### Linear Programming Models: Interior Point Algorithm

By default, the interior point algorithm is used for problems without a network component, that is, a linear programming problem. You do not need to specify the INTPOINT option in the PROC NETFLOW statement (although you will do no harm if you do).

Data for a linear programming problem resembles the data for side constraints and nonarc variables supplied to PROC NETFLOW when solving a constrained network problem. It is also very similar to the data required by the LP procedure.

#### Mathematical Description of LP

If the network component of NPSC is removed, the result is the mathematical description of the linear programming problem. If an LP has variables, and constraints, then the formal statement of the problem solved by PROC NETFLOW is

where

• is the objective function coefficient vector

• is the variable value vector

• is the constraint coefficient matrix for variables, where is the coefficient of variable in the th constraint

• is the side constraint right-hand-side vector

• is the variable value lower bound vector

• is the variable value upper bound vector

#### Interior Point Algorithmic Details

After preprocessing, the linear program to be solved is

This is the primal problem. The matrices , , and of NPSC have been renamed , , and respectively, as these symbols are by convention used more, the problem to be solved is different from the original because of preprocessing, and there has been a change of primal variable to transform the LP into one whose variables have zero lower bounds. To simplify the algebra here, assume that variables have infinite bounds, and constraints are equalities. (Interior point algorithms do efficiently handle finite bounds, and it is easy to introduce primal slack variables to change inequalities into equalities.) The problem has variables; is a variable number, is an iteration number, and if used as a subscript or superscript it denotes of iteration .

There exists an equivalent problem, the dual problem, stated as

where are dual variables, and are dual constraint slacks.

The interior point algorithm solves the system of equations to satisfy the Karush-Kuhn-Tucker (KKT) conditions for optimality:

These are the conditions for feasibility, with the complementarity condition added. Complementarity forces the optimal objectives of the primal and dual to be equal, , as

Before the optimum is reached, a solution may not satisfy the KKT conditions:

• Primal constraints may be violated, .

• Dual constraints may be violated, .

• Complementarity may not be satisfied, .

This is called the duality gap.

The interior point algorithm works by using Newton’s method to find a direction to move from the current solution toward a better solution:

where is the step length and is assigned a value as large as possible but and not so large that an or is too close to zero. The direction in which to move is found using the following:

where , , and is a vector with all elements equal to 1.

To greatly improve performance, the third equation is changed to

where , the average complementarity, and

The effect now is to find a direction in which to move to reduce infeasibilities and to reduce the complementarity toward zero, but if any is too close to zero, it is nudged out to , and any that is larger than is nudged into . A close to or equal to 0.0 biases a direction toward the optimum, and a value for close to or equal to 1.0 centers the direction toward a point where all pairwise products . Such points make up the central path in the interior. Although centering directions make little, if any, progress in reducing and moving the solution closer to the optimum, substantial progress toward the optimum can usually be made in the next iteration.

The central path is crucial to why the interior point algorithm is so efficient. This path guides the algorithm to the optimum through the interior of feasible space. Without centering, the algorithm would find a series of solutions near each other close to the boundary of feasible space. Step lengths along the direction would be small and many more iterations would probably be required to reach the optimum.

The calculation of the direction is the most time-consuming step of the interior point algorithm. Assume the th iteration is being performed, so the subscript and superscript can be dropped from the algebra:

Rearranging the second equation,

Rearranging the third equation,

where

Equating these two expressions for and rearranging,

where

Substituting into the first direction equation,

, , , and are calculated in that order. The hardest term is the factorization of the matrix to determine . Fortunately, although the values of are different for each iteration, the locations of the nonzeros in this matrix remain fixed; the nonzero locations are the same as those in the matrix . This is due to being a diagonal matrix, which has the effect of merely scaling the columns of .

The fact that the nonzeros in have a constant pattern is exploited by all interior point algorithms, and is a major reason for their excellent performance. Before iterations begin, is examined and its rows and columns are permuted so that during Cholesky Factorization, the number of fill-ins created is smaller. A list of arithmetic operations to perform the factorization is saved in concise computer data structures (working with memory locations rather than actual numerical values). This is called symbolic factorization. During iterations, when memory has been initialized with numerical values, the operations list is performed sequentially. Determining how the factorization should be performed again and again is unnecessary.

#### The Primal-Dual Predictor-Corrector Interior Point Algorithm

The variant of the interior point algorithm implemented in PROC NETFLOW is a Primal-Dual Predictor-Corrector interior point algorithm. At first, Newton’s method is used to find a direction to move , but calculated as if is zero, that is, a step with no centering, known as an affine step:

where is the step length as before.

