Mathematical Problems in Engineering

Volume 2012, Article ID 482193, 28 pages

http://dx.doi.org/10.1155/2012/482193

## A Two-Phase Support Method for Solving Linear Programs: Numerical Experiments

^{1}Department of Technology, University of Laghouat, 03000, Algeria^{2}Laboratory of Modelling and Optimization of Systems (LAMOS), University of Bejaia, 06000, Algeria

Received 6 September 2011; Accepted 7 February 2012

Academic Editor: J. Jiang

Copyright © 2012 Mohand Bentobache and Mohand Ouamer Bibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We develop a single artificial variable technique to initialize the primal support method for solving linear programs with bounded variables. We first recall the full artificial basis technique, then we will present the proposed algorithm. In order to study the performances of the suggested algorithm, an implementation under the MATLAB programming language has been developed. Finally, we carry out an experimental study about CPU time and iterations number on a large set of the NETLIB test problems. These test problems are practical linear programs modelling various real-life problems arising from several fields such as oil refinery, audit staff scheduling, airline scheduling, industrial production and allocation, image restoration, multisector economic planning, and data fitting. It has been shown that our approach is competitive with our implementation of the primal simplex method and the primal simplex algorithm implemented in the known open-source LP solver LP_SOLVE.

#### 1. Introduction

Linear programming is a mathematical discipline which deals with solving the problem of optimizing a linear function over a domain delimited by a set of linear equations or inequations. The first formulation of an economical problem as a linear programming problem is done by Kantorovich (1939, [1]), and the general formulation is given later by Dantzig in his work [2]. LP is considered as the most important technique in operations research. Indeed, it is widely used in practice, and most of optimization techniques are based on LP ones. That is why many researchers have given a great interest on finding efficient methods to solve LP problems. Although some methods exist before 1947 [1], they are restricted to solve some particular forms of the LP problem. Being inspired from the work of Fourier on linear inequalities, Dantzig (1947, [3]) developed the simplex method which is known to be very efficient for solving practical linear programs. However, in 1972, Klee and Minty [4] have found an example where the simplex method takes an exponential time to solve it.

In 1977, Gabasov and Kirillova [5] have generalized the simplex method and developed the primal support method which can start by any basis and any feasible solution and can move to the optimal solution by interior points or boundary points. The latter is adapted by Radjef and Bibi to solve LPs which contain two types of variables: bounded and nonnegative variables [6]. Later, Gabasov et al. developed the adaptive method to solve, particularly, linear optimal control problems [7]. This method is extended to solve general linear and convex quadratic problems [8–18]. In 1979, Khachian developed the first polynomial algorithm which is an interior point one to solve LP problems [19], but it’s not efficient in practice. In 1984, Karmarkar presented for the first time an interior point algorithm competitive with the simplex method on large-scale problems [20].

The efficiency of the simplex method and its generalizations depends enormously on the first initial point used for their initialization. That is why many researchers have given a new interest for developing new initialization techniques. These techniques aim to find a good initial basis and a good initial point and use a minimum number of artificial variables to reduce memory space and CPU time. The first technique used to find an initial basic feasible solution for the simplex method is the full artificial basis technique [3]. In [21, 22], the authors developed a technique using only one artificial variable to initialize the simplex method. In his experimental study, Millham [23] shows that when the initial basis is available in advance, the single artificial variable technique can be competitive with the full artificial basis one. Wolfe [24] has suggested a technique which consists of solving a new linear programming problem with a piecewise linear objective function (minimization of the sum of infeasibilities). In [25–31], crash procedures are developed to find a good initial basis.

In [32], a two-phase support method with one artificial variable for solving linear programming problems was developed. This method consists of two phases and its general principle is the following: in the first phase, we start by searching an initial support with the Gauss-Jordan elimination method, then we proceed to the search of an initial feasible solution by solving an auxiliary problem having one artificial variable and an obvious feasible solution. This obvious feasible solution can be an interior point of the feasible region. After that, in the second phase, we solve the original problem with the primal support method [5].

In [33, 34], we have suggested two approaches to initialize the primal support method with nonnegative variables and bounded variables: the first approach consists of applying the Gauss elimination method with partial pivoting to the system of linear equations corresponding to the main constraints and the second consists of transforming the equality constraints to inequality constraints. After finding the initial support, we search a feasible solution by adding only one artificial variable to the original problem, thus we get an auxiliary problem with an evident support feasible solution. An experimental study has been carried out on some NETLIB test problems. The results of the numerical comparison revealed that finding the initial support by the Gauss elimination method consumes much time, and transforming the equality constraints to inequality ones increases the dimension of the problem. Hence, the proposed approaches are competitive with the full artificial basis simplex method for solving small problems, but they are not efficient to solve large problems.

