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Posterior constraint optimal selection techniques (COSTs) are developed for nonnegative linear programming problems (NNLPs), and a geometric interpretation is provided. The posterior approach is used in both a dynamic and non-dynamic active-set framework. The computational performance of these methods is compared with the CPLEX standard linear programming algorithms, with two most-violated constraint approaches, and with previously developed COST algorithms for large-scale problems.

Consider the linear programming (LP) problem

( P ) Maximize z = c T x (1)

subject to

A x ≤ b (2)

x ≥ 0 , (3)

where c and x are n-dimensional column vectors of objective coefficients and variables respectively; A is an m × n matrix [ a i j ] with 1 × n row vectors a i , i = 1 , ⋯ , m ; b is an m × 1 column vector; and 0 is an n × 1 vector of zeros.

The non-polynomial simplex methods and the polynomial interior-point barrier-function algorithms are currently the principal two-solution approaches for solving problem P , but for either there are problem instances for which it performs poorly [

In this paper we consider the nonnegative linear programming problem (NNLP), which is the special case of P with a i ≥ 0 but a i ≠ 0 , i = 1 , ⋯ , m ; b > 0 ; and c > 0. NNLPs model a large number of linear programming applications such as determining an optimal driving path for navigation systems using traffic data [

1) the origin x = 0 is feasible,

2) x j ≤ min i = 1 , ⋯ , m { b i a i j : a i j > 0 } , j = 1 , ⋯ , n .

Thus NNLPs have a bounded feasible region and bounded objective function if and only if no column of A is a zero vector. It follows that the boundedness of an NNLP objective function is easily verifiable without computation.

We propose here an active-set method to solve nonnegative linear programming problems faster than current approaches. Our method divides the constraints of problem P into operative and inoperative constraints at each active-set iteration. Operative constraints are those active in the current relaxed subproblem P r , r = 1 , 2 , ⋯ , of P at iteration r , while the inoperative ones are constraints of the problem P not active in P r . In our active-set method we iteratively solve P r , r = 1 , 2 , ⋯ , of P after adding one or more violated inoperative constraints from (2) to P r − 1 until the solution x r * to P r is a solution to P .

Active-set methods have been studied by Stone [

More recently, Corley et al. [

References [

R A D ( a i , b i , c ) = a i c b i (4)

for NNLPs. RAD is a geometric constraint selection criterion for determining the constraints most likely to be binding at optimality. In the associated active-set algorithm of [

In this paper a posterior constraint selection metric NVRAD is developed for NNLPs. NVRAD may be considered as a posterior version of RAD. The posterior NVRAD is then implemented in the dynamic framework of [

More specifically, in Section 2 we state the posterior constraint selection metric NVRAD and provide a geometric interpretation. A dynamic version of NVRAD for NNLPs is then developed. In Section 2 we extend NVRAD to a hybrid approach HYBR, where RAD and NVRAD are alternated. In Section 3, computational results are presented. NVRAD is shown to be significantly faster for NNLPs than all CPLEX solvers, as well as faster than VIOL and RAD. HYBR appears slightly faster than NVRAD. In Section 4, we present conclusions. Throughout the paper, both a constraint selection metric and the associated active-set algorithm are identified by the same name-RAD or NVRAD, for example. The use the term should be clear from context. The active-set algorithm itself is called a COST, i.e., a “Constraint Optimal Selection Technique”.

Let x r * be the current optimal solution for some P r with a perpendicular dis-

tance d = a i x r * − b i ‖ a i ‖ to a violated hyperplane a i x = b i . It follows that

d b i ‖ a i ‖ = a i x r * − b i b i . (5)

Note that b i ‖ a i ‖ is the perpendicular distance of hyperplane a i x = b i to the

origin. Consequently, it follows that choosing a violated hyperplane a i x = b i

with a maximum value a i x r * − b i b i on the right side of (5) can be interpreted

from the left side of (5) as selecting a violated constraint giving the deepest cut

based on information derived from x r * . But from [

the right side of (4) is the distance from the origin to the hyperplane a i x ≤ b i along the vector c , i.e., the direction of steepest ascent for the objective function (1) of the NNLP P . For this reason, in [

N V R A D ( a i , b i , c , x r * ) = a i c b i 2 ( a i x r * − b i ) . (6)

Equation (6) thus incorporates global information from RAD with local information at x r * , and our posterior active-set method adds to P r an inoperative constraint i * for which

i * ∈ arg max i ∉ OPERATIVE { a i c b i 2 ( a i x r * − b i ) : a i x r * > b i } . (7)

We mention that the term b i 2 in the denominator of (7) works better than simply b i . This fact was established in computational results not reported here but obtained to support the above derivation.

