The Decomposition Algorithm

DECOMP_SUBPROB Statement

DECOMP_SUBPROB <decomp-subprob-options>;

SUBPROB <decomp-subprob-options>;

The DECOMP_SUBPROB statement controls the subproblem.

Table 15.16 summarizes the options available for the decomposition algorithm in the DECOMP_SUBPROB statement when the subproblem algorithm chosen is an LP algorithm. (As indicated, you can specify the PRINTLEVEL= option only in the OPTLP procedure.) For descriptions of these options, see the section LP Solver Options in Chapter 7: The Linear Programming Solver, and the section PROC OPTLP Statement in Chapter 12: The OPTLP Procedure. Some options have different defaults when you use the decomposition algorithm, as shown in Table 15.16.

Table 15.16: Options in the DECOMP_SUBPROB Statement Used with an LP Algorithm

Description	decomp-subprob-option	Different
		Default
Algorithm Option
Specifies the subproblem algorithm	ALGORITHM=	PS (METHOD=USER) NETWORK_PURE (METHOD=NETWORK)^†
Presolve Option
Controls the dualization of the problem	DUALIZE=	OFF
Specifies, for the first subproblem solve only, the type of presolve	INITPRESOLVER=
Specifies the type of presolve	PRESOLVER=	NONE (ALGORITHM=PS)^†
Control Options
Specifies the feasibility tolerance	FEASTOL=	1E–7
Specifies how frequently to print the solution progress	LOGFREQ=
Specifies the level of detail of solution progress to print in the log	LOGLEVEL=
Specifies the maximum number of iterations	MAXITER=
Specifies the time limit for the optimization process	MAXTIME=
Specifies the number of threads to use in the subproblem solver	NTHREADS=
Specifies the optimality tolerance	OPTTOL=	1E–7
Enables or disables printing summary (OPTLP procedure only)	PRINTLEVEL=
Specifies the initial seed for the random number generator	SEED=
Specifies whether time units are CPU time or real time	TIMETYPE=
Simplex Algorithm Options
Specifies the type of initial basis	BASIS=	WARMSTART (ALGORITHM=PS)^†
Specifies the type of pricing strategy	PRICETYPE=
Specifies the queue size for determining entering variable	QUEUESIZE=
Enables or disables scaling of the problem	SCALE=
Interior Point Algorithm Options
Enables or disables interior crossover	CROSSOVER=
Specifies the stopping criterion based on duality gap	STOP_DG=
Specifies the stopping criterion based on dual infeasibility	STOP_DI=
Specifies the stopping criterion based on primal infeasibility	STOP_PI=

† When METHOD=USER is specified in the DECOMP statement, ALGORITHM=PS, PRESOLVER=NONE, and BASIS=WARMSTART by default. These defaults are motivated by the fact that primal feasibility of the subproblem is preserved when the objective is changed, so a warm start from the previous optimal basis tends to be more efficient than solving the subproblem from scratch in each iteration. When METHOD=NETWORK, ALGORITHM=NETWORK_PURE by default because each subproblem is a pure network, causing the specialized pure network solver to usually be the most efficient choice.

Table 15.17 summarizes the options available in the DECOMP_SUBPROB statement when the subproblem algorithm chosen is a MILP algorithm. When the subproblem consists of multiple blocks (a block-diagonal structure), these settings apply to all subproblems. For descriptions of these options, see the section MILP Solver Options in Chapter 8: The Mixed Integer Linear Programming Solver, and the section PROC OPTMILP Statement in Chapter 13: The OPTMILP Procedure.

