Linear Programming Models: Interior Point Algorithm

The NETFLOW Procedure

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 g variables, and k constraints, then the formal statement of the problem solved by PROC NETFLOW is

: min {d^T z}
subject to: $Q z \geq, =, \leq r$
: $m \leq z \leq v$

where

: d is the g x 1 objective function coefficient of variables vector.
: z is the g x 1 variable value vector.
: Q is the k x g constraint coefficient matrix for variables, where Q_i,j is the coefficient of variable j in the ith constraint.
: r is the k x 1 side constraint right-hand-side vector.
: m is the g x 1 variable value lower bound vector.
: v is the g x 1 variable value upper bound vector.

Interior Point Algorithmic Details

After preprocessing, the Linear Program to be solved is

: min {c^T x}
subject to: A x = b
: $x \geq 0$

This is the primal problem. The matrices of d z Q of NP have been renamed c x and A 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 n variables. i is a variable number. k is an iteration number, and if used as a subscript or superscript denotes "of iteration k".

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

: max {b^T y}
subject to: A^T y + s = c
: $s \geq 0$
where: y are dual variables, and s are dual constraint slacks

What the Interior Point has to do is solve the system of equations to satisfy the Karush-Kuhn-Tucker (KKT) conditions for optimality:

: A x = b
: A^T y + s = c
: x^T s = 0
: $x \geq 0$
: $s \geq 0$

These are the conditions for feasibility, with the complementarity condition x^T s = 0 added. c^T x = b^T y must occur at the optimum. Complementarity forces the optimal objectives of the primal and dual to be equal, c^T x_opt = b^T y_opt, as

: 0 = x^T_opt s_opt = s^T_opt x_opt = (c - A^T y_opt)^T x_opt = c^T x_opt - y^T_opt (A x_opt) = c^T x_opt - b^T y_opt
: 0 = c^T x_opt - b^T y_opt

Before the optimum is reached, a solution (x, y, s) may not satisfy the KKT conditions.

Primal constraints can be broken, $infeas_b = b - A x \neq 0$ .
Dual constraints can be broken, $infeas_c = c - A^T y - s \neq 0$ .
Complementarity is unsatisfied, $x^T s = c^T x - b^T y \neq 0$ .This is called the duality gap.

Interior Point algorithm works by using Newtons method to find a direction to move $(\Delta x^k, \Delta y^k, \Delta s^k)$ from the current solution (x^k, y^k, s^k) toward a better solution.

: $(x^{k+1}, y^{k+1}, s^{k+1}) = (x^k, y^k, s^k) + \alpha (\Delta x^k, \Delta y^k, \Delta s^k)$

$\alpha$ is the step length and is assigned a value as large as possible but $\leq 1.0$ and not so large that a x^k+1_i or s^k+1_i is "too close" to zero. The direction in which to move is found using

: $A \Delta x^k = -infeas_b$
: $A^T \Delta y^k + \Delta s^k = -infeas_c$
: $S^k \Delta x^k + X^k \Delta s^k = - X^k S^k e$
where: S = diag(s), X = diag(x), and e = vector with all elements = 1

To greatly improve performance, the third equation is changed to

: $S^k \Delta x^k + X^k \Delta s^k = - X^k S^k e + \sigma_k \mu_k e$
where: $\mu_k = 1/n X^k S^k e$ , the average complementarity, and
: $0 \leq \sigma_k \leq 1$

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 x^k_i s^k_i is too close to zero, it is "nudged out" to $\mu$ ,any x^k_i s^k_i that is larger than $\mu$ is "nudged into" $\mu$ .A $\sigma_k$ close to or equal to 0.0 biases a direction toward the optimum, and a value for $\sigma_k$ close to or equal to 1.0 "centers" the direction toward a point where all pairwise products $x^k_i s^k_i = \mu$ .Such points make up the Central Path in the interior. Although centering directions make little, if any, progress in reducing $\mu$ 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.

That in a nutshell is the Primal-Dual Interior Point algorithm. Varieties of the algorithm differ in the way $\alpha$ and $\sigma_k$ are chosen and the direction adjusted each iteration. A wealth of information can be found in the texts by Roos, Terlaky, and Vial 1997, Wright 1996, and Ye 1996.