A dynamic version of RAD was developed by the authors in [

cos θ i = c T x r * ‖ x r * ‖ ‖ c ‖ , (8)

is nonnegative since P r is also an NNLP. We would like to decrease θ r at each active-set iteration so that x r * points more in the same direction as the gradient c of the objective function in (1). We adapt our dynamic heuristic of [

As our ideal goal, let θ r = 0 in (8) to give

c T x r * = ‖ x r * ‖ ‖ c ‖ . (9)

When θ r = 0 , it follows from (9) that

∑ j = 1 n c j x r j * ∑ j = 1 n c j 2 = ∑ j = 1 n ( x r j * ) 2 . (10)

Letting | ⋅ | denote absolute value, define

δ r ( x r * ) = | ∑ j = 1 n c j x r j * ∑ j = 1 n c j 2 − ∑ j = 1 n ( x r j * ) 2 | (11)

as a measure of the performance of our active-set method at iteration r . The value of δ r ( x r * ) decreases as θ r decreases. Such a decrease usually occurs as x r * approaches an optimal extreme point of P itself.

The dynamic COST NVRAD for solving NNLPs is described as follows. Constraints are initially ordered by the RAD constraint selection metric (4). To construct P 0 we choose constraints from (2) in descending order of RAD (since there is no x r * ) until the A 0 matrix of has no 0 column, i.e., until each variable x j has an a i j > 0. These selected constraints become the constraints of P 0 , and we say that the variables are covered by the inequality constraints of the initial problem. P 0 is then solved by the primal simplex to achieve an initial solution x 0 * , and δ 0 ( x 0 * ) is calculated. At iteration r let γ r be the number of constraints of problem P violated by x r * . Then at iteration r − 1 and r , the values of δ r − 1 ( x r − 1 * ) and δ r ( x r * ) are calculated; and the percentage of improvement made in reducing the angle between vectors x r * and c at iteration r is measured by

ω r = max { 0 , ( δ r − 1 ( x r − 1 * ) − δ r ( x r * ) δ r − 1 ( x r − 1 * ) ) } × 100 , r = 1 , 2 , ⋯ . (12)

With [ ⋅ ] denoting the greatest integer function, let

{ φ r + 1 = φ r × ( 1 + [ ( ln ω r ) − 1 ] ) , r = 1 , 2 , ⋯ , if ω r > 1 φ r + 1 = γ r , r = 1 , 2 , ⋯ , if ω r ≤ 1 , (13)

where φ 1 = 200. The value of φ r is an upper bound on the possible number of violated constraints that can be added at active-set iteration r . The actual number added is min { φ r + 1 , γ r } . The active-set function φ r increases at every iteration since the optimal value of the objective function for P r is usually less affected by a constraint with a small value of (6) than one with a large value. Hence, more violated constraints should be added as r increases. Equation (13) represents one approach for doing so. If ω r > e (Euler’s number), for example, then φ r + 1 = φ r . If ω r = 1.01 , then φ r + 1 = 101 φ r . In other words, a much larger number and perhaps all of the remaining violated constraints could be added. NVRAD stops when γ r = 0 , i.e., when there are no more violated constraints.

The pseudocode for dynamic NVRAD algorithm is as follows.

Step 1―Identify constraints to initially bound the problem.

1: a * ← 0 , BOUNDING ← ∅

2: while a * ≯ 0 do

3: Let i * ∈ arg max i ∉ EXPLORED R A D ( a i , b i , c ) .

4: if ∃ j | a j * = 0 and a i j * > 0 then

5: BOUNDING ← BOUNDING ∪ { i }

6: end if

7: a * ← a * + a i *

8. Optimized ← false

9: end while

Step 2―Using the primal simplex method, obtain an optimal x 0 * for the initial problem.

( P 0 ) Maximize z = c T x subjectto a i x ≤ b i , i ∈ BOUNDING x ≥ 0.

Step 3―Perform the following iterations until an answer to problem P is found.