In this work, we will first extend the full artificial basis technique presented in [7], to solve problems in general form, then we will combine a crash procedure with a single artificial variable technique in order to find an initial support feasible solution for the initialization of the support method. This technique is efficient for solving practical problems. Indeed, it takes advantage of sparsity and adds a reduced number of artificial variables to the original problem. Finally, we show the efficiency of our approach by carrying out an experimental study on some NETLIB test problems.

The paper is organized as follows: in Section 2, the primal support method for solving linear programming problems with bounded variables is reviewed. In Section 3, the different techniques to initialize the support method are presented: the full artificial basis technique and the single artificial variable one. Although the support method with full artificial basis is described in [7], it has never been tested on NETLIB test problems. In Section 4, experimental results are presented. Finally, Section 5 is devoted to the conclusion.

#### 2. Primal Support Method with Bounded Variables

##### 2.1. State of the Problem and Definitions

The linear programming problem with bounded variables is presented in the following standard form:
where and are -vectors; an -vector; an -matrix with ; and are -vectors. In the following sections, we will assume that and . We define the following index sets:
So we can write and partition the vectors and the matrix as follows:
(i)A vector verifying constraints (2.2) and (2.3) is called a *feasible solution* for the problem (2.1)–(2.3).(ii)A feasible solution is called *optimal* if , where is taken from the set of all feasible solutions of the problem (2.1)–(2.3).(iii) A feasible solution is said to be -*optimal* or *suboptimal* if
where is an optimal solution for the problem (2.1)–(2.3), and is a positive number chosen beforehand.(iv)We consider the set of indices such that . Then is called a *support* if .(v)The pair comprising a feasible solution and a support will be called a *support feasible solution* (SFS).(vi)An SFS is called *nondegenerate* if .

*Remark 2.1. *An SFS is a more general concept than the basic feasible solution (BFS). Indeed, the nonsupport components of an SFS are not restricted to their bounds. Therefore, an SFS may be an interior point, a boundary point or an extreme point, but a BFS is always an extreme point. That is why we can classify the primal support method in the class of interior search methods within the simplex framework [35].

where .

Theorem 2.2 (the optimality criterion [5]). *Let be an SFS for the problem (2.1)–(2.3). So the relations:
**
are sufficient and, in the case of nondegeneracy of the SFS , also necessary for the optimality of the feasible solution .*

The quantity defined by:
is called the *suboptimality estimate*. Thus, we have the following theorem [5].

Theorem 2.3 (sufficient condition for suboptimality). *Let be an SFS for the problem (2.1)–(2.3) and an arbitrary positive number. If , then the feasible solution is -optimal.*

##### 2.2. The Primal Support Method

Let be an initial SFS and an arbitrary positive number. The scheme of the primal support method is described in the following steps:(1)Compute ; .(2)Compute with the formula (2.9).(3)If , then the algorithm stops with , an optimal SFS.(4)If , then the algorithm stops with , an -optimal SFS.(5)Determine the nonoptimal index set: (6)Choose an index from such that .(7)Compute the search direction using the relations: (8)Compute , where is determined by the formula: (9)Compute using the formula (10)Compute .(11)Compute , .(12)Compute .(13)If , then the algorithm stops with , an optimal SFS.(14)If , then the algorithm stops with , an -optimal SFS.(15)If , then we put .(16)If , then we put .(17)We put and . Go to the step (1).

#### 3. Finding an Initial SFS for the Primal Support Method

Consider the linear programming problem written in the following general form: where , , , are vectors in ; is a matrix of dimension , is a matrix of dimension , , . We assume that and .

Let be the number of constraints of the problem (3.1) and , are, respectively, the constraints indices set and the original variables indices set of the problem (3.1). We partition the set as follows: , where and represent, respectively, the indices set of inequality and equality constraints. We note by the -vector of ones, that is, and the vector of the identity matrix of order .

After adding slack variables to the problem (3.1), we get the following problem in standard form: where , , is the vector of the added slack variables and its upper bound vector. In order to work with finite bounds, we set , , that is, , where is a finite, positive and big real number chosen carefully.