The calculation of the direction is the most time-consuming step of the Interior Point algorithm. Assume the kth iteration is being performed, so the subscript and superscript k can be dropped from the algebra.

: $A \Delta x = -infeas_b$
: $A^T \Delta y + \Delta s = -infeas_c$
: $S \Delta x + X \Delta s = - X S e + \sigma \mu e$

Rearranging the second equation

: $\Delta s = -infeas_c - A^T \Delta y$

Rearranging the third equation

: $\Delta s = X^{-1}(- S \Delta x - X S e + \sigma \mu e)$
: $\Delta s = - \Theta \Delta x - S e + X^{-1} \sigma \mu e$
where: $\Theta = S X^{-1}$

Equating these two expressions for $\Delta s$ and rearranging

: $-\Theta \Delta x - S e + X^{-1} \sigma \mu e = -infeas_c - A^T \Delta y$
: $-\Theta \Delta x = S e - X^{-1} \sigma \mu e - infeas_c - A^T \Delta y$
: $\Delta x = \Theta^{-1}(-S e + X^{-1} \sigma \mu e + infeas_c + A^T \Delta y)$
: $\Delta x = \rho + \Theta^{-1} A^T \Delta y$
where: $\rho = \Theta^{-1}(-S e + X^{-1} \sigma \mu e + infeas_c)$

Substituting into the first direction equation

: $A \Delta x = -infeas_b$
: $A (\rho + \Theta^{-1} A^T \Delta y) = -infeas_b$
: $A \Theta^{-1} A^T \Delta y = -infeas_b - A \rho$
: $\Delta y = (A \Theta^{-1} A^T)^{-1}(-infeas_b - A \rho)$

$\Theta$ , $\rho$ , $\Delta y$ , $\Delta x$ and $\Delta s$ are calculated in that order. The hardest term is the factorization of the $(A \Theta^{-1} A^T)$ matrix to determine $\Delta y$ . Fortunately, although the values of $(A \Theta^{-1} A^T)$ is different each iteration, the locations of the nonzeroes in this matrix remain fixed; the nonzero locations is the same as those in the matrix (A A^T). This is due to $\Theta^{-1} = X S^{-1}$ being a diagonal matrix that has the effect of merely scaling the columns of (A A^T).

The fact that the nonzeroes in $A \Theta^{-1} A^T$ has a constant pattern is exploited by all Interior Point algorithms, and is a major reason for their excellent performance. Before iterations begin, A A^T is examined and it's rows and columns permutated so that during Cholesky Factorization, the number of fillins 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.

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 d, m, and v in problem (NP). Data for an 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 envoked 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 is 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 Q and the vector r.

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 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 "Getting Started: Network Models: Interior Point algorithm" section 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 "An Introductory Example" section in the chapter on the LP procedure.

data dcon1;
   input _id_ $14.
         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 4.19.
print some_variables(j:)/short; is information about a set of variables, (in this case, those with names that start with the character string (here, the single character "j" (without the quotes)) preceeding the colon. See Figure 4.20.
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 4.21.
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 4.22.
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 4.23.

   proc netflow
      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, that appear on the SAS log, summarize the model as read by PROC NETFLOW and note the progress toward a solution:

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: 0 columns, 0 rows and 0 coefficients were added
      to the problem to handle unrestricted variables, 
      variables that are split, and constraint slack or 
      surplus variables.
NOTE: There are 8 nonzero elements in A * A transpose.
NOTE: Number of fill-ins=0.
NOTE: Of the 5 rows and columns, 0 are sparse.
NOTE: There are 0 nonzero elements in the sparse rows of 
      the factored A * A transpose. This includes fill-ins 
      in the sparse rows.
NOTE: There are 0 operations of the form 
      u[i,j]=u[i,j]-u[q,j]*u[q,i]/u[q,q] to factorize the 
      sparse rows of A * A transpose.
NOTE: Constraint feasibility attained by iteration 3.
NOTE: Bound feasibility attained by iteration 3.
NOTE: Dual feasibility attained by iteration 3.
NOTE: Primal-Dual Predictor-Corrector Interior point 
      algorithm performed 8 iterations.
NOTE: Objective = 1544.0000121.
NOTE: The data set WORK.SOLUTN1 has 8 observations and 
      6 variables.

