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

2002

### Table 1. The role of the seed point in locating maximal feasible boxes.

2005

"... In PAGE 17: ... Table1 contains details of the results on the numerical test that examines how the place of the seed point a ects the result box constructed by the program. In the following, we use only one initial value for all d(i; j) step sizes (i = 1; 2; j = 1; 2).... In PAGE 17: ... NFE stands for the number of function (F (X) and Gj(X) for j = 1; 2; : : : ; m) evaluations. The test results presented in Table1 suggest that the seed point may be chosen close to the normal of the active constraint at x , even if it is outside of X . It is interesting that the center of the maximal volume inscribed box is not an optimal seed point.... In PAGE 18: ...e more reliable. Thus both and help determine when the algorithm stops. Table 2 shows the combined e ect of these parameters on the volume of the result box and the number of function evaluations required. The problem setting was the same as in Table1 (f quot; = 2:0), except the seed point was always xseed = (0:5; 0:5)T . As it can be seen in Table 2, the total number of function evaluations NFE is growing only slightly with the decrease of .... ..."

Cited by 4

### Table 1: Lower bound from some LP-relaxations of the CV RP. From the last section it can be readily concluded, that the x ? values of the feasible points of that program must satisfy all multistar inequalities. Vice versa, a maximal ow in the network from the last section shows, that all points of CV RPMS can be extended to feasible points for this program. In order to eliminate the maximum operator, and thus to obtain a linear program the ow equalities (25) are solved for yv0i (see (34)) and these variables are substituted in (26). With these operations we obtain the following lift of CV RPMS: x( (i)) = 2

2000

"... In PAGE 8: ...Note, that in the latter program there may exist feasible points (x; y) such that yv0;i :=di + maxf0; X k2 (i);k6 =v0(?di + dk)xikg ? X j2 (i):j6 =v0 yji ? yij lt; 0: (34) By standard network ow techniques it is easy, though, to construct a feasible point (x; ~ y) satisfying ~ yv0;i 0. Table1 compares the bounds derived from this relaxation, shown in column three, to two other LP- bounds. The second column lists the optimal value of the relaxation of the LP introduced in Section 2 where r(S) has been replaced by d(S) C .... ..."

Cited by 12

### Table 1. Determination of the maximum feasible number of precision points for a N-link, SDCSC mechanism for the RBG and PG problems. RBG Problem PG Problem

1997

"... In PAGE 9: ... One additional unknown per precision point, the input link rotation angle at that con guration ( j; 3 j M), is added for both problems. Table1 shows the number of scalar equations and unknowns for an N-link mechanism for the two problems from the third position onwards (i.... In PAGE 9: ...As indicated in Table1 , when the number of scalar loop-closure equations exceeds the number of variables, the surplus variables can be assigned arbitrary numerical values before solving the equations. These surplus variables, called free choices, can be chosen such that the equations form a linear system in terms of the remaining variables.... In PAGE 10: ... Hence, we will have only N ? 1 unknowns (R1 : : :R(N?1). The linear solution for the remaining unknowns will be possible only if the number of unknowns in the matrix is less than the number of available free choices as seen from Table1 . This requirement translates to the following criterion on the maximum number of precision points for a linear solution of the synthesis equations.... In PAGE 11: ... the free choices). An increase in the number of precision points beyond N + 1 upto the upper limit in Table1 is possible. However, the gains in terms of precise match of trajectory at more points may be o set by the disadvantage of having to solve a set of nonlinear equations.... In PAGE 14: ... Increasing the number of links not only enhances the geometric capability of the SDCSC mechanism, but also raises the number of free choices available for use in optimization. For example, with a 3-link mechanism for 4 precision points, it is clear from Table1 that 8 scalar equations in 13 variables are to be solved with 5 free choices at the disposal of the optimizer. The precision point synthesis equations after subtracting equations from the... ..."