Complementarity is calculated at and compared with the complementarity at the starting point , and the success of the affine step is gauged. If the affine step was successful in reducing the complementarity by a substantial amount, the need for centering is not great, and the value of in the following linear system is assigned a value close to zero. If, however, the affine step was unsuccessful, centering would be beneficial, and the value of in the following linear system is assigned a value closer to 1.0. The value of is therefore adaptively altered depending on the progress made toward the optimum.

A second linear system is solved to determine a centering vector from

Then

where, as before, is the step length assigned a value as large as possible but not so large that an or is too close to zero.

Although the Predictor-Corrector variant entails solving two linear system instead of one, fewer iterations are usually required to reach the optimum. The additional overhead of calculating the second linear system is small, as the factorization of the matrix has already been performed to solve the first linear system.

#### Stopping Criteria

There are several reasons why PROC NETFLOW stops interior point optimization. Optimization stops when

PROC NETFLOW may stop optimization when it detects that the rate at which the complementarity or dualitygap is being reduced is too slow, that is, there are consecutive iterations when the complementarity or duality gap has stopped getting smaller and the infeasibilities, if nonzero, have also stalled. Sometimes, this indicates the problem is infeasible.

The reasons to stop optimization outlined in the previous paragraph will be termed the usual stopping conditions in the following explanation.

However, when solving some problems, especially if the problems are large, the usual stopping criteria are inappropriate. PROC NETFLOW might stop prematurely. If it were allowed to perform additional optimization, a better solution would be found. On other occasions, PROC NETFLOW might do too much work. A sufficiently good solution might be reached several iterations before PROC NETFLOW eventually stops.

You can see PROC NETFLOW’s progress to the optimum by specifying PRINTLEVEL2=2. PROC NETFLOW will produce a table on the SAS log. A row of the table is generated during each iteration and consists of values of the affine step complementarity, the complementarity of the solution for the next iteration, the total bound infeasibility (see the array in the section Interior Point: Upper Bounds), the total constraint infeasibility (see the array in the section Interior Point Algorithmic Details), and the total dual infeasibility (see the array in the section Interior Point Algorithmic Details). As optimization progresses, the values in all columns should converge to zero.

To tailor stopping criteria to your problem, you can use two sets of parameters: the STOP_x and the KEEPGOING_x parameters. The STOP_x parameters (STOP_C, STOP_DG, STOP_IB, STOP_IC, and STOP_ID) are used to test for some condition at the beginning of each iteration and if met, to stop immediately. The KEEPGOING_x parameters ( KEEPGOING_C, KEEPGOING_DG, KEEPGOING_IB, KEEPGOING_IC, and KEEPGOING_ID) are used when PROC NETFLOW would ordinarily stop but does not if some conditions are not met.

For the sake of conciseness, a set of options will be referred to as the part of the option name they have in common followed by the suffix x. For example, STOP_C, STOP_DG, STOP_IB, STOP_IC, and STOP_ID will collectively be referred to as STOP_x.

At the beginning of each iteration, PROC NETFLOW will test whether complementarity is STOP_C (provided you have specified a STOP_C parameter) and if it is, PROC NETFLOW will stop. If the duality gap is STOP_DG (provided you have specified a STOP_DG parameter), PROC NETFLOW will stop immediately. This is also true for the other STOP_x parameters that are related to infeasibilities, STOP_IB, STOP_IC, and STOP_ID.

For example, if you want PROC NETFLOW to stop optimizing for the usual stopping conditions, plus the additional condition, complementarity 100 or duality gap 0.001, then use

  proc netflow stop_c=100 stop_dg=0.001


If you want PROC NETFLOW to stop optimizing for the usual stopping conditions, plus the additional condition, complementarity 1000 and duality gap 0.001 and constraint infeasibility 0.0001, then use

  proc netflow
and_stop_c=1000 and_stop_dg=0.01 and_stop_ic=0.0001


Unlike the STOP_x parameters that cause PROC NETFLOW to stop when any one of them is satisfied, the corresponding AND_STOP_x parameters ( AND_STOP_C, AND_STOP_DG, AND_STOP_IB, AND_STOP_IC, and AND_STOP_ID) cause PROC NETFLOW to stop only if all (more precisely, all that are specified) options are satisfied. For example, if PROC NETFLOW should stop when

then use

  proc netflow
stop_c=100 stop_dg=0.001
and_stop_c=1000 and_stop_dg=0.01 and_stop_ic=0.0001


Just as the STOP_x parameters have AND_STOP_x partners, the KEEPGOING_x parameters have AND_KEEPGOING_x partners. The role of the KEEPGOING_x and AND_KEEPGOING_x parameters is to prevent optimization from stopping too early, even though a usual stopping criterion is met.