NETFLOW PROCEDURE

_N_	_NAME_	_OBJFN_	_UPPERBD	_LOWERBD	_VALUE_
1	a_heavy	-165	165	0	5.583E-7
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

NETFLOW PROCEDURE

_N_	_id_	_type_	_rhs_	_NAME_	_COST_	_CAPAC_	_LO_	_VALUE_	_COEF_
1	heating_o_conv	EQ	0	a_heavy	-165	165	0	5.583E-7	0.3
2	heating_o_conv	EQ	0	a_light	-175	110	0	110	0.39
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_heavy	-165	165	0	5.583E-7	0.075
6	naphtha_i_conv	EQ	0	a_light	-175	110	0	110	0.1
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_heavy	-165	165	0	5.583E-7	0.03
10	naphtha_l_conv	EQ	0	a_light	-175	110	0	110	0.035
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	heatingo	0	99999999	0	77.3	0.7
14	recipe_1	EQ	0	jet_1	300	99999999	0	60.65	-1
15	recipe_1	EQ	0	naphthai	0	99999999	0	21.8	0.3
16	recipe_2	EQ	0	heatingo	0	99999999	0	77.3	0.8
17	recipe_2	EQ	0	jet_2	300	99999999	0	63.33	-1
18	recipe_2	EQ	0	naphthal	0	99999999	0	7.45	0.2

Figure 4.19: print problem/short;

NETFLOW PROCEDURE

_N_	_NAME_	_COST_	_CAPAC_	_LO_	_VALUE_
1	jet_1	300	99999999	0	60.65
2	jet_2	300	99999999	0	63.33

Figure 4.20: print some_variables(j:)/short;

NETFLOW PROCEDURE

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

Figure 4.21: print some_cons(recipe_1)/short;

NETFLOW PROCEDURE

_N_	_id_	_type_	_rhs_	_NAME_	_COST_	_CAPAC_	_LO_	_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 4.22: print con_variables(_all_,brega)/short;

NETFLOW PROCEDURE

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

Figure 4.23: print con_variables(recipe:,n: jet_1)/short;

Unlike PROC LP, that 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 SOLUTION data set. It can be displayed with PROC PRINT as

   proc print data=solutn1;
      var _name_ _cost_ _capac_ _lo_ _flow_ _fcost_;
      sum _fcost_;
      title3 'LP Optimum'; run;

Notice, in the CONOUT=SOLUTION (Figure 4.24), the optimal value through each variable in the Linear program is given in the variable named _FLOW_, and the cost of value for each variable is given in the variable _FCOST_.

LP Optimum

Obs	_NAME_	_COST_	_CAPAC_	_LO_	_FLOW_	_FCOST_
1	a_heavy	-165	165	0	0.000	-0.00
2	a_light	-175	110	0	110.000	-19250.00
3	brega	-205	80	0	80.000	-16400.00
4	heatingo	0	99999999	0	77.300	0.00
5	jet_1	300	99999999	0	60.650	18195.00
6	jet_2	300	99999999	0	63.330	18999.00
7	naphthai	0	99999999	0	21.800	0.00
8	naphthal	0	99999999	0	7.450	0.00
						1544.00

Figure 4.24: CONOUT=SOLUTN1

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

data scon2;
   input _type_ $ @10 _col_ $13. @24 _row_ $16. _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_ $14.
         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, incidently, 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= options.

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 when you prefer to change the default type from $\leq$ to = or $\geq$ .The essential constraint type data in CONDATA= data set is that which overrides the DEFCONTYPE= type you specified.

data dcon4;
   input _id_ $14.
         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 follows. 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;
   input _type_ $ @10 _col_ $13. @24 _row_ $16. _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=vars4        /* 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 variables and 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 options ;

Functional Summary: Linear Programming Models: Interior Point algorithm

The following tables outline 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 4.36: Input Data Set Options

Description	Statement	Option
arcs input data set	NETFLOW	ARCDATA=
constraint input data set	NETFLOW	CONDATA=

