Preprint Stephan Dempe, Alina Ruziyeva The Karush-Kuhn-Tucker optimality conditions in fuzzy optimization ISSN

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1 Fakultät für Mathematik und Informatik Preprint Stephan Dempe, Alina Ruziyeva The Karush-Kuhn-Tucker optimality conditions in fuzzy optimization ISSN

2 Stephan Dempe, Alina Ruziyeva The Karush-Kuhn-Tucker optimality conditions in fuzzy optimization TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Institut für Numerische Mathematik und Optimierung Akademiestr. 6 (Mittelbau) FREIBERG

3 ISSN Herausgeber: Herstellung: Dekan der Fakultät für Mathematik und Informatik Medienzentrum der TU Bergakademie Freiberg

4 THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN FUZZY OPTIMIZATION S. DEMPE AND A. RUZIYEVA Abstract. In the present paper recent concepts of the solution of the fuzzy optimization problem based on the level-cut, the interval arithmetic and the Pareto set notions were generalized. Using basic methods of convex multiobjective optimization necessary and sufficient conditions were derived. 1. Introduction In many situations optimization problems with unknown or only approximately known data need to be solved. In such situations stochastic, parametric or fuzzy optimization approaches can be used. A parametric approach takes too much time to calculate a solution (and only one!) and it demands special conditions (e.g. functions should be twice differentiable). A stochastic approach demands normal distributions or for sufficiently appropriate calculations the distribution functions should be known exactly. Then it seems reasonable to use the fuzzy set theory because it is very sensible to use fuzzy functions and continuous fuzzy numbers when an ambiguity appears. In the present paper nonlinear fuzzy optimization problem is investigated where the objective function has fuzzy values and the constraint function is a crisp one, i.e.: (1) f(x) min g(x) 0 Here g = (g 1,..., g k ) : R n R k is a crisp function and f maps R n to the space of fuzzy numbers. This problem has many applications, e.g. the fuzzy multicommodity flow problem (compute optimal flows in a traffic network with fuzzy costs for passing streets, see for instance [6]) or problems of optimal planning [10]. The investigated problem can easily be generalized to more general problems where the constraint functions are fuzzy functions and / or the relations are fuzzy. Key words and phrases. fuzzy optimization; multicriteria optimization; Pareto optimal solution; necessary and sufficient conditions; Karush-Kuhn-Tucker optimality conditions. 1

5 2 S. DEMPE AND A. RUZIYEVA (2) In the linear case of (1) obtained the following problem c x min Ax = b x 0. The solution algorithm and the problem itself is well-described e.g. in [4, 5, 12, 17]. For nonlinear fuzzy optimization problems the interested reader is referred to [11, 14, 15, 16]. In [15] the authors gave sufficient optimality conditions for an optimal solution of the fuzzy optimization problem (1) under convexity assumptions. The same conditions presented in [14] for a more general fuzzy problem which also has fuzzy functions as constraints. In the paper [16] for sufficient optimality conditions of the problem (1) the authors used integrals in the Karush- Kuhn-Tucker conditions. This means they use a certain average value of the α-level sets of the membership function of the fuzzy objective function. In distinction, in the papers [15, 16] only one α-cut was used. Right- and left-hand side functions were used in the papers [14, 15, 16] to describe the α-cuts of the fuzzy objective function which then appear in the Karush-Kuhn-Tucker conditions. Also the notion of the comparable fuzzy functions was applied in the paper [11]. This enables to derive a condition which is valid for all α-cuts at the same time and to use only one of the mentioned right- and left-hand side functions. An early approach for solving fuzzy optimization problems is due to Bellman and Zadeh [2]. In this paper also the α-cuts were used to describe the membership function of the objective function and it was assumed that its leftand right-hand sides values are given by functions f L (x, α), f R (x, α) for α [0, 1]. Then using a suitable ordering of the intervals f(x)[α] := [f L (x, α), f R (x, α)] for fixed α a task of the fuzzy function minimization over a feasible set is transformed into a bicriterial optimization problem. If more than one α is used at the same time as e.g. in the paper [12] this would lead to a multiobjective optimization problem. But a generalization to this case is straightforward. Let determine an optimal solution of the fuzzy optimization problem through the Pareto-optimal solution of the bicriterial optimization problem. This will be described in Section 3. Then using some convexity assumptions was obtained that an optimal solution of the fuzzy optimization problem is also an optimal solution of some nonlinear optimization problem obtained via scalarization of the objective functions of the bicriterial optimization problem. A necessary optimality condition for this is given by using the Karush- Kuhn-Tucker conditions. This is explained in Section 4. It turns out

