### Table 4: Convergence table in Example 3 (plate with a hole). The convergence table displays the relative error in displacements (75) and the variation of plastic zones VPZ (76) per iteration step for various uniformly refined meshes.

in Newton-Like Solver for Elastoplastic Problems with Hardening and its Local Super-Linear Convergence

"... In PAGE 31: ... The displacement is multiplied by 100. Table4 reports on the convergence of the Newton-like method.... ..."

### Table 2. N FN MRV MRVF S BP RS

"... In PAGE 10: ... For E and E R the best possible result is 1; and larger values of indices indicate a better result. The results from Table 1 are summarized in Table2 . The new methods appear to be fairly competitive with the considered Newton-like methods.... ..."

### Table 3. Example 3, Scheme 1

1993

"... In PAGE 12: ...0 projecting on span fu0g. In (2.3) we set M1 = hs=jsjH ; iH. The results illustrating the behaviour of k n are given in Table3 . An interesting comparison can be made with the Newton-like method proposed in [19].... ..."

### Table 1: Numerical results for the basic semismooth solver problem n SP ksemi kr(xf)k1

"... In PAGE 14: ... Obviously, this is a restriction, but it is our impression that neither the tunneling methods nor the lled function methods are very reliable for larger problems. We run our program on a SUN SPARC station and summarize the numerical results in Tables 1 { 3, with Table1 containing the results for the basic semismooth solver from Algorithm 2.1, Table 2 containing the results for the modi cation using the two di erent tunneling approaches, and Table 3 containing the results for the two di erent lled function modi cations (here, we call the lled functions P1 and P2 the exponential and the rational lled function, respectively).... In PAGE 14: ...ctive more than once (i.e., if jglobal gt; 1), this di erence provides the cumulated number of these iterations. We next discuss the results given in Tables 1 { 3: The results in Table1 are basically there in order to compare our global optimization techniques with the basic solver. The only... In PAGE 18: ...Table1 is that the reader might get a wrong feeling about this basic solver since it fails on so many problems. However, we stress that the basic solver is in fact one of the best solvers which is currently available and that the test problems selected for this paper are just a subset of the most di cult problems from the GAMSLIB and MCPLIB collections.... In PAGE 18: ... Furthermore, it does not seem to destroy the overall e ciency of the algorithm. Second, we see that the tunneling methods are quite successful: While there are 14 failures in Table1 , there are only 3 failures left for both tunneling methods in Table 2. The failures occur on di erent test problems for the two tunneling versions: The classical tunneling function was not able to solve problems duopoly and games (third and fth starting point) while the exponential tunneling method fails on the three starting points for problem pgvon106.... ..."

### Table 1: Attributes of volumetric cutting plane methods

1999

"... In PAGE 3: ... Our nal result is a central cut volumetric cutting plane method that requires no more than 25n constraints at any time. In Table1 we summarize important attributes of four papers (including this paper) on volumetric cutting plane methods. These features are the placement of added cuts (shallow or central), the number of Newton or Newton-like steps required after a constraint addition or deletion, the maximum number of constraints required, and the value of a scalar V , de ned as the di erence between the mimimal increase in the volumetric barrier following a constraint addition, and the maximal decrease following a constraint deletion (see Section 3).... In PAGE 12: ...he assumptions of Theorem 3.2 thus hold with V = :0340 ? :0326 = :0014 . 2 The value V = :0014 demonstrated in Theorem 3.3 may seem relatively small, but it should be noted that this is the largest value of V to date for a volumetric cutting plane algorithm; see Table1 in Section 1. 4 Adding a Central Cut Let x be an interior point of P.... ..."