1: r ← 0

2: while Optimized = false do

3: Calculate δ r ( x r * ) .

4: if r > 1 then ω r = max { 0 , ( δ r − 1 ( x r − 1 * ) − δ r ( x r * ) δ r − 1 ( x r − 1 * ) ) } × 100

5: if ω r > 1 then φ r + 1 = φ r × ( 1 + [ ( ln ω r ) − 1 ] )

6: else if ω r ≤ 1 then φ r + 1 = γ r

7: end if

8: else φ r ← 0

9: end if

10: if a i x r * > b i , i = 1 , ⋯ , rowsthen

11: γ r ← #{ { a i x r * > b i , i = 1 , ⋯ , rows }

12: Let i * ∈ arg max i ∉ OPERATIVE { N V R A D ( a i , b i , c , x r * ) = a i c b i 2 ( a i x r * − b i ) : a i x r * > b i } .

13: for i = 1 , ⋯ , min { φ r + 1 , γ r } OPERATIVE ← OPERATIVE ∪ { i } end

14: Solve the following P r by the dual simplex method to obtain x r * .

15: r ← r + 1

16: Go to 3

17: elseOptimized ← true / / x r * is an optimal solution to P .

18: end if

19: end while

A reasonable conjecture is that that combining the global information of RAD and the local information of NVRAD might be advantageous. Therefore, we will also consider an approach that alternates the dynamic RAD and NVRAD metrics in a single algorithm at even and odd iterations, respectively, to yield a hybrid COST designated here as HYBR. The results obtained for HYBR demonstrate that combining posterior and prior COSTs may be superior to either a prior or posterior approach by itself.

Dynamic NVRAD is compared in this section with the CPLEX primal simplex, dual simplex, and barrier methods. It is also compared with the prior active-set method RAD and the standard posterior active-set method VIOL, as well as to a normalized version of VIOL called NVIOL that was superior to VIOL in computational results not reported here. Both dynamic and multi-bound, multi-cut versions of NVRAD were compared to dynamic and multi-bound, multi-cut versions of the other active-set methods for insight into the individual merits of the dynamic and posterior approaches.

Five sets of NNLPs from [

Two CPLEX parameters for solving linear programming are discussed here. The preprocessing pre-solve indicator (PREIND) and the preprocessing dual setting (PREDUAL) are the two parameters that CPLEX uses for solving linear programming. Preprocessing pre-solver is enabled with the parameter setting PREIND = 1 (ON), which reduces both the number of variables and the constraints before any type of algorithm is used. The pre-solver routine in CPLEX is disabled by setting PREIND = 0 (OFF). The second preprocessing parameter in CPLEX affecting the computational speed is PREDUAL. By setting parameter PREDUAL = 0 (ON) or PREDUAL = −1 (OFF), CPLEX automatically selects whether to solve the dual of the original LP or not, respectively.

Both PREIND and PREDUAL were turned off for CPLEX when CPLEX was used as part of NVRAD or HYBR. However, all computational results reported here for any individual CPLEX solver had PREIND and PREDUAL turned on. In other words, our NVRAD was compared to CPLEX at its fastest setting. CPLEX would choose automatically whether to solve either the primal or dual, whichever seemed best. Moreover, preprocessing would substantially reduce the size of any problem P by removing appropriate rows or columns of the constraint matrix A before applying the primal simplex, dual simplex, or interior-point barrier method. In fact, much of the speed of the CPLEX solvers is due to its proprietary preprocessing routines.

The experiments were performed on an Intel^{®}Core^{TM} 2 Duo X9650 3.00 GHz processor with a Linux 64-bit operating system and 8 GB of RAM. The COST NVRAD uses the IBM CPLEX 12.5 callable library to solve P 0 by the primal simplex and then P r , r = 1 , 2 , ⋯ by the dual simplex when selected constraints are added to P r − 1 . The CPU times shown in the tables below represent the average computation time of five problem instances at each density level.