6 THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN FUZZY OPTIMIZATION3 this condition to be much similar to the sufficient optimality condition used in the papers [14, 15, 16]. The distinction is the use of weighting coefficients in the objective. Sufficient optimality conditions which endow of the originality investigated in Section 5. The paper is concluded with a short summary and a numerical example. 2. Basic notions Let consider the fuzzy function f(x) and take its α-cut which is assumed to be a closed and bounded interval. This interval can be denoted by f(x)[α] = [f L (x, α), f R (x, α)], where f L (x, α) = min f(x)[α] and f R (x, α) = max f(x)[α] pro tanto. Thus f(x)[α] is fully described using the two functions f L (x, α) and f R (x, α), which are called as the left- and right-hand side functions for the certain level set α of the fuzzy function f(x). Following [11] et al. it was assumed that f L (x, α) is a bounded increasing and f R (x, α) is a bounded decreasing functions of α. Moreover it is obvious that f L (x, α) f R (x, α) for all α [0, 1]. The fuzzy function f(x) is called convex if for all α the functions x f L (x)[α] and x f R (x)[α] are convex (see for instance, [15]). Also continuity and differentiability of the fuzzy function f(x) can be defined through continuity and differentiability of the left- and righthand side functions for the fixed aspiration level α (see [15]). Let f(x) be a fuzzy function on R and assume that partial derivatives of f L (x, α) and f R (x, α) with respect to x R for fixed α [0, 1] exist and are respectively denoted by f L (x, α) and f R (x, α). Then (f L ( x, α), f R ( x, α)) defines a vector of derivatives in x for the fixed α. Please note that it is not assumed that f L ( x, α) f R ( x, α). Similarly, for the fuzzy function f( ) mapping R n to the space of fuzzy numbers, defined its gradient through the gradients of the lefthand and right-hand functions on the certain α-cut. Let f(x) be a fuzzy function on R n (x R n ) and let us assume that all partial derivatives of the functions f L (x, α) and f R (x, α) in x for this α exist. Then the gradient of f(x)[α] in x is the matrix of the pairs of the gradients ( f L ( x, α), f R ( x, α)), i.e. f L ( x,α) x 1 f L ( x,α) x n, f R( x,α) x 1..., f R( x,α) x n

7 4 S. DEMPE AND A. RUZIYEVA 3. Fuzzy optimization problem In this section the fuzzy optimization problem (1) is consider. This problem replaced with the minimization of the α-cut on the feasible set as it is usually done (see e.g. [5, 12, 18]). In the present paper for simplicity only one fixed α (0, 1) was used. So an interval optimization problem obtained: (3) [f L (x, α), f R (x, α)] min g(x) 0 To find an optimal solution of the problem (3) it is necessary to compare intervals in the objective for different values of x. For the comparison of two different intervals [a, b] and [c, d] in R the following definition [5] was taken: [a, b] [c, d] if a c and b d (with at least one strong inequality). Applying this notion to problem (3) it is easy to see that f(x 1 )[α] f(x 2 )[α] iff (4) [f L (x 1, α), f R (x 1, α)] [f L (x 2, α), f R (x 2, α)] Using this ordering of intervals the task of finding an optimal solution of the interval optimization problem (3) reduces to search a solution of the following two-objective optimization problem with the fixed α-cut: (5) f L (x, α) min f R (x, α) min g(x) 0 Due to [5] this bi-objective optimization problem is equivalent to the interval optimization problem (3) with the defined order relation. Solutions of the problem (5) are defined by means of the Pareto optimality concept [7]: The feasible point x X := {x : g(x) 0} is called Pareto optimal for (5) if there does not exist another feasible point ˇx X with [f L ( x, α), f R ( x, α)] [f L (ˇx, α), f R (ˇx, α)]. Now it is possible to define the notion of an optimal solution of the fuzzy optimization problem (1). A point x X is called an optimal solution of the problem (1) provided that it is a Pareto optimal solution of the problem (5). Note that in general using this approach an optimal solution of the fuzzy optimization problem turns out non-unique since the Pareto optimal solutions of the problem (5) form a certain set in R n. This is related to the idea of Chanas and Kuchta [5]. To compute all optimal solutions of the fuzzy optimization problem it is sufficient to compute all optimal points of the scalarized problem (6) that is defined below. Those optimal points form the sets of optimal points Ψ(λ) where λ is a coefficient of scalarization (0 λ 1). Observe that a point x Ψ(0)

