comparing two versions of nimbus optimization system

University of Jyvaskyla Department of Mathematics Laboratory of Scienti c Computing Report 23/1996

COMPARING TWO VERSIONS OF NIMBUS OPTIMIZATION SYSTEM Kaisa Miettinen

Marko M. M akel a

UNIVERSITY OF JYVA SKYLA JYVA SKYLA 1996

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COMPARING TWO VERSIONS OF NIMBUS OPTIMIZATION SYSTEM Kaisa Miettinen and Marko M. Makela

University of Jyvaskyla Department of Mathematics Laboratory of Scienti c Computing P.O. Box 35, FIN-40351 Jyvaskyla FINLAND Abstract.

The interactive NIMBUS method has been developed, especially, for nondierentiable multiobjective optimization. NIMBUS is needed while many real-world optimization applications include several con icting objectives of possible nonconvex and nondierentiable character but appropriate methods are missing. Two versions of the basic method have been developed so far. They are introduced and compared with respect to both theoretical and computational aspects. Theoretically, the versions dier in handling the information requested from the user. Numerical experiments indicate dierences in computational eciency and controllability of the solution processes. One problem is here solved in more detail with both the versions and a summary of several other tests is given.

1. Introduction

Discontinuity and nonsmoothness are characteristics of many real-life problems in economics and engineering. Examples of these cases are piecewise linear tax systems and stresses in structures. It is also typical that practical models include several, con icting goals. For example, in structural engineering, the desired construction should be durable and cheap at the same time. When modelling such applications mathematically, we are faced with nondierentiable multiobjective optimization problems with a possible large-scale nature. In order to be able to work in a reliable and satisfactory way, the algorithms used should contain interactivity. Typically, nondierentiabilities have been handled by regularization techniques (see [Haslinger, Neittaanmaki, 1988]). However, depending on smoothing parameters the regularized problem is numerically either unreliable or unstable. Multiple criteria can be transformed into one criterion. The scalarization by summing up the functions or by some other simple methods is yet often arti cial and rough. 1991 Mathematics Subject Classi cation. 90C29, 65K05. Key words and phrases. Multiobjective Optimization, Nonsmooth Optimization.

K. MIETTINEN AND M.M. MA KELA

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Further, the special nature of multiobjective optimization problems where optimality is not an explicit concept is not taken into account. In other words, both regularization and scalarization weaken the accuracy of the model. Nondierentiable optimization problems can be solved by ecient bundle-type methods (see [Makela, Neittaanmaki, 1992]). On the other hand, there exists a wide range of interactive methods for multiobjective optimization (see [Hwang, Masud, 1979]), but the combinations of these two are scarce. For this reason, a method called NIMBUS (Nondierentiable Interactive Multiobjective BUndlebased optimization System) was developed. NIMBUS is one of the few interactive methods capable of solving nondierentiable multiobjective optimization problems. As is the case in multiobjective optimization problems in general, the concept of optimality is not trivial and we need a human to select the nal solution from among a set of candidates that are optimal in some sense. In this set, improvement according to some criteria can be attained only by allowing impairment in some other criteria. NIMBUS is based on the classi cation of the objective functions into up to ve dierent classes. We move around the set and the user speci es which of the criterion values should be decreased and which could increase. NIMBUS has been designed to be easy to use and, unlike most interactive methods, it does not need any dicult or arti cial information from the user. The rst version of NIMBUS was developed in [Miettinen, 1994] and [Miettinen, Makela, 1995] and applied to structural design problems in [Miettinen, Makela, Makinen, 1996]. In this version, we formulate a new multiobjective optimization problem (vector subproblem) according to the classi cation information. For solving the subproblem, we need a special noninteractive black-box routine for nondierentiable multiobjective optimization (described in [Miettinen, Makela, 1995]). A new NIMBUS version was derived and applied to an optimal control problem of continuous casting in [Miettinen, Makela, Mannikko, 1996]. The advantage of this version is that the subproblem after the classi cation has only one objective function (scalar subproblem) and thus can be solved by any nondierentiable optimization routine. This gives more generality and applicability to NIMBUS. In addition, this version has also been designed so that the user can better control the progress of the iterative solution process. In the following, we present and compare these two versions of the NIMBUS method. Several numerical tests are reported and dierent features of the methods, for example, computational eciency and usage controllability are observed. The paper is organized as follows. In Section 2, we present the NIMBUS method, its classi cation phase, dierent subproblems, the detailed algorithm and a theoretical comparison of the two versions. Section 3 is devoted to versatile numerical experiments with a set of test problems. The paper is concluded in Section 4.