The results of

In

VIOL^{−−} | NVIOL^{−−} | NVRAD^{−−} | NVIOL^{−−} | NVRAD^{−−} | ||
---|---|---|---|---|---|---|

Multi-Cut & Multi-Bound | Dynamic | |||||

Density No.CPU Time (Sec)^{++}CPU Time (Sec)^{++} | ||||||

0.005 | 1 | 6.54 | 4.56 | 2.51 | 2.49 | 2.26 |

0.006 | 2 | 6.84 | 5.06 | 2.92 | 2.96 | 2.62 |

0.007 | 3 | 7.15 | 5.34 | 3.03 | 3.03 | 2.75 |

0.008 | 4 | 6.61 | 4.96 | 3.02 | 3.13 | 2.83 |

0.009 | 5 | 7.02 | 5.16 | 3.11 | 3.39 | 3.09 |

0.01 | 6 | 6.83 | 5.14 | 3.41 | 3.51 | 3.12 |

0.02 | 7 | 6.11 | 4.81 | 3.36 | 3.79 | 3.44 |

0.03 | 8 | 5.79 | 4.79 | 3.33 | 3.99 | 3.52 |

0.04 | 9 | 5.71 | 4.45 | 3.31 | 3.99 | 3.71 |

0.05 | 10 | 5.41 | 4.62 | 3.49 | 4.10 | 3.64 |

0.06 | 11 | 5.32 | 4.3 | 3.52 | 3.91 | 3.63 |

0.07 | 12 | 5.87 | 4.73 | 3.79 | 3.93 | 3.73 |

0.08 | 13 | 5.53 | 4.68 | 3.68 | 3.86 | 3.61 |

0.09 | 14 | 5.76 | 4.89 | 3.99 | 4.06 | 3.65 |

0.1 | 15 | 6.04 | 5.07 | 4.31 | 4.08 | 3.89 |

0.2 | 16 | 10.9 | 9.64 | 8.28 | 4.96 | 4.82 |

0.3 | 17 | 17.3 | 15.15 | 13.05 | 6.03 | 5.68 |

0.4 | 18 | 24.64 | 22.12 | 20.53 | 7.28 | 6.56 |

0.5 | 19 | 32.93 | 29.85 | 27.67 | 7.63 | 7.34 |

0.75 | 20 | 62.21 | 57.36 | 54.97 | 10.53 | 10.50 |

1 | 21 | 261.23 | 251.65 | 245.5 | 11.43 | 11.13 |

Average | 23.89 | 21.82 | 20.04 | 4.86 | 4.55 |

^{++}Average of 5 instances at each density. ^{−−} Used CPLEX presolve = OFF and predual = OFF.

NVRAD over all densities are 19.07 and 16.86 seconds, respectively. For Set 3, dynamic NVRAD is superior to RAD averaging 38.91 seconds compared to 41.87 seconds. Similarly, for Set 4 the averages are 41.87 for NVRAD as compared to 46.98 for RAD. Thus the results of

NVRAD^{−−} | RAD^{−−} | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | 1000 | 3163 | 10,000 | 14,143 | 1000 | 3163 | 10,000 | 14,143 | |||

m | 200,000 | 63,246 | 20,000 | 14,143 | 200,000 | 63,246 | 20,000 | 14,143 | |||