Table 15.17: Options in the DECOMP_SUBPROB Statement Used with a MILP Algorithm

Description	decomp-subprob-option	Different
		Default
Algorithm Option
Specifies the subproblem algorithm	ALGORITHM=
Presolve Option
Specifies, for the first subproblem solve only, the type of presolve	INITPRESOLVER=
Specifies the type of presolve	PRESOLVER=
Control Options
Specifies the stopping criterion based on absolute objective gap	ABSOBJGAP=
Emphasizes feasibility or optimality	EMPHASIS=
Specifies the maximum violation on variables and constraints	FEASTOL=	1E–7
Specifies the maximum allowed difference between an integer variable’s value and an integer	INTTOL=
Specifies how frequently to print the node log	LOGFREQ=
Specifies the level of detail of solution progress to print in the log	LOGLEVEL=
Specifies the maximum number of nodes to be processed	MAXNODES=
Specifies the maximum number of solutions to be found	MAXSOLS=
Specifies the time limit for the optimization process	MAXTIME=
Specifies the number of threads to use in the subproblem solver	NTHREADS=
Specifies the tolerance used when deciding on the optimality of nodes in the branch-and-bound tree	OPTTOL=	1E–7
Specifies whether to use the previous best primal solution as a warm start	PRIMALIN=
Specifies the probing level	PROBE=
Specifies the stopping criterion based on relative objective gap	RELOBJGAP=
Specifies the scale of the problem matrix	SCALE=
Specifies the stopping criterion based on target objective value	TARGET=
Specifies whether time units are CPU time or real time	TIMETYPE=
Heuristics Option
Specifies the primal heuristics level	HEURISTICS=
Search Options
Specifies the level of conflict search	CONFLICTSEARCH=
Specifies the node selection strategy	NODESEL=
Specifies the restarting strategy	RESTARTS=
Specifies the initial seed for the random number generator	SEED=
Specifies the number of simplex iterations performed on each variable in strong branching strategy	STRONGITER=
Specifies the number of candidates for strong branching	STRONGLEN=
Specifies the level of symmetry detection	SYMMETRY=
Specifies the rule for selecting branching variable	VARSEL=
Cut Options
Specifies the cut level for all cuts	ALLCUTS=
Specifies the clique cut level	CUTCLIQUE=
Specifies the flow cover cut level	CUTFLOWCOVER=
Specifies the flow path cut level	CUTFLOWPATH=
Specifies the Gomory cut level	CUTGOMORY=
Specifies the generalized upper bound (GUB) cover cut level	CUTGUB=
Specifies the implied bounds cut level	CUTIMPLIED=
Specifies the knapsack cover cut level	CUTKNAPSACK=
Specifies the lift-and-project cut level	CUTLAP=
Specifies the mixed lifted 0-1 cut level	CUTMILIFTED=
Specifies the mixed integer rounding (MIR) cut level	CUTMIR=
Specifies the multicommodity network flow cut level	CUTMULTICOMMODITY=
Specifies the row multiplier factor for cuts	CUTSFACTOR=
Specifies the overall cut aggressiveness	CUTSTRATEGY=
Specifies the zero-half cut level	CUTZEROHALF=

The following options, listed in Table 15.16 and Table 15.17, are specific to the DECOMP_SUBPROB statement and are not described in the LP or MILP solver sections.

ALGORITHM=string SOLVER=string SOL=string

specifies one of the algorithms shown in Table 15.18 (the valid abbreviated value for each string is shown in parentheses).

Table 15.18: Values for ALGORITHM= Option

string	Description
PRIMAL (PS)	Uses the primal simplex algorithm.
DUAL (DS)	Uses the dual simplex algorithm.
NETWORK (NS)	Uses the network simplex algorithm.
NETWORK_PURE (NSPURE)	Uses the network simplex algorithm for pure networks.
INTERIORPOINT (IP)	Uses the interior point algorithm.
MILP	Uses the mixed integer linear solver.

The default is NETWORK_PURE if METHOD=NETWORK, MILP for mixed integer linear programming subproblems, or PS for linear programming subproblems.

INITPRESOLVER=number | string INITPRESOL=number | string

specifies, for the first subproblem solve only, presolve conditions as listed in Table 15.19.

Table 15.19: Values for INITPRESOLVER= Option

number	string	Description
–1	AUTOMATIC	Applies the default level of presolve processing
0	NONE	Disables presolver
1	BASIC	Performs minimal presolve processing
2	MODERATE	Applies a higher level of presolve processing
3	AGGRESSIVE	Applies the highest level of presolve processing

The default is AUTOMATIC.

NTHREADS=number

specifies the number of threads to use in the subproblem solver (if the selected solver method supports multithreading). The value of the NTHREADS= option in the PERFORMANCE statement, which is described in the section PERFORMANCE Statement, serves as the overall capacity for the number of active threads that can run at one time. By default, the number of subproblem threads is $t = \max (1,\min (s,\left\lfloor {p/n} \right\rfloor )$ , where s is the value of the NTHREADS= option in the DECOMP_SUBPROB statement, p is the value of the NTHREADS= option in the PERFORMANCE statement, $n=\max (1,\left\lfloor {d/m}\right\rfloor )$ is the number of blocks being processed simultaneously, d is the number of block threads, and m is the number of compute nodes (which can be more than one, when you run the decomposition algorithm in distributed mode).

PRIMALIN=number | string PIN=number | string

specifies (for MILP problems only) whether the MILP solver is to use the values of the previous best solution’s variables as a starting solution (warm start). If the MILP solver finds that the input solution is feasible, then the input solution provides an incumbent solution and a bound for the branch-and-bound algorithm. If the solution is not feasible, the MILP solver tries to repair it. When it is difficult to find a good integer-feasible solution for a problem, warm start can reduce solution time significantly. Table 15.20 describes the valid values of the PRIMALIN= option.

Table 15.20: Values for PRIMALIN= Option

number	string	Description
0	OFF	Ignores the previous solution.
1	ON	Starts from the previous solution.

The default is ON.