2.1. Concepts

2. NIMBUS Method

We study a multiobjective optimization problem of the form

(1)

minimize ff1(x); f2 (x); : : : ; fk (x)g subject to x 2 S;

COMPARING TWO VERSIONS OF NIMBUS OPTIMIZATION SYSTEM

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where we have k objective functions fi : Rn ! R. The decision vector x belongs to the (nonempty) feasible set S = fx 2 Rn j gi (x)  0 for i = 1; 2; : : : ; mg. All the functions involved are assumed to be locally Lipschitz continuous (see for example [Miettinen, Makela, 1995]). The word \minimize" means that we want to minimize all the objective functions simultaneously. In the following, we denote the image of the feasible set by Z  Rk . The elements of Z are called criterion vectors and denoted by f (x) = (f1 (x); f2 (x); : : : ; fk (x))T . To avoid trivial cases, we suppose that there does not exist a single solution which is optimal with respect to every objective function. Components of an unattainable ideal criterion vector z? 2 Rk are formed by minimizing separately every fi subject to the constraints. The ideal criterion vector can be used as some kind of a reference point. As the concepts of optimality we employ local Pareto optimality and local weak Pareto optimality. We denote an open ball around a point x with radius  > 0 by B(x ; ). A decision vector x 2 S and the corresponding criterion vector f (x ) are locally Pareto optimal if there does not exist another decision vector x 2 B(x ; ) \ S (for certain  > 0) such that fi (x)  fi (x ) for all i = 1; : : : ; k and fj (x) < fj (x ) for at least one objective function fj . If in the de nition above the inequalities are all strict, the vectors are locally weakly Pareto optimal . We assume that we have a single decision maker or a unanimous group of decision makers. A decision maker is a person who has deeper insight into the problem and who can give a complete order to the Pareto optimal solutions. By solving a multiobjective optimization problem we here mean nding a feasible decision vector such that it is Pareto optimal and satis es the needs and requirements of the decision maker. Such a solution is called a nal solution . Since locally Lipschitz continuous functions may not have derivatives in a classical sense, we need a generalization of gradients. A subdierential of the locally Lipschitz continuous function f : Rn ! R evaluated at the point x 2 Rn is the set @f (x ) = conv f 2 Rn j  = llim rf (xl ); xl ! x ; rf (xl ) existsg; !1 where conv denotes a convex hull. The vectors  2 @f (x ) are called subgradients . Notice that continuous dierentiability or convexity guarantee local Lipschitz continuity.

2.2. Classi cation

We assume that  All the objective and the constraint functions are locally Lipschitz continuous.  The feasible region S is convex.  Less is preferred to more to the decision maker. The idea of NIMBUS is that the decision maker examines the values of the objective functions calculated at a decision point xh at the iteration h and classi es the objective functions into up to ve classes. The classes are functions fi whose values i 2 I < should be decreased,

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i 2 I  should be decreased down till some aspiration level, i 2 I = are satisfactory at the moment, i 2 I > are allowed to increase up till some upper bound, and i 2 I  are allowed to change freely. The decision maker is asked to specify the aspiration levels zih for i 2 I  satisfying zih < fi (xh ) and the upper bounds "hi for i 2 I > such that "hi > fi (xh ). The dierence between the classes I < and I  is that the functions in I < are to be minimized as far as possible but the functions in I  only till the aspiration level. The classi cation is the core of NIMBUS. However, the decision maker can tune the order of importance inside the classes I < and I  with optional positive weighting coecients wi summing up to one. If the decision maker does not want to specify any weighting coecients, we set wih = wjh = 1 for i 2 I < and j 2 I . Notice that the weighting coecients do not change the primary orientation speci ed in the classi cation phase. After the decision maker has classi ed the objective functions, we form two alternative subproblems, called vector and scalar subproblems. Thus, the original multiobjective optimization problem is transformed into either a new multiobjective or a single objective optimization problem, respectively. The subproblems lead into two dierent versions of NIMBUS, to be called vector version and scalar version .

