### Table 2. Unconstrained Problems

1997

"... In PAGE 14: ... This approach is suitable for large problems because it has been observed in practice that small values of m, say m 2 [3;; 20] often give satisfactory results [44], [31]. The numerical performance of the limited memory method L-BFGS is illus- trated in Table2 , where we compare it [65] with the Newton method provided by the LANCELOT package on a set of test problems from the CUTE [6] collection.... ..."

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### Table 6 Unconstrained maximum of the quadratic programming problem.

"... In PAGE 32: ... Now (69) restricted to (i; j) can be rewritten as follows: maximize 8 gt; lt; gt; : ?12 i ? i j ? j gt; Kii Kij Kji Kjj i ? i j ? j +vi( i ? i ) + vj( j ? j) ? quot;( i + i + j + j) subject to ( i ? i ) + ( j ? j) = i; i ; j; j 2 [0; C] (81) Next one has to eliminate j; j by exploiting the summation constraint. Ig- noring terms independent of ( ) i one obtains10 maximize ?12( i ? i )2(Kii + Kjj ? 2Kij) ? quot;( i + i )(1 ? s) +( i ? i )(vi ? vj ? (Kij ? Kjj)) subject to ( ) i 2 [L( ); H( )]: (82) The unconstrained maximum of (82) with respect to i or i can be found in Table6 . Here the shorthand := Kii + Kjj ? 2Kij is used.... ..."

### Table 5. Unconstrained benchmark problems to be minimized, with a = 1000, b = 100, y = O1x, and z = O2x, where O1 and O2 are orthogonal matrices.

2005

"... In PAGE 15: ... Here, these problems are considered only because they are frequently used and we want to discover how the MO-CMA-ES compares to algorithms biased towards such kind of problems. Unconstrained Test Functions with Quadratic Objectives The second group of benchmarks are functions where for each objective the objective function is quadratic (a quadratic approximation close to a local optimum is reasonable for any smooth enough tness func- tion), see Table5 . They are of the general form fm(x) = xT Qx = xT OT mAOmx, where x 2 Rn; Q; Om; A 2 Rn n with Om orthogonal and A diagonal and positive de nite.... In PAGE 18: ... The dis- tribution indices of the crossover and mutation operator were set to c = m = 20. In the case of the unconstrained benchmark functions in Table5 the boundaries of the mutation and crossover operator were set to the boundaries of the initial regions. See Appendix A for a description of the real-coded NSGA-II variation operators and their parameters.... In PAGE 19: ...00328 of xu 2 xl 2 (we rescaled the rst component of ZDT4 to [ 5; 5]). In the unconstrained problems, Table5 , we set (0) equal to 60 % of the initialization range of one component. In all algorithms the population size ( MO) was set to 100 as in the study by Deb et al.... ..."

### Table 5: Available tools | unconstrained problems. and the simple bounds

"... In PAGE 20: ... Of course, if all the bounds are in nite, then the problem is unconstrained. A summary of the available subroutines and their purpose is given in Table5 . A more detailed description is given in Appendix B.... ..."

### TABLE 4.1 Unconstrained Problem, h=1/639.

### Table 2: Numerical Results for the Unconstrained Algorithms.

2006

"... In PAGE 16: ... In all experiments we have applied an (exact) TSP algorithm to the resulting matrices to minimize the total treatment time for the given decomposition. Table2 presents the results for the unconstrained, Table 3 those for the constrained problems. All experiments were run on a Pentium 4 PC with 2.... In PAGE 16: ...Table 2: Numerical Results for the Unconstrained Algorithms. Table2 shows that Xia and Verhey (1998) is the fastest algorithm. However, it never found the optimal DT value and found the best DC value for only one instance.... ..."

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### Table 8 Unconstrained bi-objective problems. Problem Objective functions n Variable bounds Comments

2006

"... In PAGE 19: ... From them, we have selected first the following bi-objective unconstrained problems: Schaffer [24], Fonseca [25], and Kursawe [26], as well as the prob- lems ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6, which are defined in [21]. The formulation of these problems is provided in Table8 (see Appendix B for the tables describing the problems), which also shows the number of variables, their bounds, and the nature of the Pareto-optimal front for each problem. The second set is composed of the following constrained bi-objective problems: Osyczka2 [27], Tanaka [28], Srinivas [13], Constr Ex [1], and Golinski [29].... ..."

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### Table 2. Unconstrained Problems Problem n L-BFGS LANC/Newt

1997

"... In PAGE 14: ... This approach is suitable for large problems because it has been observed in practice that small values of m, say m 2 [3; 20] often give satisfactory results [44], [31]. The numerical performance of the limited memory method L-BFGS is illus- trated in Table2 , where we compare it [65] with the Newton method provided by the LANCELOT package on a set of test problems from the CUTE [6] collection.... ..."

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### Table 1: Unconstrained, Closed Queueing Network n i

1994

"... In PAGE 8: ...The e ort expended in the IPA calculations was negligible compared to the e ort required for the simulation, and the only storage requirement was for a 4 4 matrix. Table1 shows the results from unconstrained minimization of the function Xn( ) = 400T ( )?1 + 4 Xi=1 i: We chose this functional form to model a problem in which one wants to maximize throughput, but in which there is some cost (in this case, 1) for increasing the service rate of a server. The optimization problem then is to nd the best tradeo of cost against throughput.... In PAGE 8: ... We also ran the optimization code starting from i = 1; the number of iterations was larger, as would be expected. We also experimented with a constrained version of the same problem, in which we minimized 400T ( )?1 subject to the three inequality constraints 1 15; 4 14; 4 Xi=1 i 50: The results are shown in Table 2, in which the quantities shown are the same as those shown in Table1 except that z is the objective function value reduced by a di erent integer part (29 instead of 73). The starting point and the tolerance ACC for these computations were the same as for those of Table 1.... ..."

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### Table 3.1: MINPACK-2 Unconstrained Optimization Problems Name Description of the Problem EPT Elastic-Plastic Torsion

1996

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