k is the trust-region radius. Let the quadratic model of the Lagrangian function (3.5) be q k (s) =l k + lk T s st H k s + ρ k 2 U k(g k + gk T s) 2. (3.8) where l k is the Lagrangian function (3.2) and H k is the Hessian of the lagrangian function l k or approximation to it. Given the normal component s n k, we compute the tangential component s t k = Z k s t k by solving the following trust-region subproblem minimize [Z T k ( l k + H k s n k + ρ k g k U k g k )] T s t stt Z T k B kz k s t subject to Z k s t Δ k, (3.9) where Δ k = δk 2 sn k 2 and B k = H k + ρ k g k U k gk T. Once the trial step is computed, it needs to be tested to determine whether it will be accepted. To do that, a merit function is needed. We use the augmented Lagrangian (3.6) as a merit function. Once the trial step is computed, we test it to determine whether it is accepted. To test the step, estimates for the two Lagrange multipliers λ k+1 and ν k+1 are needed. We compare the actual reduction in the merit function in moving from ( x k,λ k,ν k )to( x k +s k,λ k+1,ν k+1 ) versus the predicted reduction. We define the actual reduction as Ared k =Φ( x k,λ k,ν k ; ρ k ; r k ) Φ( x k + s k,λ k+1,ν k+1 ; ρ k ; r k ). The predicted reduction in the merit function is defined to be Pred k = q k (0) q k (s k ) Δλ T k (h k + h T k s k) (3.10) Δνk T (g k + gk T s k )+r k [ h k 2 h k + h T k s k 2 ], where Δλ k = λ k+1 λ k and Δν k = ν k+1 ν k.

8 1374 B. El-Sobky and Y. Abo-Elnaga We define the tangential predicted decrease Tpred k to be the decrease at the k th iteration the quadratic model of the Lagrangian function (3.5) by the step s t k = Z k s t k. It is defined to be Tpred k = (Z T k ( l k + H k s n k ))T s t k 1 2 stt k ZT k H kz k s t k (3.11) + ρ k 2 [ U kg k 2 U k (g k + gk T Z k s t k) 2 ]. After computing a trial step and updating the Lagrange multipliers, the penalty parameter is updated to ensure that Pred k 0. To update r k,we use a scheme that has the flavor of the scheme proposed by El-Alem [2]. This scheme is described in Step 6 of algorithm (3.1) below. After that, the step is tested to know whether it is accepted. This is done by comparing Pred k against Ared k. Our way of evaluating the trial steps and updating the trust-region radius is presented in Step 7 of algorithm (3.1) below. After accepting the step, we update the parameter ρ k and the Hessian matrix H k. To update ρ k, we use a scheme suggested by Yuan [19]. In this scheme, another parameter σ k has to be updated with ρ k. This scheme is described in Step 8 of algorithm (3.1) below. Finally, the algorithm is terminated when either Z T k l k + g k U k g k + h k ɛ 1, or s k ɛ 2 for some ɛ 1 > 0 and ɛ 2 > 0. A formal description of our trust-region algorithm for solving (SCOP)problem is presented in the following algorithm. Algorithm 3.1 (A trust-region algorithm for solving (SCOP) problem ) Step 0. (Initialization) Given x 1 R n+2m. Compute U 1. Evaluate ν 1 and λ 1 (see Step 5 with k =0and λ 0 =(0, 0,..., 0) T ). Set ρ 1 =1, r 0 =1, σ 1 =1, and β 0 =0.1. Compute δ 1 Choose ɛ 1 = ɛ 2 =10 8, α 1 =0.05, α 2 =2, η 1 =10 4, and η 2 =0.5 such that 0 <α 1 < 1 <α 2, and 0 <η 1 <η 2 < 1. Set δ min =10 3 and δ max =10 5 δ 1 such that δ min δ 1 δ max. Set k =1.