8 THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN FUZZY OPTIMIZATION5 (or x Ψ(1)) is a Pareto optimal solution provided that this set reduces to a singleton [7]. In general it can only be shown that this set contains at least one Pareto optimal solution if it is bounded. Ideas to compute the set of Pareto optimal solutions can also be found e.g. in [1, 8, 9]. 4. Necessary optimality conditions Remember that a function h : R n R is called convex on R n if for all x, y R n and all γ [0, 1] we have h(γx + (1 γ)y) γh(x) + (1 γ)h(y). Now under assumption that all the functions x f L (x, α), x f R (x, α) and g i (x), i = 1,..., k are convex, the problem (5) stated as a convex bicriterial optimization problem for which the following two results are well-known. Let Ψ eff denote the set of Pareto optimal solutions of the problem (5) and Ψ(λ) for fixed 0 λ 1 denote the set of optimal solutions of the problem (6) f(x, λ)[α] := λf L (x, α) + (1 λ)f R (x, α) min g(x) 0 Theorem 4.1 ([7]). Consider the problem (5) and assume that all the functions x f L (x, α), x f R (x, α) and g i (x), i = 1,..., k are convex. Then for each x Ψ eff there exists 0 λ 1 such that x Ψ(λ). Theorem 4.2 ([7]). If x Ψ(λ) with 0 < λ < 1, then x Ψ eff. Note that the problem (6) was used by several authors to compute solutions of the fuzzy optimization problem, see e.g. [5] for linear fuzzy optimization problems. Other authors as e.g. [12] solve the problem max{f L (x, α), f R (x, α)} min s.t. x X. Using this approach just a weak Pareto optimal solution of the problem (5) could be computed. Theorem 4.3. Let x be an optimal solution of the fuzzy optimization problem (1) and assume that all the functions x f L (x, α), x f R (x, α) and g i (x), i = 1,..., k are convex and differentiable. Suppose also that Slater s constraint qualification is satisfied: x X : g i ( x) < 0 i = 1,..., k. Then there exist 0 λ 1 and µ R n, µ 0 such that (7) λ f L ( x, α) + (1 λ) f R ( x, α) + µ g( x) = 0 µ g( x) = 0 g( x) 0.

