### Table 3 Dual prices for ScarfC213s problem

in Interfaces with Other Disciplines Efficient market-clearing prices in markets with nonconvexities

2003

"... In PAGE 9: ... In the examples, the commodity price (yC3) is either the variable cost of unit 1 (the highest unit marginal operating cost), the average cost of unit 1 at full output, or the average cost of unit 2 at full output (Table 3). The prices may not be unique, depending on the level of demand, as indicated in Table3 . This results from degeneracy in the primal LP, stemming from the coincidence that demand exactly equals the sum of the capacities of the units in the solution.... In PAGE 9: ... Each also has an economic interpretation. There are three sets of prices corresponding to the dual variables in Table3 . An example using dual price set I would be: 1.... In PAGE 14: ... In these markets, the market operator explicitly asks generators to bid costs associated with nonconvexities (start-up and minimum load). In these markets, that their marginal costs are less than the commodity price and all are operating at capacity (see Table3 ). In the linear program, the resulting scarcity rents appear as positive dual variables on binding upper bounds of activities.... ..."

### Table 2: Performance of 3 popular heuristics (in italics) and their duals on 50 Comp problems (described in the text) under a 100,000-step limit. Observe how much better the duals perform on problems from this class.

"... In PAGE 2: ... For example, max- degree tends to select variables from the central component, while the decidedly untraditional min-degree tends to prefer variables from the satellite and thereby detects inconsistencies much earlier. Table2 shows how three traditional heuristics and their duals fare on Comp. Surprisingly, the simplest duals do by far the best.... ..."

### Table 1 Parallel Solution Statistics Problem % Density Dual Dim 32 Block Iters % E ciency

1994

"... In PAGE 25: ... This is not guaranteed to work, but has proven very e ective in our computations. In Table1 we report the results on the subset of the NETLIB problems that had very good values. We give problem density, the 32 block value calcu- lated by our algorithm, the number of steps that the bundle-level method took to solve the problem on 32 processors and the parallel speedup e ciency.... ..."

Cited by 14

### Table 2. Average number of nodes explored by 3 traditionally good heuristics (in italics) and their duals on 50 composed problems (described in the text) under a 100,000-step limit. Ob- serve how much better the duals can perform on problems from this class.

"... In PAGE 4: ... In contrast, the decidedly untraditional min degree heuristic tends to prefer variables from the small satellite and thereby detects inconsistencies much earlier. Table2 shows how traditional heuristics and their duals fare on one class of composed problems. Surprisingly, the duals can do better.... In PAGE 9: ... (Note that duals may perform similarly. For example, in Table2 , both min weighted degree and max weighted degrees are class- appropriate for Comp.) On challenging problems, class-inappropriate heuristics occa- sionally acquire high weights on an initial problem, and then control subsequent deci- sions, so that either the problems go unsolved or the class-inappropriate heuristics receive additional rewards.... ..."

### Table 7.1: Test problems and their best known dual and primal values

### Table 3. Total time, CPU time, and statistical measures for individual non- random (dual) problem instances. The size of the output, m,isknownapriori. Each row corresponds to individual runs. Total Time is the number of calls of procedure add next hyperedge for the generation of all transversals.

1999

"... In PAGE 9: ... This also serves as a test of the correctness of the algorithm as already explained. The results are summarized in Table3 . Notice that the problem size is now de ned by the number of nodes, n, and the number of transversals, while the number of edges, m, is now the size of the output.... ..."

Cited by 15

### Table 3. Total time, CPU time, and statistical measures for individual non-random (dual) problem instances. The size of the output, m, is known a priori. Each row corresponds to individual runs. Total Time is the number of calls of procedure add next hyperedge for the generation of all transversals.

"... In PAGE 12: ... This also serves as a test of the correctness of the algorithm as already explained. The results are summarized in Table3 . Notice that the size of the input is now de ned by the number of nodes, n, and the number of transversals, while the number of edges, m, is now the size of the output.... ..."

### Table 1: Primal-dual smooth multiplier method applied to MCPLIB problems (part 1).

1996

Cited by 2

### Table 2: Primal-dual smooth multiplier method applied to MCPLIB problems (part 2).

1996

Cited by 2

### Table 1: Primal-dual smooth multiplier method applied to MCPLIB problems (part 1).

1996

Cited by 2