When PROC NETFLOW detects that it should stop for a usual stopping condition, it performs the following tests:

• It will test whether complementarity is > KEEPGOING_C (provided you have specified a KEEPGOING_C parameter), and if it is, PROC NETFLOW will perform more optimization.

• Otherwise, PROC NETFLOW will then test whether the primal-dual gap is > KEEPGOING_DG (provided you have specified a KEEPGOING_DG parameter), and if it is, PROC NETFLOW will perform more optimization.

• Otherwise, PROC NETFLOW will then test whether the total bound infeasibility KEEPGOING_IB (provided you have specified a KEEPGOING_IB parameter), and if it is, PROC NETFLOW will perform more optimization.

• Otherwise, PROC NETFLOW will then test whether the total constraint infeasibility KEEPGOING_IC (provided you have specified a KEEPGOING_IC parameter), and if it is, PROC NETFLOW will perform more optimization.

• Otherwise, PROC NETFLOW will then test whether the total dual infeasibility KEEPGOING_ID (provided you have specified a KEEPGOING_ID parameter), and if it is, PROC NETFLOW will perform more optimization.

• Otherwise it will test whether complementarity is > AND_KEEPGOING_C (provided you have specified an AND_KEEPGOING_C parameter), and the primal-dual gap is > AND_KEEPGOING_DG (provided you have specified an AND_KEEPGOING_DG parameter), and the total bound infeasibility AND_KEEPGOING_IB (provided you have specified an AND_KEEPGOING_IB parameter), and the total constraint infeasibility AND_KEEPGOING_IC (provided you have specified an AND_KEEPGOING_IC parameter) and the total dual infeasibility AND_KEEPGOING_ID (provided you have specified an AND_KEEPGOING_ID parameter), and if it is, PROC NETFLOW will perform more optimization.

If all these tests to decide whether more optimization should be performed are false, optimization is stopped.

For example,

  proc netflow
stop_c=1000
and_stop_c=2000 and_stop_dg=0.01
and_stop_ib=1 and_stop_ic=1 and_stop_id=1
keepgoing_c=1500
and_keepgoing_c=2500 and_keepgoing_dg=0.05
and_keepgoing_ib=1 and_keepgoing_ic=1 and_keepgoing_id=1


At the beginning of each iteration, PROC NETFLOW will stop if

When PROC NETFLOW determines it should stop because a usual stopping condition is met, it will stop only if

#### Interior Point: Upper Bounds

If the LP model had upper bounds ( where is the upper bound vector), then the primal and dual problems, the duality gap, and the KKT conditions would have to be expanded.

The primal linear program to be solved is

where is split into and . Let be primal slack so that , and associate dual variables with these constraints. The interior point algorithm solves the system of equations to satisfy the Karush-Kuhn-Tucker (KKT) conditions for optimality:

These are the conditions for feasibility, with the complementarity conditions and added. Complementarity forces the optimal objectives of the primal and dual to be equal, , as

Before the optimum is reached, a solution might not satisfy the KKT conditions:

• Primal bound constraints may be violated, .

• Primal constraints may be violated, .

• Dual constraints may be violated, .

• Complementarity conditions may not be satisfied, and .

The calculations of the interior point algorithm can easily be derived in a fashion similar to calculations for when an LP has no upper bounds. See the paper by Lustig, Marsten, and Shanno (1992). An important point is that upper bounds can be handled by specializing the algorithm and not by generating the constraints and adding these to the main primal constraints .

#### Getting Started: Linear Programming Models: Interior Point Algorithm

To solve linear programming problem using PROC NETFLOW, you save a representation of the variables and the constraints in one or two SAS data sets. These data sets are then passed to PROC NETFLOW for solution. There are various forms that a problem’s data can take. You can use any one or a combination of several of these forms.

The ARCDATA= data set contains information about the variables of the problem. Although this data set is called ARCDATA, it contains data for no arcs. Instead, all data in this data set are related to variables.

The ARCDATA= data set can be used to specify information about variables, including objective function coefficients, lower and upper value bounds, and names. These data are the elements of the vectors , , and in problem ( NPSC). Data for a variable can be given in more than one observation.