9 6 S. DEMPE AND A. RUZIYEVA Proof. If x is an optimal solution of the problem (1) then, by definition, it is an Pareto optimal solution of the problem (5) for some α [0, 1]. Then, using Theorem 4.1, it is possible to find 0 λ 1 such that x is an optimal solution of the problem (6). Now the result follows from the theory of necessary optimality conditions for differentiable convex optimization problems [13]. 5. Sufficient optimality conditions Theorem 5.1. Consider the fuzzy optimization problem (1) and assume that the functions x f L (x, α), x f R (x, α) and g i (x), i = 1,..., k are convex and differentiable. Let x be feasible and assume that there exist 0 < λ < 1 and µ 0 such that the conditions (7) are satisfied. Then x is an optimal solution of the problem (1). Since the sum of two positive numbers is positive, this result is equivalent to the sufficient optimality condition given in [15]. Let add the proof for completeness. Proof. If the assumptions of the theorem are satisfied, the point x is an optimal solution of the problem (6) [13]. Then Theorem 4.2 implies that the point x is Pareto optimal for the problem (5) and, hence by the definition, is optimal for the problem (1). 6. Example To accomplish the discussion it is interesting to explain results by giving a special example - the traffic problem - with all calculations. Consider a road system as a traffic network G = (V, E) consisting of a node set V for the junctions and an edge set E (that we would define through vertices) containing all streets connecting the junctions. The streets may have different capacities. Assume that v units of a certain good should be transported with minimal overall costs from the origin s V to the destination d V through the traffic network G. The problem is to compute optimal amounts of transported goods on the streets of the network. The travel costs for traversing a street usually are not known exactly, that motivates to assume that all travel costs have fuzzy values. Let x kl denote the amount of transported units over the edge (k, l) E, that connects two vertices k and l. Let O k (I k ) denote the set of all edges leaving (entering) the node k, i.e. such designation connected to the words in and out. Assume that the flow x kl on the edge (k, l) is bounded by the capacity u kl. This is expressed in the inequation (9) given below. The constraint

10 THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN FUZZY OPTIMIZATION7 (10) is used to guarantee that the total incoming flow is equal to the total outgoing flow. Moreover, the outgoing flow in the origin equals to v (see (11)). (8) (9) (10) (11) (12) f(x) = c kl x kl min k,l V x kl u kl k, l V x kl x li = 0, l V \ {s, d} k I l i O l x ks x si = v, k I s i O s x kl 0 To simplify this problem represented by the (2), where (8) reflects the total fuzzy flow and Ax = b is an abbreviation of the constraints (9),(10) and (11). Description of a numerical example is following. Let f(x) be the total fuzzy flow that we have to minimize: f(x) = 3x x x x x x x 45 min with demands x s1 + x s2 = 90 x 4d + x 5d = 90 x s1 = x 12 + x 13 + x 14 and capacities x s2 + x 12 = x 23 + x 25 x 13 + x 23 = x 35 x 14 = x 45 + x 4d x 25 + x 35 + x 45 = x 5d 0 x s x s x x x x x x x x 4d 35 0 x 5d 90 The numerical example of the fuzzy optimization problem is illustrated in Fig. 1. Suppose that fuzzy numbers (f) are defined as continuous triangular fuzzy numbers (n 1, n 2, n 3 ) [3]. Let take an α-cut for α = 0, 5 and in the third column write the left- and right-side bounds of the fuzzy numbers as the intervals. f = (n 1, n 2, n 3 ) [ c L, c R ] 3 = (1, 3, 5) [2, 4] 4 = (2, 4, 6) [3, 5]

11 8 S. DEMPE AND A. RUZIYEVA (35) (55) ( ) ( ) [90] s 3 ( ) 3 4 ( ) d [-90] ( ) ( ) (60) 7 ( ) 2 5 (30) Figure 1. The example of the traffic network 6 = (0, 6, 22) [3, 14] 7 = (5, 7, 9) [6, 8] 8 = (0, 8, 16) [4, 12] Now the interval optimization problem (3) could be written as f(x)[0, 5] = [2, 4]x 12 + [4, 10]x 13 + [2, 15]x 14 + [6, 8]x [6, 8]x 25 + [2, 4]x 35 + [3, 5]x 45 min with the same capacities and demands. Then it is possible to reformulate this problem to the bi-objective optimization problem (5): f L (x) = 2x x x x x x x 45 min f R (x) = 4x x x x x x x 45 min with above defined capacities and demands. Now it is possible to use some of mathematical programming software tools to approximate the Pareto set of optimal solutions of the problem (6). Our choice was MATLAB, by means of that a plot of the Pareto front - the so-called image of the set of all Pareto-optimal solutions in the objective space - was also built. For the computation the step size equal to 0.01 for steps with respect to λ was chosen. For λ = 1, when the estimation are relatively optimistic that all costs are minimal, the optimal solution is x L = (0, , 35, 0, , , 0) and total function value f L = 365.