9 Fuzzy multi-objective constrained optimization problem 1375 Step 1. (Test for convergence) If Zk T l k + g k U k g k + h k ɛ 1, then terminate the algorithm. Step 2. (Compute a trial step) If h k =0, then a) Set s n k =0. b) Compute the step s t k by solving problem (3.9)with sn k = 0. c) Set s k = Z k s t k. Else a) Compute s n k by solving problem (3.7). b) If Zk T ( l k+ρ k g k U k g k +B k s n k ) =0, then set st k =0. Else, compute s t k by solving problem (3.9), end if. c) Set s k = s n k + Z k s t k and x k+1 = x k + s k. End if Step 3. (Test for termination) If s k ɛ 2, then terminate the algorithm. Step 4. (Update the active set) Compute U k+1. Step 5. (Compute the Lagrange multipliers ν k+1 and λ k+1 ) a) Compute ν k+1 by solving minimize Z T k+1 ( f k+1 + g k+1 U k+1 ν) 2 subject to U k+1 ν 0 (3.12) b) If f k+1 + h k+1 λ k + g k+1 U k+1 ν k+1 ɛ 1, then set λ k+1 = λ k. Else, compute λ k+1 by solving minimize f k+1 + g k+1 ν k+1 + h k+1 λ 2. End if.

10 1376 B. El-Sobky and Y. Abo-Elnaga Step 6. (Update the penalty parameter r k ) a) Set r k = r k 1. b) If Pred k r k 2 [ h k 2 h k + h T k s k 2 ], then set r k = 2[q k(s k ) q k (0) + Δλ T k (h k + h T k s k)+δν T k (g k + g T k s k)] h k 2 h k + h T k s k 2 +β 0, End if Step 7. (Test the step and update the trust-region radius) If Ared k Pred k <η 1 Reduce the trust-region radius by setting δ k = α 1 s k and go to step 2. Else if η 1 Ared k Pred k <η 2, then Accept the step: x k+1 = x k + s k. Set the trust-region radius: δ k+1 = max(δ k,δ min ). Else, accept the step: x k+1 = x k + s k. Set the trust-region radius: δ k+1 = min{δ max, max{δ min,α 2 δ k }}. End if. Step 8. (Update the parameters ρ k and σ k ) a) Set ρ k+1 = ρ k and σ k+1 = σ k. b) If 1Tpred 2 k Δνk T (g k+ gk T s) σ k g k U k g k min{ g k U k g k, Δ k }, then set ρ k+1 =2ρ k and σ k+1 = 1σ 2 k. Step 9. Set k = k +1and go to Step 1. Notice that, we use the above algorithm for solving (SCOP) problem to obtain α pareto optimal solutions (x,p ) of (FMCOP) problem, the weighting vector w, and the Lagrange multiplier vectors λ and ν.

11 Fuzzy multi-objective constrained optimization problem An ε α stability set of (FMCOP) problem This section deals with the stability set for the α pareto optimal solutions of (FMCOP) problem, so we start with the definition of solvability set of problem (2.4). Definition 4.1 (Solvability set): A solvability set is the set of α pareto optimal solutions of (FMCOP) problem. Now, assume that (x,p )isanα pareto optimal solutions of (FMCOP) problem and p is the α level optimal parameter of problem (2.4), then the stability set of the first kind of (FMCOP) problem corresponding to (x,p ) is given by the following definition. Definition 4.2 (Stability set) A stability set of the first kind of (FMCOP) problem corresponding to (x,p ) is defined by the set of all {[p i1,p i2 ] R 2m such that (x,p ) is an α pareto optimal solution of (FMCOP) problem. Now, our aim is to get another α pareto optimal solutions (ˆx, ˆp ) such that the norm of the difference between them and (x,p ) is limited by a small attribute ε. So we set the following definition Definition 4.3 (ε α stability set): Let (x,p ) be α pareto optimal solutions of (FMCOP) problem, then ε α stability set of (FMCOP) problem which we denoted it by G εα is defined as follows : G εα = {(ˆx, ˆp ) R n+m (ˆx, ˆp ) (x,p ) ε.} To obtain G εα set, we added the ε α stability condition (ˆx, ˆp) (x,p ) ε, to the constraints of problem (2.4). Then, we have the following singleobjective constrained optimization problem: minimize subject to f(ˆx, ˆp, ŵ) = m i=1 ŵif i (ˆx, ˆp i ), m i=1 ŵi 1=0, c j (ˆx) 0, j=1,...,me, p i1 ˆp i p i2 i=1,...,m, (ˆx, ˆp) (x,p ) ε, ŵ i 0, i=1,...,m. (4.1) We can obtain some α pareto optimal solutions (ˆx, ˆp ) of (FMCOP) problem which lie in the range of α pareto optimal solutions (x,p ) by solving problem (4.1) by using the trust-region algorithm (3.1)above. Then we obtain G εα set.