When the data for a constrained network problem is being provided, the ARCDATA= data set always contains information necessary for arcs, their tail and head nodes, and optionally the supply and demand information of these nodes. When the data for a linear programming problem is being provided, none of this information is present, as the model has no arcs. This is the way PROC NETFLOW decides which type of problem it is to solve.

PROC NETFLOW was originally designed to solve models with networks, so an ARCDATA= data set is always expected. If an ARCDATA= data set is not specified, by default the last data set created before PROC NETFLOW is invoked is assumed to be an ARCDATA= data set. However, these characteristics of PROC NETFLOW are not helpful when a linear programming problem is being solved and all data are provided in a single data set specified by the CONDATA= data set, and that data set is not the last data set created before PROC NETFLOW starts. In this case, you must specify that an ARCDATA= data set and a CONDATA= data set are both equal to the input data set. PROC NETFLOW then knows that a linear programming problem is to be solved, and the data reside in one data set.

The CONDATA= data set describes the constraints and their right-hand sides. These data are elements of the matrix and the vector .

Constraint types are also specified in the CONDATA= data set. You can include in this data set variable data such as upper bound values, lower value bounds, and objective function coefficients. It is possible to give all information about some or all variables in the CONDATA= data set.

A variable is identified in this data set by its name. If you specify a variable’s name in the ARCDATA= data set, then this name is used to associate data in the CONDATA= data set with that variable.

If you use the dense constraint input format, these variable names are names of SAS variables in the VAR list of the CONDATA= data set.

If you use the sparse constraint input format, these variable names are values of the COLUMN list SAS variable of CONDATA= data set.

When using the interior point algorithm, the execution of PROC NETFLOW has two stages. In the preliminary (zeroth) stage, the data are read from the ARCDATA= data set (if used) and the CONDATA= data set. Error checking is performed. In the next stage, the linear program is preprocessed, then the optimal solution to the linear program is found. The solution is saved in the CONOUT= data set. This data set is also named in the PROC NETFLOW, RESET, and SAVE statements.

See the section Getting Started: Network Models: Interior Point Algorithm for a fuller description of the stages of the interior point algorithm.

#### Introductory Example: Linear Programming Models: Interior Point Algorithm

Consider the linear programming problem in the section An Introductory Example in the chapter on the LP procedure.

data dcon1;
input _id_ $17. a_light a_heavy brega naphthal naphthai heatingo jet_1 jet_2 _type_$ _rhs_;
datalines;
profit            -175 -165 -205  0  0  0 300 300 max     .
naphtha_l_conv    .035 .030 .045 -1  0  0   0   0  eq     0
naphtha_i_conv    .100 .075 .135  0 -1  0   0   0  eq     0
heating_o_conv    .390 .300 .430  0  0 -1   0   0  eq     0
recipe_1             0    0    0  0 .3 .7  -1   0  eq     0
recipe_2             0    0    0 .2  0 .8   0  -1  eq     0
available          110  165   80  .  .  .   .   . upperbd .
;


To find the minimum cost solution and to examine all or parts of the optimum, you use PRINT statements.

• print problem/short; outputs information for all variables and all constraint coefficients. See Figure 6.19.

• print some_variables(j:)/short; is information about a set of variables, (in this case, those with names that start with the character string preceding the colon). See Figure 6.20.

• print some_cons(recipe_1)/short; is information about a set of constraints (here, that set only has one member, the constraint called recipe_1). See Figure 6.21.

• print con_variables(_all_,brega)/short; lists the constraint information for a set of variables (here, that set only has one member, the variable called brega). See Figure 6.22.

• print con_variables(recipe:,n: jet_1)/short; coefficient information for those in a set of constraints belonging to a set of variables. See Figure 6.23.

proc netflow
arcdata=dcon1
condata=dcon1
conout=solutn1;
run;
print problem/short;

print some_variables(j:)/short;
print some_cons(recipe_1)/short;
print con_variables(_all_,brega)/short;
print con_variables(recipe:,n: jet_1)/short;


The following messages, which appear on the SAS log, summarize the model as read by PROC NETFLOW and note the progress toward a solution:

 NOTE: ARCDATA (or the last data set created if ARCDATA was not specified) and CONDATA are the same data set WORK.DCON1 so will assume a Linear Programming problem is to be solved. NOTE: Number of variables= 8 . NOTE: Number of <= constraints= 0 . NOTE: Number of == constraints= 5 . NOTE: Number of >= constraints= 0 . NOTE: Number of constraint coefficients= 18 . NOTE: After preprocessing, number of <= constraints= 0. NOTE: After preprocessing, number of == constraints= 0. NOTE: After preprocessing, number of >= constraints= 0. NOTE: The preprocessor eliminated 5 constraints from the problem. NOTE: The preprocessor eliminated 18 constraint coefficients from the problem. NOTE: After preprocessing, number of variables= 0. NOTE: The preprocessor eliminated 8 variables from the problem. NOTE: The optimum has been determined by the Preprocessor. NOTE: Objective= 1544. NOTE: The data set WORK.SOLUTN1 has 8 observations and 6 variables.