*Remark 3.1. *The upper bounds, , , of the added slack variables can also be deduced as follows:, where is a -vector computed with the formula
Indeed, from the system (3.3), we have , . By using the bound constraints (3.4), we get , . However, the experimental study shows that it’s more efficient to set , to a given finite big value, because for the large-scale problems, the deduction formula (3.6) given above takes much CPU time to compute bounds for the slack variables.

The initialization of the primal support method consists of finding an initial support feasible solution for the problem (3.2)–(3.5). In this section, being inspired from the technique used to initialize interior point methods [36] and taking into account, the fact that the support method can start with a feasible point and a support which are independent, we suggest a single artificial variable technique to find an initial SFS. Before presenting the suggested technique, we first extend the full artificial basis technique, originally presented in [7] for standard form, to solve linear programs presented in the general form (3.1).

##### 3.1. The Full Artificial Basis Technique

Let be a -vector chosen between and and an -vector such that . We consider the following subsets of : , , and we assume without loss of generality that , with and . Remark that and form a partition of ; and .

We make the following partition for , and : , where

, where and , where Hence, the problem (3.2)–(3.5) becomes

After adding artificial variables to the equations of the system (3.11), where is the number of elements of the artificial index set , we get the following auxiliary problem: where represents the artificial variables vector, where is a real nonnegative number chosen in advance.

If we put , , , , , , , , , then we get the following auxiliary problem: The variables indices set of the auxiliary problem (3.17)–(3.19) is Let us partition this set as follows: , where Remark that the pair , where with and , is a support feasible solution (SFS) for the auxiliary problem (3.17)–(3.19). Indeed, lies between its lower and upper bounds: for we have Furthermore, the -matrix is invertible because and verifies the main constraints: by replacing , , , and with their values in the system (3.18), we get Therefore, the primal support method can be initialized with the SFS to solve the auxiliary problem. Let be the obtained optimal SFS after the application of the primal support method to the auxiliary problem (3.17)–(3.19), where If , then the original problem (3.2)–(3.5) is infeasible.

Else, when does not contain any artificial index, then will be an SFS for the original problem (3.2)–(3.5). Otherwise, we delete artificial indices from the support , and we replace them with original or slack appropriate indices, following the algebraic rule used in the simplex method.

In order to initialize the primal support method for solving linear programming problems with bounded variables written in the standard form, in [7], Gabasov et al. add artificial variables, where represents the number of constraints. In this work, we are interested in solving the problem written in the general form (3.1), so we have added artificial variables only for equality constraints and inequality constraints with negative components of the vector .

*Remark 3.2. *We have . Since in the relationship (3.22), we have and, we get .

*Remark 3.3. *If we choose such that or , then the vector , given by the relationship (3.22), will be a BFS. Hence, the simplex algorithm can be initialized with this point.

*Remark 3.4. *If , and , two cases can occur.

*Case 1. *If , then we put .

*Case 2. *If , then we put , and .

In the two cases, we get , , therefore . Hence , so we add only artificial variables for the equality constraints.

##### 3.2. The Single Artificial Variable Technique

In order to initialize the primal support method using this technique, we first start by searching an initial support, then we proceed to the search of an initial feasible solution for the original problem.

The application of the Gauss elimination method with partial pivoting to the system of equations (3.3) can give us a support , where . However, the experimental study realized in [33, 34] reveals that this approach takes much time in searching the initial support, that is, why it’s important to take into account the sparsity property of practical problems and apply some procedure to find a triangular basis among the columns corresponding to the original variables, that is, the columns of the matrix . In this work, on the base of the crash procedures presented in [26, 28–30], we present a procedure to find an initial support for the problem (3.2)–(3.5).

*Procedure 1 (searching an initial support). *(1) We sort the columns of the matrix according to the increasing order of their number of nonzero elements. Let be the list of the sorted columns of .

(2) Let (the column of *L*) be the first column of the list verifying: such that
where is a given tolerance. Hence, we put , , .

(3) Let be the index corresponding to the column of the list , that is, . If has zero elements in all the rows having indices in and if such that
then we put , .

(4) We put . If , then go to step (3), else go to step (5).

(5) We put , , .

(6) For all , if the constraint is originally an inequality constraint, then we add to an index corresponding to the slack variable , that is, . If the constraint is originally an equality constraint, then we put and add to this latter an artificial variable for which we set the lower bound to zero and the upper bound to a big and well-chosen value . Thus, we put , , .

