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## MULTIPLE CRITERIA OPTIMIZATION AND STATISTICAL DESIGN FOR ELECTRONIC CIRCUITS (1979)

### Citations

481 | A 1950 Nonlinear programming
- Kuhn, Tucker
(Show Context)
Citation Context ... 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... |

240 |
A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,"
- Powell
- 1978
(Show Context)
Citation Context ...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... |

99 |
Proper Efficiency and the Theory of Vector Maximization,”
- Geoffrion
- 1968
(Show Context)
Citation Context ...(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... |

86 |
Optimality and Non-Scalar-Valued Performance Criteria,
- Zadeh
- 1963
(Show Context)
Citation Context ... £ 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 ... |

84 |
The convergence of variable metric methods for nonlinearly constrained optimization calculations,
- Powell
- 1978
(Show Context)
Citation Context ...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... |

78 | Augmented Lagrange multiplier functions and duality in nonconvex programming,” - Rockafellar - 1974 |

61 |
Superlinearly Convergent Variable Metric Algorithms for General Nonlinear Programming Problems
- Han
- 1976
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Citation Context ...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... |

61 | A class of solutions for group decision problems. - Yu - 1973 |

46 |
Introduction to minimax.
- Dem'Yanov, Malozemov
- 1974
(Show Context)
Citation Context ...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 ... |

41 |
Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives,
- Yu
- 1974
(Show Context)
Citation Context ... 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... |

38 |
A review and evaluation of multiobjective programing techniques.
- Cohon, Marks
- 1975
(Show Context)
Citation Context ...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... |

32 |
Diagonalized multiplier methods and quasi-Newton methods for constrained optimization.
- Tapia
- 1977
(Show Context)
Citation Context ...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... |

27 | Multiplier methods: a survey,” - Bertsekas - 1976 |

23 |
G.D.: The simplicial approximation approach to design centering. Circuits and Systems,
- Director, Hachtel
- 1977
(Show Context)
Citation Context ...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... |

18 | On the Relationship of the Tchebycheff Norm and the Efficient Frontier of Multiple-Criteria Objectives, - Bowman - 1976 |

10 |
Constrained Minimization Under Vector-Valued Criteria in Linear Topological Spaces
- Cunha, o, et al.
- 1967
(Show Context)
Citation Context ...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... |

10 |
Multiple-Objective Problems: Pareto-Optimal Solutions by Method of Proper Equality Constraints
- 'Lin
- 1976
(Show Context)
Citation Context ...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... |

9 |
Maximal Vectors and Multi-Objective Optimization
- Lin
- 1976
(Show Context)
Citation Context ...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... |

8 |
Nonlinear Programming Using Minimax Techniques
- Bandler, Charalambous
- 1974
(Show Context)
Citation Context ..., 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... |

8 |
Linearly constrained mini-max optimization
- Madsen, Schjaer-Jacobsen
- 1978
(Show Context)
Citation Context ...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... |

6 | Compromise solutions, domination structures and Salukwadze’s solution - Yu, Leitmann - 1974 |

1 |
On Noninferior Index Vectors
- Reid, Citron
- 1971
(Show Context)
Citation Context ...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 ... |

1 |
Multiple Criteria Optimization and Statistical Design for Electronic Circuits
- Lightner
- 1978
(Show Context)
Citation Context ...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... |

1 |
On the Approximation of Solutions to Multiple Crieteria Decision Making Problems
- Polak
(Show Context)
Citation Context ...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 ... |

1 |
uAn Algorithm for the Chcbyshev Problem - With an Application to Concave Programming
- Zangwill
- 1967
(Show Context)
Citation Context ...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... |