Figure 6.19: PRINT PROBLEM/SHORT;

The NETFLOW Procedure

_N_ _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_
1 a_heavy -165 165 0 0
2 a_light -175 110 0 110
3 brega -205 80 0 80
4 heatingo 0 99999999 0 77.3
5 jet_1 300 99999999 0 60.65
6 jet_2 300 99999999 0 63.33
7 naphthai 0 99999999 0 21.8
8 naphthal 0 99999999 0 7.45

The NETFLOW Procedure

_N_ _id_ _type_ _rhs_ _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_ _COEF_
1 heating_o_conv EQ 0 a_light -175 110 0 110 0.39
2 heating_o_conv EQ 0 a_heavy -165 165 0 0 0.3
3 heating_o_conv EQ 0 brega -205 80 0 80 0.43
4 heating_o_conv EQ 0 heatingo 0 99999999 0 77.3 -1
5 naphtha_i_conv EQ 0 a_light -175 110 0 110 0.1
6 naphtha_i_conv EQ 0 a_heavy -165 165 0 0 0.075
7 naphtha_i_conv EQ 0 brega -205 80 0 80 0.135
8 naphtha_i_conv EQ 0 naphthai 0 99999999 0 21.8 -1
9 naphtha_l_conv EQ 0 a_light -175 110 0 110 0.035
10 naphtha_l_conv EQ 0 a_heavy -165 165 0 0 0.03
11 naphtha_l_conv EQ 0 brega -205 80 0 80 0.045
12 naphtha_l_conv EQ 0 naphthal 0 99999999 0 7.45 -1
13 recipe_1 EQ 0 naphthai 0 99999999 0 21.8 0.3
14 recipe_1 EQ 0 heatingo 0 99999999 0 77.3 0.7
15 recipe_1 EQ 0 jet_1 300 99999999 0 60.65 -1
16 recipe_2 EQ 0 naphthal 0 99999999 0 7.45 0.2
17 recipe_2 EQ 0 heatingo 0 99999999 0 77.3 0.8
18 recipe_2 EQ 0 jet_2 300 99999999 0 63.33 -1

Figure 6.20: PRINT SOME_VARIABLES(J:)/SHORT;

The NETFLOW Procedure

_N_ _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_
1 jet_1 300 99999999 0 60.65
2 jet_2 300 99999999 0 63.33

Figure 6.21: PRINT SOME_CONS(RECIPE_1)/SHORT;

The NETFLOW Procedure

_N_ _id_ _type_ _rhs_ _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_ _COEF_
1 recipe_1 EQ 0 naphthai 0 99999999 0 21.8 0.3
2 recipe_1 EQ 0 heatingo 0 99999999 0 77.3 0.7
3 recipe_1 EQ 0 jet_1 300 99999999 0 60.65 -1

Figure 6.22: PRINT CON_VARIABLES(_ALL_,BREGA)/SHORT;

The NETFLOW Procedure

_N_ _id_ _type_ _rhs_ _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_ _COEF_
1 heating_o_conv EQ 0 brega -205 80 0 80 0.43
2 naphtha_i_conv EQ 0 brega -205 80 0 80 0.135
3 naphtha_l_conv EQ 0 brega -205 80 0 80 0.045

Figure 6.23: PRINT CON_VARIABLES(RECIPE:,N: JET_1)/SHORT;

The NETFLOW Procedure

_N_ _id_ _type_ _rhs_ _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_ _COEF_
1 recipe_1 EQ 0 naphthai 0 99999999 0 21.8 0.3
2 recipe_1 EQ 0 jet_1 300 99999999 0 60.65 -1
3 recipe_2 EQ 0 naphthal 0 99999999 0 7.45 0.2

Unlike PROC LP, which displays the solution and other information as output, PROC NETFLOW saves the optimum in output SAS data sets you specify. For this example, the solution is saved in the SOLUTN1 data set. It can be displayed with PROC PRINT as


proc print data=solutn1;
var _name_ _objfn_ _upperbd _lowerbd _value_ _fcost_;
sum _fcost_;
title3 'LP Optimum';
run;


Notice, in the CONOUT=SOLUTN1 (Figure 6.24), the optimal value through each variable in the linear program is given in the variable named _VALUE_, and the cost of value for each variable is given in the variable _FCOST_.