m/n | 200 | 20 | 2 | 1 | 200 | 20 | 2 | 1 | |||

Dynamic Active-Set | Multi-Cut & Multi Bound | ||||||||||

Density | No. | CPU Time (Sec)^{++} | CPU Time (Sec)^{++} | ||||||||

0.005 | 1 | 2.26 | 25.00 | 88.36 | 106.27 | 2.10 | 30.82 | 108.70 | 127.55 | ||

0.006 | 2 | 2.62 | 27.23 | 88.73 | 97.31 | 2.42 | 31.48 | 104.87 | 114.03 | ||

0.007 | 3 | 2.75 | 25.79 | 82.04 | 90.65 | 2.65 | 29.41 | 92.45 | 104.18 | ||

0.008 | 4 | 2.83 | 27.49 | 78.04 | 78.92 | 2.54 | 30.63 | 88.20 | 90.73 | ||

0.009 | 5 | 3.09 | 27.43 | 74.65 | 75.18 | 2.78 | 30.10 | 83.53 | 85.21 | ||

0.01 | 6 | 3.12 | 26.07 | 68.23 | 73.29 | 2.79 | 27.81 | 77.90 | 80.43 | ||

0.02 | 7 | 3.44 | 22.48 | 45.06 | 46.24 | 3.09 | 24.69 | 47.63 | 49.95 | ||

0.03 | 8 | 3.52 | 18.59 | 34.78 | 38.25 | 3.22 | 20.49 | 36.68 | 38.33 | ||

0.04 | 9 | 3.71 | 16.92 | 29.96 | 30.75 | 3.33 | 19.06 | 32.74 | 32.53 | ||

0.05 | 10 | 3.64 | 15.47 | 26.31 | 28.01 | 3.34 | 16.97 | 28.23 | 28.59 | ||

0.06 | 11 | 3.63 | 13.62 | 24.62 | 25.62 | 3.20 | 14.94 | 27.58 | 27.27 | ||

0.07 | 12 | 3.73 | 12.93 | 22.24 | 23.19 | 3.41 | 14.88 | 23.59 | 23.79 | ||

0.08 | 13 | 3.61 | 11.99 | 20.74 | 22.30 | 3.32 | 13.57 | 23.44 | 24.19 | ||

0.09 | 14 | 3.65 | 11.39 | 20.47 | 21.64 | 3.38 | 12.67 | 23.09 | 23.80 | ||

0.1 | 15 | 3.89 | 10.81 | 19.65 | 20.18 | 3.39 | 12.92 | 22.93 | 20.85 | ||

0.2 | 16 | 4.82 | 8.98 | 16.31 | 19.44 | 4.30 | 11.09 | 18.87 | 20.31 | ||

0.3 | 17 | 5.68 | 8.84 | 15.66 | 19.09 | 4.97 | 10.58 | 18.11 | 19.46 | ||

0.4 | 18 | 6.56 | 9.77 | 15.76 | 17.23 | 5.76 | 12.31 | 18.55 | 18.88 | ||

0.5 | 19 | 7.34 | 10.60 | 15.82 | 17.89 | 6.98 | 11.92 | 18.00 | 19.89 | ||

0.75 | 20 | 10.50 | 11.24 | 15.80 | 16.73 | 8.26 | 12.01 | 17.19 | 18.06 | ||

1 | 21 | 11.13 | 11.34 | 13.85 | 11.12 | 8.39 | 12.20 | 17.71 | 18.50 | ||

Average | ^{ } | 4.55 | 16.86 | 38.91 | 41.87 | 3.98 | 19.07 | 44.28 | 46.98 | ||

^{++}Average of 5 instances of LP at each density. ^{−−}Used CPLEX presolve = OFF and predual = OFF.

erage CPU across all densities was approximately 15 to 50 times greater than NVRAD over the different ratios. However, the CPLEX barrier method was slightly faster than NVRAD in problem instances with m / n = 20 and with densities less than 0.02. On the other hand, when the density reached 0.08 for m / n = 20 , NVRAD was already more than ten times faster than the barrier solver. Furthermore, note that average CPU times in