2.3. Vector Subproblem

According to the classi cation and the connected information, we form in the original NIMBUS a vector subproblem 

(2)

h

i

  minimize fi (x) (i 2 I fi (x)  "hi ;

x 2 S:

As far as the problem (2) is concerned, we can state the following theorems. Theorem 1. Any locally Pareto optimal solution of the original problem (1) can be found with an appropriate classi cation in the problem (2). Proof. The classi cation phase of NIMBUS can be performed as if the "-constraint method were used. In addition, it is well-known that any Pareto optimal solution can be found by the "-constraint method (see, e.g., [Miettinen, 1994]) when the problem is formulated as follows. Let x be any Pareto optimal solution of the problem. We set I < = fj g with wj = 1 for some arbitrary j and I > = f1; : : : ; kgn fj g with "i = fi (x ).  Theorem 2. If the set I < is nonempty, then the locally Pareto optimal solution h x^ of the problem (2) is a locally weakly Pareto optimal solution of the original problem (1). Proof. Let x^h be a locally Pareto optimal solution of the problem (2) with some sets I and I  , where I < 6= ;. In other words, there does not exist another decision vector x 2 B(^xh ; ) \ S (for certain  > 0) such that fi (x)  fi (^xh )

COMPARING TWO VERSIONS OF NIMBUS OPTIMIZATION SYSTEM i

h

h

  I < and maxj2I  max fj (x) ? zjh ; 0

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 maxj2I  max fj (^xh ) ? for all i 2 i  zjh ; 0 and at least one of the inequalities is strict. Let us assume that x^h is not locally weakly Pareto optimal to the original problem. Then there exists a point x 2 B(^xh ; ) \ S such that fi (x ) < fi (^xh ) for all i = 1; : : : ; k. Because x^h is a feasible solution of the problem (2), we have fi (x ) < fi (^xh )  fi (xh ) for i 2 I = and fi (x ) < fi (^xh )  "i for i 2 I > . Thus also x is a feasible solution of the problem (2).   For all i 2 I , fi (x  ) ? zih < fi (^xh ) ? zih . It implies that max fi (x ) ? zih; 0  max fi (^xh ) ? zih ; 0 for i 2 I  , and further h

i

h

i

max max fi (x ) ? zih ; 0  max max fi (^xh ) ? zih ; 0 : i2I i2I While, in addition, fi (x ) < fi (^xh ); for all i 2 I or I  are empty.  The vector subproblem seems to be even more complicated than the original problem. The advantage of this formulation (when compared to many other multiobjective optimization methods) is the fact that the opinions and hopes of the decision maker are taken carefully into account. Notice that if I  6= ;, we have a nondierentiable problem regardless of the dierentiability of the original problem. This fact does not bring any additional diculties while we are prepared for handling nondierentiabilities. In order to be able to solve the subproblem, we need an MPB (Multiobjective Proximal Bundle) method. Here we brie y sketch the MPB method. For details, see [Makela, 1993]. Theoretically, we minimize an unconstrained (real-valued) improvement function H : Rn  Rn ! R de ned by (3)  H (x; y) = max fi (x)=wih ? fi (y)=wih ; (i 2 I ); o

gl;j (x); (l = 1; : : : ; m) : Now we have an unconstrained single objective optimization problem minimize H^ s(xs + d) + 21 uskdk2 (4) subject to d 2 Rn; to be solved, where us > 0 is some weighting parameter. The stabilizing term 1 s 2 2 u kdk is added to guarantee the existence of solutions and to keep the approximation local enough. Notice that (4) is still a nondierentiable problem, but due to its minmax-nature it is equivalent to a certain (dierentiable) quadratic problem (see [Makela, Neittaanmaki, 1992]). For details, (see [Miettinen, Makela, 1995]).

2.4. Scalar Subproblem

In the second version of NIMBUS we, after the classi cation, form a scalar subproblem h i minimize max< wih (fi (x) ? zi? ); wjh max [fj (x) ? zjh ; 0] (5)

i2I j 2I  subject to fi (x)  fi (xh ); i 2 I < [ I  [ I = fi (x)  "hi ; i 2 I >

x 2 S;