Figure 6.24: CONOUT=SOLUTN1

 LP Optimum

Obs _NAME_ _OBJFN_ _UPPERBD _LOWERBD _VALUE_ _FCOST_
1 a_heavy -165 165 0 0.00 0
2 a_light -175 110 0 110.00 -19250
3 brega -205 80 0 80.00 -16400
4 heatingo 0 99999999 0 77.30 0
5 jet_1 300 99999999 0 60.65 18195
6 jet_2 300 99999999 0 63.33 18999
7 naphthai 0 99999999 0 21.80 0
8 naphthal 0 99999999 0 7.45 0
1544

The same model can be specified in the sparse format as in the following scon2 data set. This format enables you to omit the zero coefficients.

data scon2;
format _type_ $8. _col_$8. _row_ $16. ; input _type_$ _col_ $_row_$ _coef_;
datalines;
max      .             profit                    .
eq       .             napha_l_conv              .
eq       .             napha_i_conv              .
eq       .             heating_oil_conv          .
eq       .             recipe_1                  .
eq       .             recipe_2                  .
upperbd  .             available                 .
.        a_light       profit                 -175
.        a_light       napha_l_conv           .035
.        a_light       napha_i_conv           .100
.        a_light       heating_oil_conv       .390
.        a_light       available               110
.        a_heavy       profit                 -165
.        a_heavy       napha_l_conv           .030
.        a_heavy       napha_i_conv           .075
.        a_heavy       heating_oil_conv       .300
.        a_heavy       available               165
.        brega         profit                 -205
.        brega         napha_l_conv           .045
.        brega         napha_i_conv           .135
.        brega         heating_oil_conv       .430
.        brega         available                80
.        naphthal      napha_l_conv             -1
.        naphthal      recipe_2                 .2
.        naphthai      napha_i_conv             -1
.        naphthai      recipe_1                 .3
.        heatingo      heating_oil_conv         -1
.        heatingo      recipe_1                 .7
.        heatingo      recipe_2                 .8
.        jet_1         profit                  300
.        jet_1         recipe_1                 -1
.        jet_2         profit                  300
.        jet_2         recipe_2                 -1
;


To find the minimum cost solution, invoke PROC NETFLOW (note the SPARSECONDATA option which must be specified) as follows:

proc netflow
sparsecondata
condata=scon2
conout=solutn2;
run;


A data set that is used as an ARCDATA= data set can be initialized as follows:

data vars3;
input _name_ $profit available; datalines; a_heavy -165 165 a_light -175 110 brega -205 80 heatingo 0 . jet_1 300 . jet_2 300 . naphthai 0 . naphthal 0 . ;  The following CONDATA= data set is the original dense format CONDATA= dcon1 data set with the variable information removed. (You could have left some or all of that information in CONDATA as PROC NETFLOW merges data, but doing that and checking for consistency uses time.) data dcon3; input _id_$17.
a_light a_heavy brega naphthal naphthai
heatingo jet_1 jet_2
_type_ $_rhs_; datalines; naphtha_l_conv .035 .030 .045 -1 0 0 0 0 eq 0 naphtha_i_conv .100 .075 .135 0 -1 0 0 0 eq 0 heating_o_conv .390 .300 .430 0 0 -1 0 0 eq 0 recipe_1 0 0 0 0 .3 .7 -1 0 eq 0 recipe_2 0 0 0 .2 0 .8 0 -1 eq 0 ;  It is important to note that it is now necessary to specify the MAXIMIZE option; otherwise, PROC NETFLOW will optimize to the minimum (which, incidentally, has a total objective = -3539.25). You must indicate that the SAS variable profit in the ARCDATA=vars3 data set has values that are objective function coefficients, by specifying the OBJFN statement. The UPPERBD must be specified as the SAS variable available that has as values upper bounds. proc netflow maximize /* ***** necessary ***** */ arcdata=vars3 condata=dcon3 conout=solutn3; objfn profit; upperbd available; run;  The ARCDATA=vars3 data set can become more concise by noting that the model variables heatingo, naphthai, and naphthal have zero objective function coefficients (the default) and default upper bounds, so those observations need not be present. data vars4; input _name_$ profit available;
datalines;
a_heavy  -165 165
a_light  -175 110
brega    -205  80
jet_1     300   .
jet_2     300   .
;


The CONDATA=dcon3 data set can become more concise by noting that all the constraints have the same type (eq) and zero (the default) rhs values. This model is a good candidate for using the DEFCONTYPE= option.