Finally, for large-scale, low-density test problems with n = 5000 and

RAD^{−−} | HYBR^{−−} | ||||||||
---|---|---|---|---|---|---|---|---|---|

n | 1000 | 3163 | 10,000 | 14,143 | 1000 | 3163 | 10,000 | 14,143 | |

m | 200,000 | 63,246 | 20,000 | 14,143 | 200,000 | 63,246 | 20,000 | 14,143 | |

m/n | 200 | 20 | 2 | 1 | 200 | 20 | 2 | 1 | |

Dynamic Active-Set | Dynamic Active-Set | ||||||||

Density | No. | CPU Time (Sec)^{++} | CPU Time (Sec)^{++} | ||||||

0.005 | 1 | 2.02 | 29.51 | 106.42 | 127.70 | 2.19 | 27.14 | 94.26 | 113.63 |

0.006 | 2 | 2.32 | 30.20 | 107.61 | 116.08 | 2.53 | 28.51 | 95.25 | 103.78 |

0.007 | 3 | 2.49 | 29.07 | 95.87 | 104.79 | 2.78 | 26.14 | 84.13 | 95.47 |

0.008 | 4 | 2.47 | 29.80 | 89.37 | 93.12 | 2.76 | 27.63 | 78.88 | 83.75 |

0.009 | 5 | 2.67 | 28.59 | 80.46 | 86.46 | 2.93 | 27.21 | 77.09 | 81.11 |

0.01 | 6 | 2.65 | 27.09 | 75.60 | 81.28 | 3.02 | 25.68 | 70.59 | 75.25 |

0.02 | 7 | 2.85 | 22.01 | 45.39 | 48.01 | 3.28 | 20.90 | 44.45 | 46.91 |

0.03 | 8 | 2.83 | 17.30 | 33.40 | 36.20 | 3.19 | 17.46 | 34.44 | 36.18 |

0.04 | 9 | 2.82 | 14.97 | 29.29 | 27.47 | 3.18 | 15.08 | 27.58 | 29.31 |

0.05 | 10 | 2.97 | 13.91 | 24.38 | 24.70 | 3.04 | 13.68 | 24.61 | 24.94 |

0.06 | 11 | 2.85 | 11.43 | 22.31 | 22.82 | 3.19 | 11.81 | 23.03 | 23.32 |

0.07 | 12 | 2.93 | 11.04 | 19.40 | 20.48 | 3.31 | 11.55 | 19.99 | 20.88 |

0.08 | 13 | 2.91 | 10.37 | 18.75 | 19.87 | 3.30 | 10.36 | 19.19 | 20.72 |

0.09 | 14 | 3.16 | 9.16 | 18.11 | 18.93 | 3.34 | 9.37 | 18.52 | 19.54 |

0.1 | 15 | 3.06 | 9.54 | 17.35 | 17.34 | 3.51 | 9.18 | 17.74 | 17.89 |

0.2 | 16 | 4.34 | 8.12 | 14.39 | 15.99 | 4.36 | 7.92 | 14.01 | 15.55 |

0.3 | 17 | 5.70 | 8.57 | 13.31 | 15.32 | 5.40 | 7.86 | 12.92 | 14.66 |

0.4 | 18 | 7.00 | 9.28 | 13.55 | 14.58 | 6.50 | 8.95 | 12.65 | 13.93 |

0.5 | 19 | 7.95 | 9.85 | 13.60 | 16.35 | 7.84 | 9.60 | 13.38 | 14.52 |

0.75 | 20 | 10.64 | 12.05 | 14.43 | 15.56 | 9.81 | 10.77 | 12.38 | 14.13 |

1 | 21 | 12.66 | 11.71 | 12.60 | 14.72 | 11.00 | 9.76 | 11.39 | 11.59 |

Average | 4.25 | 16.84 | 41.22 | 44.66 | 4.31 | 16.03 | 38.40 | 41.76 |

^{++}Average of 5 instances of LP at each density. ^{−−}Used CPLEX presolve = OFF and predual = OFF.

m = 1 , 000 , 000.

Primal^{+}_{ } | Dual^{+} | Barrier^{+} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | 1000 | 3163 | 10,000 | 14,143 | 1000 | 3163 | 10,000 | 14,143 | 1000 | 3163 | 10,000 | 14,143 | |

m | 200,000 | 63,246 | 20,000 | 14,143 | 200,000 | 63,246 | 20,000 | 14,143 | 200,000 | 63,246 | 20,000 | 14,143 | |

m/n | 200 | 20 | 2 | 1 | 200 | 20 | 2 | 1 | 200 | 20 | 2 | 1 | |

Density | No. | CPU Time (Sec)^{++} | |||||||||||

0.005 | 1 | 7.01 | 71.02 | 228.51 | 309.83 | 54.84 | 762.62 | 1597.24 | 1169.04 | 2.36 | 14.52 | 240.17 | 650.83 |

0.006 | 2 | 10.36 | 77.28 | 245.60 | 291.07 | 60.29 | 803.97 | 1607.16 | 2413.42 | 2.39 | 16.30 | 224.08 | 666.54 |

0.007 | 3 | 12.98 | 75.84 | 219.72 | 265.09 | 91.39 | 876.85 | 1483.20 | 1702.47 | 3.04 | 18.34 | 233.55 | 671.56 |

0.008 | 4 | 15.72 | 82.01 | 206.45 | 239.30 | 100.06 | 912.75 | 1445.54 | 1236.76 | 3.90 | 20.70 | 232.38 | 668.82 |

0.009 | 5 | 19.25 | 80.35 | 196.72 | 216.23 | 114.95 | 898.99 | 1375.73 | 427.95 | 4.76 | 22.66 | 232.23 | 649.26 |

0.01 | 6 | 21.92 | 78.50 | 182.47 | 216.60 | 123.49 | 912.63 | 1252.05 | 436.31 | 5.53 | 24.29 | 228.76 | 650.30 |

