### Managing the LP Relaxation

The majority of the computational effort of BCP is spent solving LPs and hence a major emphasis in the development was to make this process as efficient as possible. Besides using a good LP engine, the primary way in which this is done is by controlling the size of each relaxation, both in terms of number of active variables and number of active constraints.

The number of constraints is controlled through use of a local pool and through purging of ineffective constraints. When a cut is generated by the cut generator, it is first sent to the local cut pool. In each iteration, up to a specified number of the strongest cuts (measured by degree of violation) from the local pool are added to the problem. Cuts that are not strong enough to be added to the relaxation are eventually purged from the list. In addition, cuts are purged from the LP itself when they have been deemed ineffective for more than a specified number of iterations, where ineffective is defined as either (1) the corresponding slack variable is positive, (2) the corresponding slack variable is basic, or (3) the dual value corresponding to the row is zero (or very small). Cuts that have remained effective in the LP for a specified number of iterations are sent to the global pool where they can be used in later search nodes. Cuts that have been purged from the LP can be made active again if they later become violated.

The number of variables (columns) in the relaxation is controlled through reduced cost fixing and dynamic column generation. Periodically, each active variable is priced to see if it can be fixed by reduced cost. That is, the LP reduced cost is examined in an effort to determine whether fixing that variable at one of its bounds would remove improving solutions; if not, the variable is fixed and removed from consideration. If the matrix is full at the time of the fixing, meaning that all unfixed variables are active, then the fixing is permanent for that subtree. Otherwise, it is temporary and only remains in force until the next time that columns are dynamically generated.

Because SYMPHONY was originally designed for combinatorial problems with relatively small numbers of variables, techniques for performing dynamic column generation are somewhat unrefined. Currently, variables are priced out sequentially by index, which can be costly. To improve the process of pricing variables, we plan to increase the symmetry between our methods for handling variables and those for handling cuts. This includes (1) allowing user-defined, abstract representations for variables, (2) allowing the use of ``variable generators'' analogous to cut generators, (3) implementing both global and local pools for variables, (4) implementing heuristics that help determine the order in which the indexed variables should be priced, and (5) allowing for methods of simultaneously pricing out large groups of variables. Much of this is already implemented in COIN/BCP.

Because pricing is computationally burdensome, it currently takes place only either (1) before branching (optional), or (2) when a node is about to be pruned (depending on the phase--see the description of the two-phase algorithm in Sect. 4.4.3.3). To use dynamic column generation, the user must supply a subroutine which generates the column corresponding to a particular user index, given the list of active constraints in the current relaxation. When column generation occurs, each column not currently active that has not been previously fixed by reduced cost is either priced out immediately, or becomes active in the current relaxation. Only a specified number of columns may enter the problem at a time, so when that limit is reached, column generation ceases. For further discussion of column generation, see Sect. 4.4.3.3, where the two-phase algorithm is described.

Since the matrix is stored in compressed form, considerable computation may be needed to add and remove rows and columns. Hence, rows and columns are only physically removed from the problem when there are sufficiently many to make it ``worthwhile.'' Otherwise, deleted rows and columns remain in the matrix but are simply ignored by the computation. Note that because ineffective rows left in the matrix increase the size of the basis unnecessarily, it is usually advisable to adopt an aggressive strategy for row removal.

Ted Ralphs
2011-05-30