The DEFCONTYPE= option can be useful not only when all constraints have the same type as is the case here, but also when most constraints have the same type, or if you prefer to change the default type from to = or . The essential constraint type data in CONDATA= data set is that which overrides the DEFCONTYPE= type you specified.

data dcon4;
input _id_ $17. a_light a_heavy brega naphthal naphthai heatingo jet_1 jet_2; datalines; naphtha_l_conv .035 .030 .045 -1 0 0 0 0 naphtha_i_conv .100 .075 .135 0 -1 0 0 0 heating_o_conv .390 .300 .430 0 0 -1 0 0 recipe_1 0 0 0 0 .3 .7 -1 0 recipe_2 0 0 0 .2 0 .8 0 -1 ; proc netflow maximize defcontype=eq arcdata=vars3 condata=dcon3 conout=solutn3; objfn profit; upperbd available; run;  Several different ways of using an ARCDATA= data set and a sparse format CONDATA= data set for this linear program follow. The following CONDATA= data set is the result of removing the profit and available data from the original sparse format CONDATA=scon2 data set. data scon5; format _type_$8. _col_ $8. _row_$16. ;
input _type_ $_col_$ _row_ $_coef_; datalines; eq . napha_l_conv . eq . napha_i_conv . eq . heating_oil_conv . eq . recipe_1 . eq . recipe_2 . . a_light napha_l_conv .035 . a_light napha_i_conv .100 . a_light heating_oil_conv .390 . a_heavy napha_l_conv .030 . a_heavy napha_i_conv .075 . a_heavy heating_oil_conv .300 . brega napha_l_conv .045 . brega napha_i_conv .135 . brega heating_oil_conv .430 . naphthal napha_l_conv -1 . naphthal recipe_2 .2 . naphthai napha_i_conv -1 . naphthai recipe_1 .3 . heatingo heating_oil_conv -1 . heatingo recipe_1 .7 . heatingo recipe_2 .8 . jet_1 recipe_1 -1 . jet_2 recipe_2 -1 ; proc netflow maximize sparsecondata arcdata=vars3 /* or arcdata=vars4 */ condata=scon5 conout=solutn5; objfn profit; upperbd available; run;  The CONDATA=scon5 data set can become more concise by noting that all the constraints have the same type (eq) and zero (the default) rhs values. Use the DEFCONTYPE= option again. Once the first 5 observations of the CONDATA=scon5 data set are removed, the _type_ SAS variable has values that are missing in the remaining observations. Therefore, this SAS variable can be removed. data scon6; input _col_$ _row_&\$16. _coef_;
datalines;
a_light  napha_l_conv           .035
a_light  napha_i_conv           .100
a_light  heating_oil_conv       .390
a_heavy  napha_l_conv           .030
a_heavy  napha_i_conv           .075
a_heavy  heating_oil_conv       .300
brega    napha_l_conv           .045
brega    napha_i_conv           .135
brega    heating_oil_conv       .430
naphthal napha_l_conv             -1
naphthal recipe_2                 .2
naphthai napha_i_conv             -1
naphthai recipe_1                 .3
heatingo heating_oil_conv         -1
heatingo recipe_1                 .7
heatingo recipe_2                 .8
jet_1    recipe_1                 -1
jet_2    recipe_2                 -1
;

proc netflow
maximize
defcontype=eq
sparsecondata
arcdata=vars3     /* or arcdata=vars4 */
condata=scon6
conout=solutn6;
objfn profit;
upperbd available;
run;


#### Interactivity: Linear Programming Models: Interior Point algorithm

PROC NETFLOW can be used interactively. You begin by giving the PROC NETFLOW statement, and you must specify the CONDATA= data set. If necessary, specify the ARCDATA= data set.

The variable lists should be given next. If you have variables in the input data sets that have special names (for example, a variable in the ARCDATA= data set named _COST_ that has objective function coefficients as values), it may not be necessary to have many or any variable lists.

The PRINT, QUIT, SAVE, SHOW, RESET, and RUN statements follow and can be listed in any order. The QUIT statements can be used only once. The others can be used as many times as needed.

The CONOPT and PIVOT are not relevant to the interior point algorithm and should not be used.