12 1378 B. El-Sobky and Y. Abo-Elnaga 5 Test Problem In this section, we introduce a fuzzy multi-objective constrained optimization test problem. Our programs are written in MATLAB and run under MATLAB Version 7.0 with machine epsilon The fuzzy multi-objective constrained optimization test problem is minimize (x 1 + p 1,x 2 + p 2 ), subject to (x 1 3) 2 +(x 2 2) 2 4, 2x 1 + x 2 0. (5.1) With the membership functions (2.1) of the fuzzy numbers p 1 and p 2 for i =1, 2 are p 1 =(1, 2, 4, 5) and p 2 =(3, 5, 9, 10). At α =0.36, we get, 1.36 p , and 3.72 p Then, α multi-objective constrained optimization problem is minimize (x 1 + p 1,x 2 + p 2 ), subject to : (x 1 3) 2 +(x 2 2) 2 4, 2x 1 + x 2 0, 1.36 p , 3.72 p (5.2) By using the weighting approach, the α multi-objective constrained optimization problem (5.2) converted to the following single objective constrained optimization problem minimize w 1 (x 1 + p 1 )+w 2 (x 2 + p 2 ), subject to : w 1 + w 2 =1, (x 1 3) 2 +(x 2 2) 2 4, 2x 1 + x 2 0, 1.36 p , 3.72 p , w 1 0, w 2 0, (5.3) To solve the above problem, we use the trust-region algorithm (3.1). Then α pareto optimal solutions of problem(5.1) is (x 1,x 2,p 1,p 2 )=(2.44, 0.08, 1.36, 3.72) T, and the values of wights is w =( , ) T. To obtain ε α stability set at α =0.36 and ε =0.1, we added the ε α stability condition (ˆx, ˆp) (x,p ) ε, to the constraints of problem (5.3) and solve

13 Fuzzy multi-objective constrained optimization problem 1379 this problem again by using the trust-region algorithm (3.1). This problem has the form minimize ŵ 1 (ˆx 1 +ˆp 1 )+ŵ 2 (ˆx 2 +ˆp 2 ), subject to ŵ 1 +ŵ 2 =1, (ˆx 1 3) 2 +(ˆx 2 2) 2 4, 2ˆx 1 +ˆx 2 0, 1.36 ˆp , 3.72 ˆp , (ˆx 1 2) 2 +(ˆx ) 2 +(ˆp ) 2 +(ˆp ) 2 0.1, ŵ 1 0, ŵ 2 0. (5.4) Using the trust-region algorithm (3.1) to solve the above problem, we have the following ε α stability set of (FMCOP) problem which is obviously in the following table. ˆx 1 ˆx 2 ˆp 1 ˆp 2 ŵ 1 ŵ Table 1: ε α stability set of(fmcop) problem. From the previous results we conclude that p , p So the ε α stability set at α =0.36 and ε =0.1 is as follows G εα = {(x, p) p , p , x , x }.