Cited by 11

### Table 2 Summary of the Jacobian

1993

"... In PAGE 8: ... Thus we will present a row of the Jacobian, J, but will show the row in blocks (the rst subscript refers to the training data point i, and the second to the actual parameter or column of the Jacobian). The form of the resulting Jacobian is summarized in Table2 and the third column in the table explains the origin of the term. We can now write the ith row of the Jacobian (using its inherent block structure) for the 2-3-2 network in Figure 1: Ji;1::3 = [x18x11 0(Q(i) 1 ) + x19x15 0(Q(i) 2 )] 0(P(i) 1 )(1; v(i) 1 ; v(i) 2 )T Ji;4::6 = [x18x12 0(Q(i) 1 ) + x19x16 0(Q(i) 2 )] 0(P(i) 2 )(1; v(i) 1 ; v(i) 2 )T Ji;7::9 = [x18x13 0(Q(i) 1 ) + x19x17 0(Q(i) 2 )] 0(P(i) 3 )(1; v(i) 1 ; v(i) 2 )T Ji;10::13 = x18 0(Q(i) 1 )(1; (P(i) 1 ); (P(i) 2 ); (P(i) 3 ))T Ji;14::17 = x19 0(Q(i) 2 )(1; (P(i) 1 ); (P(i) 2 ); (P(i) 3 ))T Ji;18::19 = ( (Q(i) 1 ); (Q(i) 2 ))T: Here Ji;j denotes the element in the ith row and jth column of the Jacobian.... In PAGE 13: ... The condition numbers of the Jacobian are of order 107, and 1055, respectively. Hence the condition number of JT J, used in the Newton-like methods of Table2 , is close to or smaller than the inverse of the machine epsilon for a 48-bit mantissa computer. As an example of a larger network, Figure 8 shows the singular values of the initial Jacobian for four cases for a 5-7-2 network with weights chosen randomly in the region (?1; 1) and ti sampled randomly in the cube [0; 1]5.... ..."

Cited by 31

### Table 1. Comparison of iterations and times when observation data is added. Interior-Point Semismooth

2004

"... In PAGE 17: ... The solution values x for the new observations are initialized to zero. The iterations and times for solving the problem from scratch and restarting are compared in Table1 . The results indicate that if all the data is known a priori, the best strategy is to solve the entire problem.... ..."

### Table 3: number of semi-smooths between B 2i and B 2i+1.

### Table 3: Numerical results for the lled function methods exponential lled function rational lled function

"... In PAGE 14: ... We run our program on a SUN SPARC station and summarize the numerical results in Tables 1 { 3, with Table 1 containing the results for the basic semismooth solver from Algorithm 2.1, Table 2 containing the results for the modi cation using the two di erent tunneling approaches, and Table3 containing the results for the two di erent lled function modi cations (here, we call the lled functions P1 and P2 the exponential and the rational lled function, respectively). The columns of these tables have the following meanings: problem: name of the test problem in GAMSLIB/MCPLIB n: dimension of the test problem SP: number of starting point used ksemi: number of iterations used in the basic semismooth solver ktotal: total number of iterations used jglobal: number of times we switch to the global optimization technique kr(xf)k1: norm of the natural residual at the nal iterate xf.... In PAGE 18: ... In view of our limited numerical results, it is therefore our feeling that the exponential tunneling method is slightly superior to the classical tunneling approach. Finally, the results in Table3 clearly indicate that the lled function methods are less successful than the tunneling methods. Both lled functions seem to have a similar be- haviour, and they were able to solve four/ ve more problems than the basic semismooth solver, so the improvement is much worse than the one we obtained with the two tunneling approaches.... ..."

### Table 2: Numerical results for the tunneling methods classical tunneling exponential tunneling

"... In PAGE 14: ... We run our program on a SUN SPARC station and summarize the numerical results in Tables 1 { 3, with Table 1 containing the results for the basic semismooth solver from Algorithm 2.1, Table2 containing the results for the modi cation using the two di erent tunneling approaches, and Table 3 containing the results for the two di erent lled function modi cations (here, we call the lled functions P1 and P2 the exponential and the rational lled function, respectively). The columns of these tables have the following meanings: problem: name of the test problem in GAMSLIB/MCPLIB n: dimension of the test problem SP: number of starting point used ksemi: number of iterations used in the basic semismooth solver ktotal: total number of iterations used jglobal: number of times we switch to the global optimization technique kr(xf)k1: norm of the natural residual at the nal iterate xf.... In PAGE 18: ... In fact, the overall behaviour of the basic solver is much better, and it is able to solve all other problems basically without any di culties. From the results in Table2 we can deduce a couple of things: First of all, the global optimization technique usually does not become active if the basic method itself was able to solve the underlying problem. The only exception is problem vonthmcp.... ..."