0.02 | 7 | 39.90 | 78.80 | 118.28 | 127.59 | 203.08 | 963.66 | 807.29 | 362.34 | 17.13 | 32.08 | 242.54 | 711.26 |

0.03 | 8 | 45.42 | 79.75 | 98.02 | 108.60 | 217.18 | 1207.76 | 545.91 | 723.98 | 28.79 | 45.03 | 266.90 | 727.61 |

0.04 | 9 | 50.30 | 78.78 | 89.75 | 88.32 | 248.75 | 1489.40 | 450.08 | 539.92 | 41.50 | 62.28 | 292.15 | 806.80 |

0.05 | 10 | 55.16 | 78.92 | 81.09 | 82.14 | 256.49 | 1746.46 | 418.69 | 519.50 | 53.72 | 81.32 | 327.01 | 837.67 |

0.06 | 11 | 60.34 | 77.49 | 77.28 | 78.27 | 251.39 | 2124.31 | 378.71 | 409.47 | 67.58 | 100.48 | 359.53 | 897.58 |

0.07 | 12 | 62.07 | 78.93 | 70.44 | 70.37 | 251.74 | 2446.69 | 310.89 | 544.15 | 84.70 | 125.49 | 401.72 | 948.01 |

0.08 | 13 | 62.92 | 76.96 | 70.21 | 69.81 | 264.48 | 2799.62 | 307.25 | 388.94 | 99.51 | 149.37 | 454.01 | 1038.86 |

0.09 | 14 | 66.57 | 79.07 | 71.46 | 72.37 | 258.14 | 2523.03 | 718.04 | 427.95 | 119.26 | 186.06 | 495.28 | 1153.31 |

0.1 | 15 | 71.00 | 74.57 | 67.43 | 62.64 | 287.36 | 2251.10 | 267.14 | 436.31 | 138.67 | 207.54 | 539.64 | 1194.56 |

0.2 | 16 | 87.49 | 83.12 | 64.38 | 62.99 | 294.39 | 1450.82 | 201.73 | 362.34 | 379.68 | 691.77 | 1298.76 | 2529.97 |

0.3 | 17 | 94.57 | 77.91 | 67.14 | 66.61 | 341.44 | 1280.71 | 175.16 | 267.16 | 657.45 | 1333.29 | 2418.75 | ^{b } |

0.4 | 18 | 99.33 | 78.46 | 73.58 | 71.48 | 384.10 | 1236.30 | 146.09 | 233.39 | 985.86 | 2076.09 | ^{b} | ^{b} |

0.5 | 19 | 111.30 | 84.30 | 86.50 | 75.62 | 427.16 | 1173.49 | 133.49 | 208.65 | 1350.82 | ^{b} | ^{b} | ^{b} |

0.75 | 20 | 128.26 | 99.35 | 115.00 | 102.51 | 410.98 | 1056.18 | 132.25 | 181.95 | ^{b} | ^{b} | ^{b} | ^{b} |

1 | 21 | 207.55 | 94.09 | 393.54 | 145.96 | 375.89 | 411.19 | 148.90 | 165.45 | ^{b} | ^{b} | ^{b} | ^{b} |

Average | 63.30 | 80.26 | 134.46 | 134.45 | 238.93 | 1396.60 | 662.03 | 626.55 | n/a | n/a | n/a | n/a |

^{+}CPLEX presolve = ON and predual = ON. ^{++}Average of 5 instances at each density. ^{b} Runs with CPU times > 3000 s are not report.

No. | Density | NVRAD^{−−} | RAD^{−−} | VIOL^{−−} | NVIOL^{−−} | NVRAD^{−−} | CPLEX Primal^{+} | CPLEX Dual^{+} | CPLEX Barrier^{+} | |
---|---|---|---|---|---|---|---|---|---|---|

Dynamic | Multi-Cut & Multi-Bound | Not Active-Set Methods | ||||||||

CPU Time (Sec)^{++} | CPU Time (Sec)^{++} | |||||||||

1 | 0.0004 | 6.21 | 7.54 | 157.92 | 73.09 | 72.19 | 11.90 | 14.08 | 12.31 | |

2 | 0.0005 | 9.22 | 12.26 | 177.96 | 100.86 | 106.31 | 23.41 | 29.83 | 16.61 | |

3 | 0.0006 | 11.94 | 16.51 | 252.74 | 75.41 | 76.12 | 13.45 | 107.61 | 20.45 | |

4 | 0.0007 | 15.45 | 22.19 | 282.95 | 92.70 | 93.64 | 18.99 | 176.50 | 24.60 | |

5 | 0.0008 | 20.16 | 27.66 | 325.51 | 108.42 | 95.22 | 28.88 | 257.06 | 27.43 | |