14 1380 B. El-Sobky and Y. Abo-Elnaga 6 Concluding Remarks A trust-region algorithm for solving a single objective constrained optimization problem is used to get the stability set of the third kind for fuzzy multiobjective constrained optimization problem, which represents the set of all fuzzy parameter for which a set of α pareto optimal solutions for one fuzzy parameters has been introduced. The following are the significant contributions of this paper The stability set is determined for α pareto optimal solutions not for an α pareto optimal point The stability set is determined numerically by using trust-region algorithm. This approach seems to be an interactive approach where the decision maker specifies the epsilon value his needs. Allowing the decision maker to control the resolutions of the pareto set by choosing the epsilon value according his needs. References [1] J.Dennis, M.El-Alem, and K.Williamson, A trust-region approach to nonlinear systems of equalities and inequalities, SIAM J Optimization, 9(1999), [2] M. El-Alem, A global convergence theory for a class of trust-region algorithms for constrained optimization, PhD thesis, Department of Mathematical Sciences, Rice University, Houston, Texas, [3] M. El-Alem, M.Abdel-Aziz, and A.El-Bakry, A projected Hessian gaussnewton algorithm for solving systems of nonlinear equations and inequalities, International J of Mathematics and Mathematica Science, [4] B. El-Sobky, A robust trust-region algorithm for general nonlinear constrained optimization problems, PhD thesis, Department of Mathematics, Alexandria University, Alexandria, Egypt, 1998.

15 Fuzzy multi-objective constrained optimization problem 1381 [5] B. El-Sobky, A global convergence theory for an active trust region algorithm for solving the general nonlinear programming problem, Applied Mathematics and Computation Archive,144(1)(2003), [6] B.El-Sobky, A global convergence theory for trust-region algorithm for general nonlinear programming problem, Publicities in fourth Saudi science conference Contribution of Science Faculties in the development process of Kingdom of Saudi Arabia sponsored by Taibah University, Al-Madinah Al-Munawarah, Kingdom of Saudi Arabia, March, 21st -24th, [7] B.El-Sobky and Y.Abo-Elnaga, An active-set trust-region algorithm for solving constrained multi-objective optimization problem, Applied mathematics science, 6(33)(2012), [8] J. Moré, Recent developments in algorithms and software for trust-region methods, In A. Bachem, M. Grotschel, B. Korte, editors, Mathematical Programming, , Springer-Verlag, New York, [9] E. Omojokun, Trust-region strategies for optimization with nonlinear equality and inequality constraints, PhD thesis, Department of Computer Science, University of Colorado, Boulder, Colorado, [10] M.Osman and El-Banna, Stability of multi-objective non-linear programming problems with fuzzy parameters, Mathematics and Computers in Simulation, 35(1993), [11] M.Osman, M. abo-sinna, and Y. Abo-Elnaga, On recent approaches in solving multi-objective programming problems, PhD thesis, Department of basic science, Faculty of engineering, Ain Shames university, Cairo, Egypt, [12] S. Orlovski, Multi-objective programming problems with fuzzy parameters, International institute for applied systems analysis, Austria, [13] M. Powell, Convergence properties of a class of minimization algorithms, In O. L.Mangasarian, R. R. Meyer, and S. M. Robinson, editors, Nonlinear Programming 2, 1:27, Academic Press, New York, 1975.

16 1382 B. El-Sobky and Y. Abo-Elnaga [14] M. Sakawa, An interactive fuzzy satisfying method for multi-objective non-linear programming problems with fuzzy parameters, Electron. Commun. Japan, [15] M. Sakawa, An interactive fuzzy decision-making in multi-objective linear programming problem and its application, I.E.C.E. Japan, (1982), [16] M. Sakawa, H. Yano, and J. Takahashi, Pareto optimality for multiobjective linear fractional programming problems with fuzzy parameters, Information Sciences, 63(1992), [17] H. Tanaka and K. Asai, Formulation of linear programming problem by fuzzy function, Systems and Control 6, (1981). [18] H. Tanaka, H. Ichihashi and K. Asai, Formulation of fuzzy linear programming problem by fuzzy objective function, J. Oper. Res. Soc. Japan 2, (1984). [19] Y. Yuan, On the convergence of a new trust region algorithm, Numer. Math. 70(1995), Received: December, 2011

A PROJECTED HESSIAN GAUSS-NEWTON ALGORITHM FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES

IJMMS 25:6 2001) 397 409 PII. S0161171201002290 http://ijmms.hindawi.com Hindawi Publishing Corp. A PROJECTED HESSIAN GAUSS-NEWTON ALGORITHM FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES