### Table 1: Comparing the cuts Our previous computational experience suggests that the simple MIR inequalities are more e ective when they are coupled with partition inequalities and the mixed partition inequalities are 11

1999

"... In PAGE 12: ...panning-tree inequalities only. Next, we also used the partition inequalities. Finally, we included simple MIR and mixed-partition inequalities as well. In Table1 , we present the results of this study for nine sample problem instances (described in Section 6). The cost data is scaled so that the initial LP-relaxations have value 0 and the optimal integer programs (or the best known upper bound) have value 100.... ..."

Cited by 25

### Table 2 Sizes of Mixed Integer Programs

"... In PAGE 23: ...0. Sizes of the reduced mixed-integer programs, as well as their computational times (in CPU seconds) are given in Table2 . In Table 3 we show the initial (default) solution, the heuristic solution derived from the mixed-integer program (15) - (18), and the optimal solution for each problem we considered.... ..."

### Table 2: Decomposing matrices of mixed integer programs.

"... In PAGE 22: ...nder consideration. The rst interesting case in this context are two blocks and we set := 2. We used := (#rows) 1:05 2 rounded up as the block capacity, which allows a deviation of 10% of the actual block sizes in the decomposition. Table2 shows the results that we obtained for matrices of mixed integer programs taken from the Mipliby and preprocessed with the presolver of the general purpose MIP-solver SIP that is currently under devel- opment at the Konrad-Zuse-Zentrum. We again considered all instances with up to 1,000 rows.... ..."

### Table 1: Decomposing matrices of mixed integer programs.

1997

"... In PAGE 9: ... We used := (#rows) 1:05 2 rounded up as the block capacity, which allows a deviation of 10% of the actual block sizes in the decomposition. Table1 reports the results of our computational experiments for all instances with up to 400 rows. The format is as follows: Column 1 provides the name of the problem, Columns 2 to 4 contain the number of rows, columns and non-zeros of the matrix to be decomposed.... ..."

Cited by 4

### TABLE I NUMBER OF BINARY VARIABLES, AUXILIARY CONTINUOUS VARIABLES AND MIXED-INTEGER INEQUALITIES IN TERMS OF THE NUMBER OF TANKS

### Table 3: Decomposing transposed matrices of mixed integer programs.

"... In PAGE 24: ... The test set consists of all problems with up to 1,000 rows (1,000 columns in the original problem). Table3 shows the results of our computations. Surprisingly, the performance of our algorithm is not only similar to the \primal quot; case, in fact it is even better! We can solve almost all problems with up to 400 rows.... ..."

### Table 1: The size of the formulated 0-1 integer programming problem. # of nodes # of variables # of inequality constraints

"... In PAGE 8: ... (Note that each variable corresponds to a node group, while each inequality constraint corresponds to a partition of V .) Table1 shows the size of the 0-1 integer programming problem for networks of up to 12 nodes. The number of variables is the same as that of node groups, i.... In PAGE 13: ... In the case of Network 7, for example, the number of variables is 219 and that of inequalities is 1519. Comparing these values with Table1 , one can see that the number of variables is reduced by half and that of the inequality constraints is decreased by more than 90%. (Notice that Network 7 has nine nodes.... ..."

### Table 2: Duality for closed conic convex programs

"... In PAGE 23: ...d #03 = inf 8 #3E #3C #3E : s 5 #0C #0C #0C #0C #0C #0C #0C 2 6 4 0 1 0 1 s 2 s 5 = p 2 0 s 5 = p 2 0 3 7 5 #17 0 9 #3E = #3E ; = 1: Finally, the possibility of the entries in Table2 where weak infeasibility is not involved, can be demonstrated by a 2-dimensional linear programming problem: Example 5 Let n =2,c2#3C 2 ,K=K #03 =#3C 2 + and A = f#28x 1 ;x 2 #29jx 1 =0g; A ? =f#28s 1 ;s 2 #29js 2 =0g: We see that #28P#29 is strongly feasible if c 1 #3E 0,weakly feasible if c 1 =0and strongly infeasible if c 1 #3C 0. Similarly, #28D#29 is strongly feasible if c 2 #3E 0,weakly feasible if c 2 =0and strongly infeasible if c 2 #3C 0.... In PAGE 27: ... #0F The regularizedprogram CP#28b; c; A; K 0 #29 is dual strongly infeasible if and only if F D = ;. Combining Theorem 8 with Table2 , we see that the regularized conic convex program is in perfect duality: Corollary 7 Assume the same setting as in Theorem 8. Then there holds #0F If d #03 = 1, then the regularized primal CP#28b; c; A; K 0 #29 is either infeasible or unbounded.... ..."

### Table 1: Comparison of BALANCE+RANDOM ROUND versus mixed integer programming.

2004

Cited by 2

### Table 1: Comparison of SIP with and without feasible set inequalities ( apos;FS apos;).

1998

"... In PAGE 9: ...Table 1: Comparison of SIP with and without feasible set inequalities ( apos;FS apos;). Table1 summarizes our results over all 18 problem instances. The last col- umn gives the sum of the gaps (in percentage) between the lower and upper bounds.... In PAGE 9: ... The last col- umn gives the sum of the gaps (in percentage) between the lower and upper bounds. Table1 shows that the time decreases slightly (5%) and the gap de- creases by around 14% when adding feasible set inequalities. Based on these results we conclude that feasible set inequalities are a tool that helps solving mixed integer programs.... ..."

Cited by 3