6 | 0.0009 | 23.70 | 33.24 | 346.76 | 120.57 | 91.06 | 40.17 | 339.49 | 29.80 | |

7 | 0.0010 | 28.01 | 39.81 | 374.06 | 141.35 | 107.34 | 50.91 | 427.60 | 31.73 | |

8 | 0.0020 | 70.01 | 89.57 | 393.48 | 222.63 | 174.78 | 173.03 | 1775.03 | 48.79 | |

9 | 0.0030 | 90.09 | 104.83 | 368.92 | 245.17 | 190.37 | 244.01 | ^{b } | 61.31 | |

10 | 0.0040 | 99.32 | 113.40 | 346.56 | 224.35 | 183.58 | 316.53 | ^{b } | 85.60 | |

11 | 0.0050 | 103.78 | 113.17 | 322.98 | 215.49 | 172.42 | 366.80 | ^{b } | 91.11 | |

12 | 0.0060 | 112.15 | 122.85 | 320.97 | 217.81 | 171.40 | 443.43 | ^{b } | 112.46 | |

13 | 0.0070 | 106.61 | 116.00 | 283.16 | 214.48 | 160.63 | 474.40 | ^{b } | 136.03 | |

14 | 0.0080 | 100.14 | 113.05 | 258.56 | 184.76 | 148.74 | 529.44 | ^{b } | 158.54 | |

15 | 0.0090 | 94.43 | 104.68 | 229.32 | 165.47 | 138.51 | 566.20 | ^{b } | 198.31 | |

16 | 0.0100 | 100.91 | 112.82 | 233.08 | 171.28 | 137.64 | 629.59 | ^{b } | 239.87 | |

17 | 0.0200 | 76.77 | 83.83 | 142.85 | 106.45 | 90.60 | 1134.77 | ^{b } | 899.87 | |

18 | 0.0300 | 69.41 | 76.69 | 114.25 | 86.83 | 76.77 | 1740.28 | ^{b } | ^{b } | |

19 | 0.0400 | 65.87 | 67.36 | 103.22 | 79.26 | 71.60 | 1865.70 | ^{b } | ^{b } | |

20 | 0.0500 | 63.71 | 64.58 | 100.60 | 80.35 | 71.87 | 2159.55 | ^{b } | ^{b } | |

21 | 0.0600 | 64.57 | 65.62 | 102.05 | 82.42 | 74.41 | ^{b} | ^{b } | ^{b } | |

Average | 63.45 | 71.79 | 249.42 | 143.29 | 119.29 | |||||

^{++}Average of 5 instances at each density.^{b} Runs with CPU times > 2400 s are not reported. ^{−−}Used CPLEX presolve = OFF and predual = OFF. ^{+}Used CPLEX presolve = ON and predual = ON.

An efficient posterior COST called NVRAD was developed here for NNLPs to utilize both prior global information and posterior local information. The associated constraint selection metric NVRAD is a heuristic, so a geometric interpretation was presented to offer insight into its performance. NVRAD’s inherent active-set efficiency was enhanced by a dynamic approach varying the number of constraints added at each iteration. In addition to NVRAD, adynamic active-set approach HYBR was also proposed. HYBR alternates between the posterior method NVRAD and the prior method RAD. To check their performance, both NVRAD and HYBR were used to solve five sets of large-scale NNLPs. Dynamic NVRAD outperformed the previously developed COST RAD, as well as the standard posterior cutting-plane method VIOL. Dynamic NVRAD significantly outperformed the CPLEX primal simplex, dual simplex, and barrier solvers. On the other hand, HYBR appears slightly faster than NVRAD or RAD. The results of this paper provide further evidence that active-set methods may be the fastest approach for solving linear programming problems.

Corley, H.W., Noroziroshan, A. and Rosenberger, J.M. (2017) Posterior Constraint Selection for Nonnegative Linear Programming. American Journal of Operations Research, 7, 26- 40. http://dx.doi.org/10.4236/ajor.2017.71002