*Remark 3.5. *The inputs of Procedure 1 are:(i)the -matrix of constraints, ;(ii)a pivoting tolerance fixed beforehand, ;(iii)a big and well chosen value .

The outputs of Procedure 1 are:(i): the initial support for the problem (3.2)–(3.5);(ii): the indices of the constraints for which we have added artificial variables;(iii): the indices of artificial variables added to the problem (3.2)–(3.5);(iv): the number of artificial variables added to the problem (3.2)–(3.5).

After the application of the procedure explained above (Procedure 1) for the problem (3.2)–(3.5), we get the following linear programming problem: where is the -vector of the artificial variables added during the application of Procedure 1, with , is an -matrix.

Since we have got an initial support, we can start the procedure of finding a feasible solution for the original problem.

*Procedure 2 (searching an initial feasible solution). *Consider the following auxiliary problem:
where is an artificial variable, , and is a -vector chosen between and . We remark that
is an obvious feasible solution for the auxiliary problem. Indeed,
Hence, we can apply the primal support method to solve the auxiliary problem (3.31) by starting with the initial SFS , where is the support obtained with Procedure 1. Let
be, respectively, the optimal solution, the optimal support, and the optimal objective value of the auxiliary problem (3.31).

If , then the original problem (3.2)–(3.5) is infeasible.

Else, when does not contain any artificial index, then will be an SFS for the original problem (3.2)–(3.5). Otherwise, we delete the artificial indices from the support and we replace them with original or slack appropriate indices, following the algebraic rule used in the simplex method.

*Remark 3.6. *The number of artificial variables of the auxiliary problem (3.31) verifies the inequality: .

The auxiliary problem will have only one artificial variable, that is, , when the initial support found by Procedure 1 is constituted only by the original and slack variable indices (), and this will hold in two cases.

*Case 1. *When all the constraints are inequalities, that is, .

*Case 2. *When and step (4) of Procedure 1 ends with .

The case holds when the step (4) of Procedure 1 stops with .

*Remark 3.7. *Let’s choose a -vector between and and two nonnegative vectors and . If we put in the auxiliary problem (3.31), , with , then the vector
is a feasible solution for the auxiliary problem. Indeed,
We remark that if we put , we get , then and we obtain the evident feasible point that we have used in our numerical experiments, that is,
It’s important here, to cite two other special cases.*Case 1. *If we choose the nonbasic components of equal to their bounds, then we obtain a BFS for the auxiliary problem, therefore we can initialize the simplex algorithm with it.*Case 2. *If we put and , then . Hence, the vector
is a feasible solution for the auxiliary problem (3.31), with .

*Numerical Example*

Consider the following LP problem:
where
We put , and we apply the two-phase primal support method using the single artificial variable technique to solve the problem (3.39).

*Phase 1. *After adding the slack variables and to the problem (3.39), we get the following problem in standard form:
where
The application of Procedure 1 to the problem (3.41) gives us the following initial support: . In order to find an initial feasible solution to the original problem, we add an artificial variable to problem (3.41), and we compute the vector : . Thus, we obtain the following auxiliary problem:
Remark that the SFS , where , and is an obvious support feasible solution for the auxiliary problem. Hence, the primal support method can be applied to solve the problem (3.43) starting with the SFS .

*Iteration 1. *The initial support is , ; the initial feasible solution and the corresponding objective function value are: and .

The vector of multipliers is , and . Hence, the reduced costs vector is .

The suboptimality estimate is . So the current solution is not optimal. The set of nonoptimal indices is .

In order to improve the objective function, we compute the search direction : we have , so ; . Hence, the search direction is .

Since , . So , , and . Hence, the primal step length is . The new solution is , and the new objective value is is optimal for the auxiliary problem. Therefore, the pair , where and , is an SFS for the problem (3.41).

*Phase 2. **Iteration 1. *The initial support is , and .

The initial feasible solution is , and the objective value is .

The multipliers vector and the reduced costs vector are: and . The suboptimality estimate is . So the initial solution is not optimal.

The set of nonoptimal indices is the entering index is .

The search direction is .

The primal step length is . So the leaving index is . The new solution is , and .*Iteration 2. *The current support is , and .

The current feasible solution is , and the objective value is .

The multipliers vector and the reduced costs vector are: and . The suboptimality estimate is .

Therefore, the optimal solution and the optimal objective value of the original problem (3.39) are

#### 4. Experimental Results

In order to perform a numerical comparison between the simplex method and the different variants of the support method, we have programmed them under the MATLAB programming language version 7.4.0 (R2007a).