Use the RESET or SAVE statement to change the name of the output data set. There is only one output data set, the CONOUT= data set. With the RESET statement, you can also indicate the reasons why optimization should stop, (for example, you can indicate the maximum number of iterations that can be performed). PROC NETFLOW then has a chance to either execute the next statement or, if the next statement is one that PROC NETFLOW does not recognize (the next PROC or DATA step in the SAS session), do any allowed optimization and finish. If no new statement has been submitted, you are prompted for one. Some options of the RESET statement enable you to control aspects of the interior point algorithm. Specifying certain values for these options can reduce the time it takes to solve a problem. Note that any of the RESET options can be specified in the PROC NETFLOW statement.

The RUN statement starts optimization. Once the optimization has started, it runs until the optimum is reached. The RUN statement should be specified at most once.

The QUIT statement immediately stops PROC NETFLOW. The SAVE statement has options that enable you to name the output data set; information about the current solution is saved in this output data set. Use the SHOW statement if you want to examine the values of options of other statements. Information about the amount of optimization that has been done and the STATUS of the current solution can also be displayed using the SHOW statement.

The PRINT statement instructs PROC NETFLOW to display parts of the problem. The ways the PRINT statements are specified are identical whether the interior point algorithm or the simplex algorithm is used; however, there are minor differences in what is displayed for each variable or constraint coefficient.

PRINT VARIABLES produces information on all arcs. PRINT SOME_VARIABLES limits this output to a subset of variables. There are similar PRINT statements for constraints:

  PRINT CONSTRAINTS;
PRINT SOME_CONS;


PRINT CON_VARIABLES enables you to limit constraint information that is obtained to members of a set of variables that have nonzero constraint coefficients in a set of constraints.

For example, an interactive PROC NETFLOW run might look something like this:

  proc netflow
condata=data set
other options;
variable list specifications;     /* if necessary */
reset options;
print options;    /* look at problem              */
run;                 /* do some optimization         */
print options;    /* look at the optimal solution */
save options;     /* keep optimal solution        */


If you are interested only in finding the optimal solution, have used SAS variables that have special names in the input data sets, and want to use default setting for everything, then the following statement is all you need:

proc netflow condata= data set ;

#### Functional Summary: Linear Programming Models: Interior Point Algorithm

The following table outlines the options available for the NETFLOW procedure when the interior point algorithm is being used to solve a linear programming problem, classified by function.

Table 6.8: Functional Summary, Linear Programming Models

Description

Statement

Option

Input Data Set Options:

Arcs input data set

Constraint input data set

Output Data Set Option:

Solution data set

CONDATA has sparse data format

Default constraint type

Special COLUMN variable value

Special COLUMN variable value

Data for a constraint found once in CONDATA

Data for a coefficient found once in CONDATA

Data are grouped, exploited during data read

Problem Size Specification Options:

Approximate number of variables

Approximate number of coefficients

Approximate number of constraints

Network Options:

Default variable objective function coefficient

Default variable upper bound

Default variable lower bound

Memory Control Options:

Issue memory usage messages to SAS log

Number of bytes to use for main memory

Proportion of memory for arrays

maximum bytes for a single array

Interior Point Algorithm Options:

Use interior point algorithm

Factorization method

Allowed amount of dual infeasibility

Allowed amount of primal infeasibility

Allowed total amount of dual infeasibility

Allowed total amount of primal infeasibility

Cut-off tolerance for Cholesky factorization

Density threshold for Cholesky processing

Step-length multiplier

Preprocessing type

Print optimization progress on SAS log

Write optimization time to SAS log

Interior Point Stopping Criteria Options:

Maximum number of interior point iterations

Primal-dual (duality) gap tolerance

Stop because of complementarity

Stop because of duality gap

Stop because of

Stop because of

Stop because of

Stop because of complementarity

Stop because of duality gap

Stop because of

Stop because of

Stop because of

Stop because of complementarity

Stop because of duality gap

Stop because of

Stop because of

Stop because of

Stop because of complementarity

Stop because of duality gap

Stop because of

Stop because of

Stop because of

PRINT Statement Options:

Display everything

Display variable information

Display constraint information

Display information for some variables

Display information for some constraints

Display information for some constraints associated with some variables

PRINT Statement Qualifiers:

Produce a short report

Produce a long report

Display arcs/variables with zero flow/value

Display arcs/variables with nonzero flow/value

SHOW Statement Options:

Show problem, optimization status

Show LP model parameters

NETSTMT

Show data sets that have been or will be created

Miscellaneous Options:

Infinity value

Scale constraint row, variable column coefficients, or both