Cited by 7

### Tables 6.20{6.21 give results for this set. SNOPT solved 54 of the 56 problems attempted. The successes include two problems that SNOPT identi ed as having in- feasible nonlinear constraints (discs and nystrom5). The nal sum of the nonlinear constraint violations for these problems was 4:00, 3:193 10?3 and 1:72 10?2 respec- tively. To our knowledge, no feasible point has ever been found for these problems. SNOPT was unable to solve problems cresc132 and leaknet in 1000 major iterations. For leaknet, the run gives a point that appears to be close to optimality, with a nal nonlinear constraint violation of 6:3 10?9. MINOS declared 7 problems to be infeasible (cresc132 , discs, lakes, nystrom5, robot, truspyr1 and truspyr2). Feasible points found by SNOPT imply that this diagnosis is correct only for discs and nystrom5. Unbounded iterations occurred in 8 cases (brainpc3 , brainpc7 , brainpc9 , errinbar, tenbars1, tenbars2, tenbars3 and tenbars4). The major iteration limit was enforced for problem reading6.

1997

Cited by 146

### Table 1: Results of the Three Methods Random Feasible Problem Method Var Eq apos;s Ineq apos;s Points Points All-time One-time

1998

"... In PAGE 4: ... Assuming a coe cient of 1 in (2),we would need to generate approximately 1012 random points to nd one approximate feasible point. As seen in Table1 in x6, processing of 106 randomly generated points needs 6846.16 STU (Standard Times Units; see x5).... In PAGE 10: ... On the system used, an STU is approximately 0:0417124 CPU seconds. 6 Test Results In Table1 , we present test results of random search, the iterative tech- nique with equality constraints and slack variables, and the iterative tech- nique with inequality constraints treated directly. The column labels of the table are as follows.... In PAGE 12: ... We tried 106 random points without nding any approximate feasible points. In Table1 , we use y to represent the expected average time for nding one feasible point, if this happened. In Table 1, * indicates a number that is meaningless to compute, since random search is impractical for some of the problems.... In PAGE 12: ... In Table 1, we use y to represent the expected average time for nding one feasible point, if this happened. In Table1 , * indicates a number that is meaningless to compute, since random search is impractical for some of the problems. The following analysis of the di erent techniques is made with regard to average time for nding one feasible point.... ..."

Cited by 3

### Table B.1: Comparing the two potential functions on the Satis ability problem To get an idea what a nal unrounded solution may look like, see Figure B.1; this solution is obtained after one iteration. We observe that a lot of variables already approach ?1 or +1, when we compare it to the interior solution of an instance of the RLFAP after one iteration (Figure B.2). This seems to be due to the di erence in starting points: for the satis ability problem we can use a starting point that lies central in the feasible region, while for the RLFAP the starting point is far away from any feasible solution.

### Table 2: Number of ampli ers needed for the various ampli er-placement schemes. (Note that N = number of stations and M = number of stars for the lower bound computation. A \* quot; in column 4 indicates that the NLP solver could not do better than the LP solution, even when it was given multiple feasible starting points, including the solutions found in [4] and [8].)

1997

"... In PAGE 5: ... Local minima: The non-linear program solver might terminate at a point corresponding to a local mini- mum for the objective function. This happens, for example, when the starting point corresponds to the Linear Program solution (see Table2 and the exam- ple in Fig. 1).... In PAGE 6: ... This led to a very simple ampli er- placement algorithm. Unfortunately, as was shown in [8] and can also be seen in Table2 , this approach does not minimize the number of ampli ers needed in the network. The transmitter powers can be adjusted to avoid placing ampli ers on the links which originate at a station.... In PAGE 6: ... This network has many stars and yet it needs no ampli ers to function. Table2 reveals that the new method was indeed able to come up with the solution of not needing any ampli- ers. This is the type of network where the unequally- powered-wavelengths solution is clearly superior to the previous two ampli er placement methods.... In PAGE 6: ... This network was designed in a semi- random fashion with some heuristics to guide the de- sign. Table2 shows that the new method was able to nd a solution which required fewer ampli ers than the methods in [4] and [8]. Fig.... In PAGE 7: ... 4 except that the distances have been scaled up and down, respectively, by a factor of 10. As we see in Table2 , the results seem to verify our earlier pre- dictions. The new method is not able to nd a better solution than the equally-powered-wavelengths solution for the larger (\scaled-up quot;) network, even when it was given multiple feasible starting points (including the so- lutions found in [4] and [8]).... In PAGE 8: ... We predicted this because the more nodes a network has, the more variables the solver is manipulating and the more local minima the solver can get stuck at. As Table2 shows, the solver was unable to come up with a better solution than the LP solution, even when given multiple feasible starting points including the solutions found in [4] and [8]. 4 Future Work 4.... ..."

