### Table 2: Extreme Points

1994

"... In PAGE 2: ...1 De nition and Calculation Like the Convex-Hull and Convex Sets, there are many ways to de ne Extreme Points of a Convex Set. Table2 shows one of the classical de nitions of Extreme Points for conventional Convex sets, and presents its generalization for B-Convex sets. We denote the set of Extreme Points of a given Convex Set Y by E(Y ) and the set of Extreme Points of a given... ..."

Cited by 1

### Table 1. Efficient Extreme Points

1999

"... In PAGE 5: ... In [10], the relation be- tween the number of extreme points, dimension and number of constraints was studied. Table1 was copied from [10] to illustrate the relationship between constraints and dimension. Consider the case of a design alternative with 10 attributes and 20 preference con- straints.... In PAGE 5: ... Consider the case of a design alternative with 10 attributes and 20 preference con- straints. Referring to Table1 , we have at most 28 extreme points. Let D8 CT be the amount of time needed to calculate the extreme points with ADBASE, and D8 BC be the time to com- pute the fitness value at each extreme point (which is the amount of time for performing 10 multiplications and 9 ad- ditions), and D8 BD be the amount of time for comparing two fitness values.... ..."

Cited by 7

### Table 1. Extreme points when there are three attributes

2002

### Table 1: Average Number of Efficient Extreme Points (3 Objective Function)

1999

"... In PAGE 12: ... In [15], the relationship between the number of extreme points, dimension of W n , and number of preference constraint is studied. We adopt a table from [15] to illustrate this relationship (See Table1 ). The number of extreme points in this table are for three objective functions while there are no objective function in our case.... ..."

Cited by 7

### Table 1: Average Number of Efficient Extreme Points (3 Objective Function)

"... In PAGE 12: ... In [15], the relationship between the number of extreme points, dimension of Wn, and number of preference constraint is studied. We adopt a table from [15] to illustrate this relationship (See Table1 ). The number of extreme points in this table are for three objective functions while there are no objective function in our case.... ..."

### Table 3: Confidence and Prediction Intervals for Number of Efficient Extreme Points

2003

"... In PAGE 14: ...Table 3: Confidence and Prediction Intervals for Number of Efficient Extreme Points From the preceding analysis and tables we can observe several interesting things: a. The point estimates for number of efficient extreme points in Table3 vary from 10 to 92,420. b.... In PAGE 14: ... b. The upper limits for 90% prediction intervals for the number of efficient extreme points in Table3 vary from 55 to 505,347. This indicates that the set of efficient extreme points can grow quite large.... In PAGE 14: ... The predictions for number of efficient extreme points tend to increase with the number of objectives. This is evident by looking at the first entry for each number of objectives in Table3 . For these entries, the number of objectives is varied while all other values are held constant.... In PAGE 15: ... g. As the point estimates for number of efficient extreme points in Table3 and CPU time in Table 4 increase so does the confidence and prediction interval widths. This is a result of the increase in variance detected for MOLPs with high-valued configuration parameters.... ..."

### Table 2. Feasible extreme points ( }) 2 , 1 { }, 3 , 2 ({ R )

2002

### Table 1: The 6 extreme points of M(W; X; Y ) for the rst version of Example 5.

2001

"... In PAGE 18: ... In the rst version of the example, M(W; X; Y ) consists of all probability distributions P such that P (S = h) = 1=2 when S is either W , X or Y . The extreme points of M(W; X; Y ) are listed in Table1 . The modi ed version of the example adds the constraints: P (S = sjU = u; V = v) 1=3 whenever (S; U; V ) is a permutation of (W; X; Y ) and s, u, v take values in fh; tg.... ..."

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### Table 5. Extreme points searched for each object by simplex algorithm compared with total number of bottlenecks.

"... In PAGE 19: ...Complexity of the search is shown in Table5 . It shows different combinations of number of resources and classes, K and C, respectively.... In PAGE 19: ... The column labeled Bottlenecks shows the number of extreme points (or bottlenecks) as a function of C+K, as given by the equation (6). The practical complexity of the linear simplex is 3K/2 (Chvatal 1983]) and shown in the last column of Table5 . We note that this number is much lower than the total number of bottlenecks (which, although it is smaller than the number of workload mixes, is still combinatorial).... ..."

### Table 2. Estimated covariance matrix for the compositive noise on extremal points (expressed in the local frame).

in A Framework for Uncertainty and Validation of 3-D Registration Methods based on Points and Frames

1997

Cited by 65