Table 4.37: Options for Networks

Description	Statement	Option
default variable objective function coefficient	NETFLOW	DEFCOST=
default variable upper bound	NETFLOW	DEFCAPACITY=
default variable lower bound	NETFLOW	DEFMINFLOW=

Table 4.38: Miscellaneous Options

Description	Statement	Option
infinity value	NETFLOW	INFINITY=
do constraint row and/or variable column coefficient	NETFLOW	SCALE=
scaling, or neither
maximization instead of minimization	NETFLOW	MAXIMIZE

Table 4.39: Data Set Read Options

Description	Statement	Option
CONDATA has sparse data format	NETFLOW	SPARSECONDATA
default constraint type	NETFLOW	DEFCONTYPE=
special COLUMN variable value	NETFLOW	TYPEOBS=
special COLUMN variable value	NETFLOW	RHSOBS=
data for an constraint found in only one obs of CONDATA	NETFLOW	CON_SINGLE_OBS
data for a coefficient found once in CONDATA	NETFLOW	NON_REPLIC=
data is grouped, exploited during data read	NETFLOW	GROUPED=

Table 4.40: Problem Size (approx.) Options

Description	Statement	Option
number of variables	NETFLOW	NNAS=
number of coefficients	NETFLOW	NCOEFS=
number of constraints	NETFLOW	NCONS=

Table 4.41: Memory Control Options

Description	Statement	Option
issue memory usage messages to SASLOG	NETFLOW	MEMREP
number of bytes to use for main memory	NETFLOW	BYTES=
proportion of memory used by frequently	NETFLOW	COREFACTOR=
accessed arrays
maximum bytes for a single array	NETFLOW	MAXARRAYBYTES=

Table 4.42: Output Data Set Options: RESET

Description	Statement	Option
solution data set	RESET	CONOUT=

Table 4.43: PRINT Statement Options

Description	Statement	Option
display everything	PRINT	PROBLEM
display variables information	PRINT	VARIABLES
display constraint information	PRINT	CONSTRAINTS
display information for some variables	PRINT	SOME_VARIABLES
display information for some constraints	PRINT	SOME_CONS
display information for some constraints	PRINT	CON_VARIABLES
associated with some variables

Table 4.44: PRINT Statement Qualifiers

Description	Statement	Option
produce a short report	PRINT	/ SHORT
produce a long report	PRINT	/ LONG
only variables with zero value	PRINT	/ ZERO
only variables with nonzero value	PRINT	/ NONZERO

Table 4.45: SHOW Statement Options

Description	Statement	Option
show problem, optimization status	SHOW	STATUS
show LP model parameters	SHOW	NETSTMT
show data sets that have, will be created	SHOW	DATA SETS

Table 4.46: Output Data Set Options: SAVE

Description	Statement	Option
constrained solution data set	SAVE	CONOUT=

Table 4.47: Interior Point algorithm Options

Description	Statement	Option
use Interior Point algorithm	NETFLOW	INTPOINT
allowed amount of dual infeasibility	RESET	TOLDINF=
allowed amount of primal infeasibility	RESET	TOLPINF=
cut-off tolerance for Cholesky factorization	RESET	CHOLTINYTOL=
density threshold for Cholesky processing	RESET	DENSETHR=
maximum number of Interior Point algorithm iterations	RESET	MAXITERB=
Primal-Dual (Duality) gap tolerance	RESET	PDGAPTOL=
step-length multiplier	RESET	PDSTEPMULT=
preprocessing type	RESET	PRSLTYPE=

Syntax: Linear Programming Models: Interior Point algorithm

Below are statements used in PROC NETFLOW, listed in alphabetical order as they appear in the text that follows.

PROC NETFLOW options ;: CAPACITY variable ;; COEF variables ;; COLUMN variable ;; COST variable ;; DEMAND variable ;; ID variables ;; LO variable ;; NAME variable ;; PRINT options ;; QUIT;; RESET options ;; ROW variables ;; RHS variables ;; RUN;; SAVE options ;; TYPE variable ;; VAR variables ;

${ . {PROC \; NETFLOW {\: options};}. } & { {\rm required \: statement} } \ & \... ...\\;\;\;\;\;\;\;\;\;\;\: \} } & { {\rm optional \:interactive \:statements} } \$

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