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... the next I minimization. In succeeding iterations the designer will pick known noninferior solutions among which he wishes to trade-off and gives preferences g.. Then the weights are generated using =-=(11)-=- and (12). This procedure continues until the designer finds a noninferior design with which he is satisfied. This I norm method of generating noninferior solutions has proved oo completely satisfacto...

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...f the yield over the i simplex is estimated by taking the gradient of (19), i.e., 3 3f(?iki'xo) ^x ) {f(V*o>T .n ,.snii . j n af(£. . ,x ) o-»sA ( ^ a i ^ 2 nsi o 2, .n ,. ..n 3A.(x ) j n Notice that =-=(20)-=- is the exact gradient of (19). Further since we have reduced the work of calculating A.(x ) to an inner product, and V A.(x ) to checking o a sign, the major work in evaluating (19) and (20) is the e...

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...(xI) £ f±v'x> for all i < f/(x) for some j. The image f(x) of a noninferior point x will be called a noninferior solution. Alternate terminology for noninferiority is Pareto optimality'[3], efficiency=-=[4]-=-, admissability and a form of nondomination [5]. Many methods have been developed for generating noninferior solutions [2-15], reviews are found in [9] and [10]. We will present two standard methods i...

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... £ 3. = 1 we generate the point f * .th where f. is the point found by minimizing the i objective. Now based upon the ideas embodied in the canonic weight we choose the weight W.. = - i = 1, . . ., m =-=(12)-=- K for the next I minimization. In succeeding iterations the designer will pick known noninferior solutions among which he wishes to trade-off and gives preferences g.. Then the weights are generated ...

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...esign problem. IV. Numerical Example In order to illustrate an MCO problem including yield we will consider the following problem Min f± - (x^l.5) 2 + (x2 - 3 ) 2 X1 > X2 f2 - ( x r 7 ) 2 + (x2-3.5)2 =-=(21)-=- l-Y(x) subject to (Xl-6)2 (x?-6)2 1s9 + - ^ < 1 (22) (5.5)Z (2)Z where Y(x) is the yield and x, and x2 are independent gaussian with equal variance. The first step is to generate a simplicial approxi...

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...f.j^ In order to solve (9) it is imperative that a powerful constrained optimization method be available. We use the new method proposed by Powell [20-22] which is similar to methods discussed by Han =-=[23]-=- and Tapia [24]. Powellfs method consists of solving a sequence of quadratic programs of the form k T i T min F(x ) + d G + i_d Bd d l (10) subject to dTVgi.(xk) + g±(xk) £ 0 i = 1, . . ., I where F(x...

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...I o I o , coordinates of the vertices of the i interior simplex exclusive of x , the nominal. This matrix is independent of the nominal. It can be shown [10] that A.(x ) = -1-. I det(X.) {1-x TX."1e} =-=(17)-=- i o n! ' I o I v y a n d V x V X J = h s S n Uet(X ){l-x TX "S}] (-X "Xe) (18) is. X U II. X U X X o where n is the number of designable parameters det(X.) is the determinant of of X., e is a column ...

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... image f(x) of a noninferior point x will be called a noninferior solution. Alternate terminology for noninferiority is Pareto optimality'[3], efficiency[4], admissability and a form of nondomination =-=[5]-=-. Many methods have been developed for generating noninferior solutions [2-15], reviews are found in [9] and [10]. We will present two standard methods in order to illustrate the problems and ideas of...

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...ninferiority is Pareto optimality'[3], efficiency[4], admissability and a form of nondomination [5]. Many methods have been developed for generating noninferior solutions [2-15], reviews are found in =-=[9]-=- and [10]. We will present two standard methods in order to illustrate the problems and ideas of MCO, then a new family of solutions will be described. Historically the first method used to generate a...

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...o solve (9) it is imperative that a powerful constrained optimization method be available. We use the new method proposed by Powell [20-22] which is similar to methods discussed by Han [23] and Tapia =-=[24]-=-. Powellfs method consists of solving a sequence of quadratic programs of the form k T i T min F(x ) + d G + i_d Bd d l (10) subject to dTVgi.(xk) + g±(xk) £ 0 i = 1, . . ., I where F(xk) is the value...