Since the test problems available in the NETLIB library present bounds on the variables which can be infinite, we have built a sample of 68 test problems having finite bounds and a same optimal objective value as those of NETLIB. The principle of the building process is as follows: let and be the obtained bounds after the conversion of a given test problem from the “mps” format to the “mat” format and the optimal solution. We put and . Thus, the constraint matrix and the objective coefficients vector of the new problem remains the same as those of NETLIB, but the new lower bounds and upper bounds will be changed as follows: Table 1 represents the listing of the main characteristics of the considered 68 test problems, where NC, NV, NEC, and represent, respectively, the number of constraints, the number of variables, the number of equality constraints, and the density of the constraints matrix (the ratio between the number of nonzero elements and the number of total elements of the constraints matrix, multiplied by 100).

There exist many LP solvers which include a primal simplex algorithm package for solving LP problems, we cite: commercial solvers such as CPLEX [37], MINOS [28] and open-source solvers such as LP_SOLVE [38], GLPK [39], and LPAKO [30]. The LP_SOLVE solver is well documented and widely used in applications and numerical experiments [40, 41]. Moreover, the latter has a mex interface called mxlpsolve which can be easily installed and used with the MATLAB language. That is why we compare our algorithm with the primal simplex algorithm implemented in LP_SOLVE. However, MATLAB is an interpreted language, so it takes much time in solving problems than the native language C++ used for the implementation of LP_SOLVE. For this reason, the numerical comparison carried out between our method and LP_SOLVE concerns only the iterations number. In order to compare our algorithm with the primal simplex algorithm in terms of CPU time, we have developed our own simplex implementation. The latter is based on the one given in [42].

In order to compare the different solvers, we have given them the following names.

*Solver1. SupportSav*

The two-phase support method, where we use the suggested single artificial variable technique in the first phase, that is, the initial support is found by applying Procedure 1, and the feasible solution is found by applying Procedure 2.

*Solver2. SupportFab*

The two-phase support method, where we use the full artificial basis technique presented above in the first phase.

*Solver3. SimplexFab*

The two-phase primal simplex method, where we use the full artificial basis technique presented above in the first phase (we use the BFS presented in Remark 3.3, with ).

*Solver4. LP_Solve_PSA*

The primal simplex method implemented in the open-source LP solver LP_SOLVE. The different options are (Primal-Primal, PRICER_DANTZIG, No Scaling, No Presolving). For setting these options, the following MATLAB commands are used.(i)mxlpsolve(“set_simplextype”, lp, 5); (Phase 1: Primal, Phase 2: Primal).(ii)mxlpsolve(“set_pivoting”, lp, 1); (Dantzig’s rule).(iii)mxlpsolve(“set_scaling”, lp, 0); (No scaling).

The Lagrange multipliers vector and the basic components of the search direction in the first three solvers are computed by solving the two systems of linear equations: and , using the factorization of [43]. The updating of the and factors is done, in each iteration, using the Sherman-Morrison-Woodbury formula [42].

In the different implementations, we have used the Dantzig’s rule to choose the entering variable. To prevent cycling, the EXPAND procedure [44] is used.

We have solved the 68 NETLIB test problems listed in Table 1 with the solvers mentioned above on a personal computer with Intel (R) Core (TM) 2 Quad CPU Q6600 2.4 GHz, 4 GB of RAM, working under the Windows XP SP2 operating system, by setting the different tolerances appropriately. We recall that the initial point , needed in the three methods (Solvers 1, 2 and 3), must be located between its lower and upper bounds, so after giving it some values and observing the variation of the CPU time, we have concluded that it’s more efficient to set . The upper bounds for the slack and artificial variables added for the methods SupportSav, SupportFab, and SimplexFab are put to .

Numerical results are reported in Tables 2, 3, 4, and 5, where nit_{1}, nit, and cput represent respectively the phase one iterations number, the total iterations number, and the CPU time in seconds of each method necessary to find the optimal objective values presented in [25] or [45]; cput_{1}, shown in columns 2 and 7 of Table 2, represents the CPU time necessary to find the initial support with Procedure 1. The number of artificial variables added to the original problem in our implementations is listed in Table 6.