Cited by 7

### Table 2: Relative performance of the various ampli er-placement schemes. A \* quot; in column 4 indicates that the NLP solver could not perform better than the LP solution, even when it was given multiple feasible starting points, including the solutions found in [14] and [20]. Column 6 shows the total CPU time taken by the nonlinear solver running on an otherwise-unloaded DEC 5000/240 to solve each problem.

1998

"... In PAGE 6: ... Local minima: The nonlinear program solver might terminate at a point corresponding to a local mini- mum for the objective function. This happens, for example, when the starting point corresponds to the Linear Program solution (see Table2 and the exam- ples in Figs. 1 and 11).... In PAGE 7: ... This led to a very simple ampli er-placement algo- rithm. Unfortunately, as was shown in [20] and can also be seen in Table2 , this approach does not minimize the number of ampli ers needed in the network. The trans- mitter powers can be adjusted to avoid placing ampli ers on the links which originate at a station.... In PAGE 7: ... The lower bound on the number of ampli ers required using the Linear-Program (LP) method in [20] is thus M ?1, where M is the number of stars in the network. We show the results of this algorithm for various networks in column 3 of Table2 (see [20] for details). The method described in this paper (see Section 2) is a global one too; however, unlike the LP method in [20], it allows the wavelengths at any point in the network to operate at unequal powers.... In PAGE 7: ...7 The solution obtained to the ampli er-placement problem is not guaranteed to be the optimum because of the presence of local minima. We show the results of this algorithm for various networks in column 4 of Table2 . The absolute lower bound was de- veloped in [11] by rst utilizing the number of wavelengths on each link and the physical constraints on the ampli ers to derive the maximum gain available from each ampli er on a given link.... In PAGE 7: ... These values were then included in a LP- solvable solution to derive the lower bound on the number of ampli ers required in the network. We show the results of the lower bound computation for various networks in column 5 of Table2 (see [11] for more details). Next, we compare the results of these three approaches to ampli er placement on certain sample networks (see Table 2).... In PAGE 8: ... Both of these networks are the same as the network in Figure 10 except that the distances have been scaled up and down, respectively, by a factor of 10. As we see in Table2 , the results seem to verify our ear- lier predictions. The new method is not able to nd a better solution than the equally-powered-wavelengths so- lution for the larger network in Figure 11, even when it was given multiple feasible starting points (including the solutions found in [14] and [20]).... In PAGE 9: ... the solver can get stuck at. As Table2 shows, the solver was unable to come up with a better solution than the LP solution, even when given multiple feasible starting points including the solutions found in [14] and [20]. The \denser-MAN quot; network in Figure 13 di ers from the MAN network in Figure 10 in that there are 12 addi- tional stations in it, 3 in each of the four groups of stations.... In PAGE 9: ... Note, however, that the NLP method performs better than the other two schemes and remains closest to the absolute lower bound on the number of am- pli ers.For each of the previous example networks, column 6 in Table2 shows the total CPU time taken by the nonlinear solver running on an otherwise-unloaded DEC 5000/240. In general, the running time is found to increase with (1) increasing number of network components (which leads to more constraints) and (2) increasing link spans (which leads to a greater choice in feasible solutions).... ..."

Cited by 9