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...esent a new method for generating solutions to the MCO problem and a suggestion for effective implementation of this method. In Section III we present a method based upon the Simplicial Approximation =-=[1]-=- for efficiently estimating the circuit yield and the gradient of the yield with respect to the nominal point. In Section IV we present a geometric example of a MCO problem with yield as a criteria. F...

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...1, such that fi(xI) £ f±v'x> for all i < f/(x) for some j. The image f(x) of a noninferior point x will be called a noninferior solution. Alternate terminology for noninferiority is Pareto optimality'=-=[3]-=-, efficiency[4], admissability and a form of nondomination [5]. Many methods have been developed for generating noninferior solutions [2-15], reviews are found in [9] and [10]. We will present two sta...

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...s with nominal parameter vector x . With Q as the feasible region in input space we have n Y(x ) = JF(x-x )dx. (13) n By replacing Q with the simplicial approximation, SA, we have Y(x ) = 'F(x,x )dx. =-=(14)-=- SA The key to the method is to realize that, by construction, each face of the simplicial approximation is a simplex. Thus, as shown in Figure 5, each nominal point x induces a unique interior simpli...

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...ensively using the Simplicial Approximation. Let F(x., x ) be the p.d.f. of the parameters with nominal parameter vector x . With Q as the feasible region in input space we have n Y(x ) = JF(x-x )dx. =-=(13)-=- n By replacing Q with the simplicial approximation, SA, we have Y(x ) = 'F(x,x )dx. (14) SA The key to the method is to realize that, by construction, each face of the simplicial approximation is a s...

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..., the nominal. This matrix is independent of the nominal. It can be shown [10] that A.(x ) = -1-. I det(X.) {1-x TX."1e} (17) i o n! ' I o I v y a n d V x V X J = h s S n Uet(X ){l-x TX "S}] (-X "Xe) =-=(18)-=- is. X U II. X U X X o where n is the number of designable parameters det(X.) is the determinant of of X., e is a column vector of all ones, and sgn is the sign function. Thus we need to find X. e and...

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...I is the number of segments desired, x^m the vertices of the SA, we estimate the yield over the i simplex as Y.(XO) The gradient of the yield over the i simplex is estimated by taking the gradient of =-=(19)-=-, i.e., 3 3f(?iki'xo) ^x ) {f(V*o>T .n ,.snii . j n af(£. . ,x ) o-»sA ( ^ a i ^ 2 nsi o 2, .n ,. ..n 3A.(x ) j n Notice that (20) is the exact gradient of (19). Further since we have reduced the work...

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...he image of ft by f = (f.(x), , f (x)), i.e., 1 m A-{f|fef(G)}, will be called the feasible region in output space. The solution concept for the MCO problem that we will use is that of noninferiority =-=[2]-=-. Definition. A point x e ft is called a noninferior point if and only if there does not exist another point x1 e ft, x £ x1, such that fi(xI) £ f±v'x> for all i < f/(x) for some j. The image f(x) of ...

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...rity is Pareto optimality'[3], efficiency[4], admissability and a form of nondomination [5]. Many methods have been developed for generating noninferior solutions [2-15], reviews are found in [9] and =-=[10]-=-. We will present two standard methods in order to illustrate the problems and ideas of MCO, then a new family of solutions will be described. Historically the first method used to generate a noninfer...

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...te the gradient of the yield over i region as the gradient of the approximation over that region - which we can calculate exactly. Generically the gradient will have the form k=l o o k=l By examining =-=(15)-=- and (16) we see that to be able to efficiently estimate yield and its gradient over the i region we must efficiently evaluate A.(x ) and V A.(x ). Let X. be a matrix whose columns are the1 o x I o I ...

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...adient of the yield over i region as the gradient of the approximation over that region - which we can calculate exactly. Generically the gradient will have the form k=l o o k=l By examining (15) and =-=(16)-=- we see that to be able to efficiently estimate yield and its gradient over the i region we must efficiently evaluate A.(x ) and V A.(x ). Let X. be a matrix whose columns are the1 o x I o I o , coord...