In order to compare the different solvers, we have ordered the problems according to the increasing order of the sum of the constraints and variables numbers (NC+NV), as they are presented in the different tables and we have computed the CPU time ratio (Table 7) and the iteration ratio (Table 8) of the different solvers over the solver SupportSav (Solver1), where for each test problem, we have The above ratios (see [46]) indicate how many times SupportSav is better than the other solvers. Ratios greater than one mean that our method is more efficient for the considered problem: let S be one of the solvers 2, 3, and 4. If , (resp., ), then our solver (Solver1) is competitive with SolverS in terms of CPU time, (resp., in terms of iterations number). Ratios greater or equal to one are mentioned with the bold character in Tables 7 and 8.

We plot the CPU time ratios of the solvers SupportFab and SimplexFab over SupportSav (Figures 1(a) and 1(c)), and the iteration ratios of each solver over SupportSav (Figures 1(b), 1(d), and 1(e)). Ratios greater than one correspond to the points which are above the line in the graphs of Figure 1.

The analysis of the different tables and graphs leads us to make the following observations.(i)In terms of iterations number, SupportSav is competitive with SupportFab in solving 74% of the test problems with an average iteration ratio of 1.33. In terms of CPU time, SupportSav is competitive with SupportFab in solving 66% of the test problems with an average CPU time ratio of 1.20. Particularly, for the LP “scagr25” (Problem number 32), SupportSav is 5.14 times faster than SupportFab in terms of iterations number and solves the problem 4.55 times faster than SupportFab in terms of CPU time.(ii)In terms of iterations number, SupportSav is competitive with SimplexFab in solving 79% of the test problems with an average iteration ratio of 1.57. In terms of CPU time, SupportSav is competitive with SimplexFab in solving 68% of the test problems with an average CPU time ratio of 1.35. The peaks of the graph ratios correspond to the problems where our approach is 2 up to 5 times faster than SimplexFab. Particularly, for the LP “capri” (Problem number 21), SupportSav is 5.29 times faster than SimplexFab in terms of iterations number and solves the problem 3.99 times faster than SimplexFab in terms of CPU time. (iii)In terms of iterations number, SupportSav is competitive with LP_SOLVE_PSA in solving 72% of the test problems with an average iteration ratio of 1.49 (the majority of iteration ratios are above the line in Figure 1(e)). Particularly, for the LP “scsd8” (Problem number 55), our method is 9.58 times faster than the primal simplex implementation of the open-source solver LP_SOLVE.(iv)We remark from the last row of Table 6 that the total number of artificial variables added in order to find the initial support for SupportSav (13234) is considerably less than the total number of artificial variables added for SimplexFab (27389) and SupportFab (27389).(v)If we compute the total number of iterations necessary to solve the 68 LPs for each solver, we find (143737) for SupportSav, (159306) for SupportFab, (163428) for SimplexFab, and (158682) for LP_SOLVE_PSA. Therefore, the total number of iterations is the smallest for our solver.

#### 5. Conclusion

In this work, we have proposed a single artificial variable technique to initialize the primal support method with bounded variables. An implementation under the MATLAB environnement has been developed. In the implementation, we have used the *LU* factorization of the basic matrix to solve the linear systems and the Sherman-Morrison-Woodbury formula to update the *LU* factors. After that, we have compared our approach (SupportSav) to the full artificial basis support method (SupportFab), the full artificial basis simplex method (SimplexFab), and the primal simplex implementation of the open-source solver LP_SOLVE (LP_SOLVE_PSA). The obtained numerical results are encouraging. Indeed, the suggested method (SupportSav) is competitive with SupportFab, SimplexFab, and LP_SOLVE_PSA.

Note that during our experiments, we have remarked that the variation of the initial support and the initial point affects the performances (CPU time and iterations number) of the single artificial variant of the support method. Thus, how to choose judiciously the initial point and the initial support in order to improve the efficiency of the support method? In future works, we will try to apply some crash procedure like those proposed in [25, 27] in order to initialize the support method with a good initial support. Furthermore, we will try to implement some modern sparse algebra techniques to update the *LU* factors [47].