12 THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN FUZZY OPTIMIZATION f R (x) f L (x) Figure 2. The Pareto front On the contrary, the pessimistic estimation, which is represented by λ = 0, has the optimal solution is x R = (0, 0, 0, 60, 30, 60, 0) and f R = 960. For the illustrative point it is important to focus on the Fig. 2. It is obvious that any attempt for decreasing one value of the functions f L (x) and f R (x) has as consequence increasing the value the other. As a final decision the decision-maker has to select one of the Pareto optimal solutions (i.e. one of the optimal solutions of the problem (1)) according to an additional rule. This rule can be based e.g. on the graphic. 7. Conclusions In the present work the fuzzy optimization problem was solved by its reformulation into the interval optimization problem for some fixed level-cut. By the means of a certain interval ordering the bi-objective

13 10 S. DEMPE AND A. RUZIYEVA optimization problem was developed. This problem, in turn, was solved by methods of the multiobjective optimization problem s scalarization technique due to [7]. The Pareto set of the multiobjective optimization problem was interpreted as the optimal solutions of the initial fuzzy problem. It was also discussed, that the set of the optimal solutions depends on linearization parameters as soon as each different single objective optimization problem can determine a different solution set. Certainly, the approximation of the set of Pareto-optimal solutions was used in the numerical example. As it was shown above, necessary and sufficient conditions for the optimal solution of the the fuzzy nonlinear optimization problem under certain convexity and differentiability assumptions were derived and explained through the necessary and sufficient conditions for the Pareto optimal solution of the bi-objective optimization problem. More complicated formulation of the fuzzy nonlinear optimization problem will be investigated in future. References 1. C. Audet, G. Savard, and W. Zghal, Multiobjective optimization though a series of single-objective formulations, SIAM J. Optim. 19 (2008), R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Science (1970), J.J. Buckley, Joint solution to fuzzy programming problems, Fuzzy Sets and Systems 72 (1995), S. Chanas, The use of parametric programming in fuzzy linear programming, Fuzzy Sets and Systems 11 (1983), S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions - a generalized approach, European Journal of Operational Research 94 (1996), S. Dempe, D. Fanghänel, and T. Starostina, Optimal toll charges: fuzzy optimization approach, Methods of Multicriteria Decision Theory and Applications (F. Heyde and C. Tammer A. Lhne, eds.), Shaker Verlag, Aachen, 2009, pp M. Ehrgott, Multicriteria optimization, Springer Verlag, Berlin, J. Fliege, Gap-free computation of Pareto-points by quadratic scalarizations, Mathematical Methods of Operations Research 59 (2004), Y. Li, G. M. Fadel, M. Wiecek, and V. Y. Blouin, Minimum effort approximation of the Pareto space of convex bi-criteria problems, Optimization and Engineering 4 (2003), S. A. Orlovski, Mathematical programming problems with fuzzy parameters, Management Decision Support Systems using Fuzzy Sets and Possibility Theory (J. Kacprzyk and R. R. Yager, eds.), Verlag TV, Kln, 1985, pp

14 THE KARUSH-KUHN-TUCKER OPTIMALITY CONDITIONS IN FUZZY OPTIMIZATION M. Panigrahi, G. Panda, and S. Nanda, Convex fuzzy mapping with differentiability and its application to fuzzy optimization, European Journal of Operational Research (2006). 12. H. Rommelfanger, R. Hanuscheck, and J. Wolf, Linear programming with fuzzy objectives, Fuzzy Sets and Systems 29 (1989), A. Ruszczyński, Nonlinear optimization, Princeton University Press, Princeton, H.-C. Wu, An (α, β)-optimal solution concept in fuzzy optimization problems, Optimization 53 (2004), , The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, European Journal of Operational Research 176 (2007), , The optimality conditions for optimization problems with fuzzy-valued objective functions, Optimization 57 (2008), H.-J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978), , Fuzzy set theory and its applications, Kluwer Academic Publishers, Dordrecht, TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Freiberg, Germany address: alina.ruziyeva@student.tu-freiberg.de