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where zi? for i 2 I < are components of the ideal criterion vector. The optimality results of the scalar subproblem are equal to those of the vector subproblem. Theorem 1 is valid also here. Thus, any locally Pareto optimal solution of the original problem can be found with an appropriate classi cation while constants do not have any eect in the objective function. Theorem 3. If the set I < is nonempty, then the local solution x^h of the problem (5) is a locally weakly Pareto optimal solution of the original problem (1). Proof. Let us denote the objective function of the problem (5) by F (x) to be minimized and the feasible region by S^. Let x^h be a locally optimal solution of the problem (5) with some sets I < , I  , I =, I > , and I  , where I < 6= ;. In other words, there exists an open ball B(^xh ; ) with  > 0 such that F (^xh )  F (x) for all x 2 B(^xh ; ) \ S^. Let us assume that x^ h is not locally weakly Pareto optimal. This means that there exists a vector x 2 B(^xh ; ) \ S such that fi (x ) < fi (^xh ) for all i = 1; : : : ; k. Because x^h 2 S^, we have fi (x ) < fi (^xh )  fi (xh ) for i 2 I < [ I  [ I = and fi (x ) < fi (^xh )  "i for i 2 I > . Thus, also x 2 B(^xh ; ) \ S^. Since zi?  fi (x ) < fi (^xh ) for all i 2 I < 6= ;, we have fi (^xh ) ? zi? > 0 for all i 2 I , and I  at the point f (xh ) such that I > [ I  6= ; and I < [ I  6= ;. If either of the unions is empty, go to step 9. Ask the aspiration levels zih for i 2 I  and the upper bounds "hi for i 2 I > from the decision maker. Ask also the optional weighting coecients wih > 0 for i 2 I < [ I  , summing up to one. (3) Calculate x^ h by solving the subproblem. If x^h = xh , ask the decision maker whether (s)he wants to try another classi cation. If yes, set xh+1 = xh , h = h + 1, and go to step 2; if no, go to step 9. (4) Present f (xh ) and f (^xh ) to the decision maker. If the decision maker wants to see dierent alternatives between f (xh ) and f (^xh ), set dh = x^ h ? xh and go to step 6. If the decision maker prefers f (xh ), set xh+1 = xh and h = h + 1, and go to step 2. (5) Now the decision maker wants to continue from f (^xh ). If I < 6= ;, set xh+1 = x^ h and h = h + 1, and go to step 2. Otherwise< (I < = ;), the weak Pareto optimality must be guaranteed by setting I = f1; 2; : : : ; kg and solving the subproblem. Let the solution be x h . Set xh+1 = xh and h = h + 1, and go to step 2.   j ?1 h h (6) Calculate vectors f (x + tj d ), j = 1; : : : ; P , where tj = P ?1 and the number P is given by the decision maker. (7) Produce weakly Pareto optimal criterion vectors from the above vectors with the classi cation I < = f1; 2; : : : ; kg. (8) Present the P alternatives to the decision maker and let her or him choose the most preferred one among them. Denote the corresponding decision vector by xh+1 and set h = h + 1. If the decision maker wants to continue, go to step 2. (9) Check the Pareto optimality of xh by solving the auxiliary problem (7). Let the solution be (~x; ~ ). (10) Stop with the nal solution x~.

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Any starting point can be projected into the feasible region by solving the auxiliary problem (6)

maximize min[0; g1(x); g2 (x); : : : ; gm (x)] subject to x 2 Rn:

Since the Pareto optimality of the solutions produced cannot be guaranteed, we check it in the end by solving an auxiliary problem maximize

(7)

k X

i

i=1 subject to fi (x) + i  fi (xh ) for all i = 1; : : : ; k; i  0 for all i = 1; : : : ; k;

x 2 S;

with respect to x 2 Rn and i 2 R for i = 1; : : : k. If xh is not Pareto optimal, then the solution x~ of the problem (7) is. For clarity of notations, it has not been mentioned in the algorithm that the decision maker may check the Pareto optimality of any current solution during the solution process. Notice that, if the subproblem (5) is employed in the algorithm, we have to calculate the components of the ideal criterion vector z? in the rst step. We must remember that we cannot guarantee global optimality. If the solution obtained is not completely satisfactory, one can always solve the problem again from a dierent starting point. It is also advised if the decision maker has to stop the solution process after the step 3.

2.6. Comparison of the Versions

The two versions of NIMBUS introduced in previous sections dier in the form of the subproblem used. The rst has several objective functions (vector subproblem) and the second only one (scalar subproblem). The origin of the development of the scalar version was the drawbacks discovered in the vector subproblem. Theoretically, the solution of the vector subproblem has to be locally Pareto optimal in order to guarantee locally weakly Pareto optimal solutions to the original multiobjective optimization problem. This assumption is quite demanding. With the scalar subproblem we do not have problems of this kind. The vector version needs a special solution tool { MPB. In addition, the role of the weighting coecients is not commensurable between the classes I < and I  . This implies that the controllability of the method suers. The advantage of having a single objective function is that we can employ any ecient optimization routine of nondierentiable optimization. In the scalar subproblem, we treat the functions in I < and I  in a consistent way and, thus, the roles of the weighting coecients are identical. In all, this means that the decision maker can better direct the solution process. Notice that in addition to the dierence in the objective functions of the subproblems, there is also deviation in the constraint part. Due to the goal of the classes I < and I  , we have to make sure that the values of these functions do not

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increase. This is the reason for modifying the constraints of the scalar subproblem (5). In the vector subproblem, the black-box routine MPB does not allow increment in I