#### References

- L. V. Kantorovich,
*Mathematical Methods in the Organization and Planning of Production*, Publication House of the Leningrad State University, 1939, Translated in Management Science, vol. 6, pp. 366–422, 1960. - G. B. Dantzig, “Maximization of a linear function of variables subject to linear inequalities,” in
*Activity Analysis of Production and Allocation*, R. C. Koopmans, Ed., pp. 339–347, Wiley, New York, NY, USA, 1951. View at Google Scholar · View at Zentralblatt MATH - G. B. Dantzig,
*Linear Programming and Extensions*, Princeton University Press, Princeton, NJ, USA, 1963. - V. Klee and G. J. Minty, “How good is the simplex algorithm?” in
*Inequalities III*, O. Shisha, Ed., pp. 159–175, Academic Press, New York, NY, USA, 1972. View at Google Scholar · View at Zentralblatt MATH - R. Gabasov and F. M. Kirillova,
*Methods of Linear Programming*, vol. 1–3, The Minsk University, 1977, 1978 and 1980. - S. Radjef and M. O. Bibi, “An effective generalization of the direct support method,”
*mathematical Problems in Engineering*, vol. 2011, Article ID 374390, 18 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Gabasov, F. M. Kirillova, and S. V. Prischepova,
*Optimal Feedback Control*, Springer, London, UK, 1995. - R. Gabasov, F. M. Kirillova, and O. I. Kostyukova, “Solution of linear quadratic extremal problems,”
*Soviet Mathematics Doklady*, vol. 31, pp. 99–103, 1985. View at Google Scholar · View at Zentralblatt MATH - R. Gabasov, F. M. Kirillova, and O. I. Kostyukova, “A method of solving general linear programming problems,”
*Doklady AN BSSR*, vol. 23, no. 3, pp. 197–200, 1979 (Russian). View at Google Scholar - R. Gabasov, F. M. Kirillova, and A. I. Tyatyushkin,
*Constructive Methods of Optimization. Part1. Linear Problems*, Universitetskoje, Minsk, Russia, 1984. - E. A. Kostina and O. I. Kostyukova, “An algorithm for solving quadratic programming problems with linear equality and inequality constraints,”
*Computational Mathematics and Mathematical Physics*, vol. 41, no. 7, pp. 960–973, 2001. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Kostina, “The long step rule in the bounded-variable dual simplex method: numerical experiments,”
*Mathematical Methods of Operations Research*, vol. 55, no. 3, pp. 413–429, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. O. Bibi, “Support method for solving a linear-quadratic problem with polyhedral constraints on control,”
*Optimization*, vol. 37, no. 2, pp. 139–147, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. Brahmi and M. O. Bibi, “Dual support method for solving convex quadratic programs,”
*Optimization*, vol. 59, no. 6, pp. 851–872, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Bentobache and M. O. Bibi, “Two-phase adaptive method for solving linear programming problems with bounded variables,” in
*Proceedings of the YoungOR 17*, pp. 50–51, University of Nottingham, UK, 2011. - M. Bentobache and M. O. Bibi, “Adaptive method with hybrid direction: theory and numerical experiments,” in
*Proceedings of the Optimization*, pp. 24–27, Universidade Nova de Lisboa, Portugal, 2011. - M. O. Bibi and M. Bentobache, “The adaptive method with hybrid direction for solving linear programming problems with bounded variables,” in
*Proceedings of the colloque International sur l’Optimisation et les Systèmes d’Information (COSI’ 11)*, pp. 80–91, University of Guelma, Algeria, 2011. - M. O. Bibi and M. Bentobache, “An hybrid direction algorithm for solving linear programs,” in
*Proceedings of the International Conference on Discrete Mathematics & Computer Science (DIMACOS’11)*, pp. 28–30, University of Mohammedia, Morocco, 2011. - L. G. Khachian, “A polynomial algorithm for linear programming,”
*Soviet Mathematics Doklady*, vol. 20, pp. 191–194, 1979. View at Google Scholar - N. Karmarkar, “A new polynomial-time algorithm for linear programming,”
*Combinatorica*, vol. 4, no. 4, pp. 373–395, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. I. Gass,
*Linear Programming: Methods and Applications*, McGraw-Hill, New York, NY, USA, 1964. - C. M. Shetty, “A simplified procedure for quadratic programming,”
*Operations Research*, vol. 11, pp. 248–260, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. B. Millham, “Fast feasibility methods for linear programming,”
*Opsearch*, vol. 13, pp. 198–204, 1976. View at Google Scholar - P. Wolfe, “The Composite Simplex Algorithm,”
*SIAM Review*, vol. 7, no. 1, pp. 42–54, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. E. Bixby, “Implementing the simplex method: the initial basis,”
*ORSA Journal on Computing*, vol. 4, no. 3, pp. 1–18, 1992. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - N. I. M. Gould and J. K. Reid, “New crash procedures for large systems of linear constraints,”
*Mathematical Programming*, vol. 45, no. 1–3, pp. 475–501, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - I. Maros and G. Mitra, “Strategies for creating advanced bases for large-scale linear programming problems,”
*INFORMS Journal on Computing*, vol. 10, no. 2, pp. 248–260, 1998. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. A. Murtagh and M. A. Saunders, “MINOS 5.5 User’s Guide,” Tech. Rep. SOL 83–20, Systems Optimization Lab., Stanford University, Stanford, Calif, USA, 1998. View at Google Scholar
- W. Orchard-Hays,
*Advanced Linear-Programming Computing Techniques*, McGraw-Hill, New York, NY, USA, 1968. - S. Lim and S. Park, “LPAKO: a simplex-based linear programming program,”
*Optimization Methods and Software*, vol. 17, no. 4, pp. 717–745, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. D. Sherali, A. L. Soyster, and S. G. Baines, “Nonadjacent extreme point methods for solving linear programs,”
*Naval Research Logistics Quarterly*, vol. 30, no. 1, pp. 145–161, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Bentobache,
*A new method for solving linear programming problems in canonical form and with bounded variables*, M.S. thesis, University of Bejaia, Algeria, 2005. - M. Bentobache and M. O. Bibi, “Two-phase support method for solving linear programming problems with nonnegative variables: numerical experiments,” in
*Proceedings of the Colloque International sur l’Optimisation et les Systèmes d’Information (COSI’08)*, pp. 314–325, University of Tizi Ouzou, Algeria, 2008. - M. Bentobache and M. O. Bibi, “Two-phase support method for solving linear programming problems with bounded variables: numerical experiments,” in
*Proceedings of the Colloque International sur l’Optimisation et les Systèmes d’Information (COSI’09)*, pp. 109–120, University of Annaba, Algeria, 2009. - G. Mitra, M. Tamiz, and J. Yadegar, “Experimental investigation of an interior search method within a simplex framework,”
*Communications of the ACM*, vol. 31, no. 12, pp. 1474–1482, 1988. View at Publisher · View at Google Scholar · View at Scopus - R. J. Vanderbei,
*Linear Programming: Foundations and Extensions*, Kluwer Academic Publichers, Princeton University, 2001. - ILOG CPLEX, “9.0 User’s Manual,” 2003, http://www.ilog.com.
- “LP_SOLVE,” http://sourceforge.net/projects/lpsolve/files/lpsolve.
- A. Makhorin, “GNU linear programming Kit,” Reference Manual Version 4.9, Draft Edition, 2006, http://www.gnu.org/software/glpk/glpk.html.
- P. G. García and Á. Santos-Palomo, “A deficient-basis dual counterpart of Paparrizos, Samaras and Stephanides’s primal-dual simplex-type algorithm,” in
*Proceedings of the 2nd Conference on Optimization Methods & Software and 6th EUROPT Workshop on Advances in Continuous Optimization*, Prague, Czech Republic, 2007. - S. R. Thorncraft, H. R. Outhred, and D. J. Clements, “Evaluation of open-source LP optimization codes in solving electricity spot market optimization problems,” in
*Proceedings of the 19th Mini-Euro Conference on Operations Research Models and Methods in the Energy Sector*, Coimbra, Portugal, 2006. View at Publisher · View at Google Scholar - M. C. Ferris, O. L. Mangasarian, and S. J. Wright, “Linear programming with MATLAB,”
*MPSSIAM Series on Optimization*, 2007. View at Google Scholar - R. H. Bartels and G. H. Golub, “The simplex method of linear programming using LU decomposition,”
*Communications of the ACM*, vol. 12, no. 5, pp. 266–268, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “A practical anti-cycling procedure for linearly constrained optimization,”
*Mathematical Programming*, vol. 45, no. 1-3, pp. 437–474, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Gay, “Electronic mail distribution of linear programming test problems,” Mathematical Programming Society COAL, Bulletin no. 13, 1985, http://www.netlib.org/lp/data.
- K. Paparrizos, N. Samaras, and G. Stephanides, “An efficient simplex type algorithm for sparse and dense linear programs,”
*European Journal of Operational Research*, vol. 148, no. 2, pp. 323–334, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. J. H. Forrest and J. A. Tomlin, “Updated triangular factors of the basis to maintain sparsity in the product form simplex method,”
*Mathematical Programming*, vol. 2, no. 1, pp. 263–278, 1972. View at Publisher · View at Google Scholar · View at Scopus