A New Product Form of the Inverse
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1 Applied Mathematical Sciences, Vol 6, 2012, no 93, A New Product Form of the Inverse Souleymane Sarr Department of mathematics and computer science University Cheikh Anta Diop Dakar, Senegal ssarr@voilafr Youssou Gningue Department of mathematics and computer science Laurentian s University Ontario, Canada Abstract Many algorithms of solving linear programs are based on the revised simplex method The product form of the inverse is used to inverse the base in the revised simplex methodwe present in this paper an inversion of matrix which complexity is quadratic This method is more efficient than the product form of the inverse We tested the revised simplex method and the algorithm proposed about 55 linear problems The densities of the LPs are 10%, 5% and 25% they are randomly generated Keywords: the revised simplex method, product form of the inverse, linear programming, simplex method 1 Introduction The revised simplex is based on the different steps of the simplex tableau In the tableau simplex, the next iteration is determined by the Gauss-Jordan row operations but in the revised simplex, it is used the base and its inverse the inverse is determined by using the product form of the inverse Noting that the inversion of the base is the costliest [7] step among the steps of the revised simplex The method we present here is obtained from the product form of the inverse It s complexity is quadratic In fact, the new base is inverted directly
2 4642 S Sarr and Y Gningue every 20 iterations Many variants of the revised simplex exist Among them we can cite the Bartels-Golub s method, the sparse Bartels-Golub s method, the Reid s method, and the forrest-tomlin s method This paper is subdivided in five sections We present the revised simplex method in the second section In the third section, the new product form is exposed The numerical results are presented in the four th section and we conclude in the fifth section 2 Algorithm of the revised simplex method The algorithm solves the following linear problem Max Z = c j x j subject to aij x j b i i =1, 2,,mandj =1, 2,n x j 0 c 1,c 2,,c n are the cost coefficients x 1,x 2,,x n are the decision variables a ij for i =1, 2,,m and j =1, 2,,n are called the technological coefficients The computations in the revised simplex are based on the original data (C, A, b) and the inverse of the basis In each iteration the inverse of the basis is computed by using the product form of the inverse 21 Product form of the inverse Let P r the entering vector and P s the leaving vector in the current simplex iteration Suppose that B 1 is the inverse of the basis, to compute Bnext 1 we use the formula : B 1 next = EB 1 E is an m-identity matrix whose sth column is replaced by :
3 A new product form of the inverse 4643 ξ = 1 (B 1 P r ) s (B 1 P r ) 1 (B 1 P r ) 2 (B 1 P r ) (s 1) 1 (B 1 P r ) (s+1) (B 1 P r ) m with (B 1 P r ) s > 0 22 Algorithm Let C B the row vector of cost coefficients associated to the basic variables, P j the jth column of A associated to x j we have A =[P 1,P 2,,P n ] Step 1 Initialization Put the problem in a standard form : B = I m = B 1, ˆb = b Step 2 optimality conditions For each nonbasic variable x j determine the cost coefficient associated Ĉ j = C B B 1 P j C j Then, let Ĉr = min{ĉj} If Ĉr 0, then the current basic feasible solution is optimal, Z = C Bˆb stop Otherwise Step 3 entering variable x r is the entering variable Step 4 leaving variable Let a r = B 1 P r, the leaving variable x s is determined as follows : min { } ˆbi /(a r ) i > 0 = ˆb s with (a r ) i (a r ) ˆb = B 1 b s If (a r ) i 0 i then the optimal solution is unbounded Otherwise let γ = ˆb s (a r) s
4 4644 S Sarr and Y Gningue Step 5 update We get ˆb = ˆb γa r, then ˆb(s) =γ The inverse of the new base is determined by the product form of the inverse Bnext 1 = EB 1 with E =(e 1,e 2,,e s 1,ξ,e s+1,,e m ), where e i is a vector of zeros except for 1 at the ith position go to step 2 ξ = 1 (a r ) s (a r ) 1 (a r ) 2 (a r ) (s 1) 1 (a r ) (s+1) (a r ) m 3 The new product form of the inverse Suppose that x s is the leaving variable, B 1 next is determined by the following formula : B 1 next = B 1 +(ξ e s )L s e s is a vector (dimension m) of zeros except for 1 at the sth position L s =(l s1,l s2,,l sm ) is the sth row of B 1 31 Algorithm Step 1 Initialization Put the problem in a standard form : B = I m = B 1, ˆb = b Step 2 optimality conditions For each nonbasic variable x j determine the cost coefficient associated Ĉ j = C B B 1 P j C j Then, let Ĉr = min{ĉj} If Ĉr 0, then the current basic feasible solution is optimal, Z = C Bˆb stop Otherwise
5 A new product form of the inverse 4645 Step 3 entering variable x r is the entering variable Step 4 leaving variable Let a r = B 1 P r, the leaving variable x s is determined as follows : min { } ˆbi /(a r ) i > 0 = ˆb s with (a r ) i (a r ) ˆb = B 1 b s If (a r ) i 0 i then the optimal solution is unbounded Otherwise let γ = ˆb s (a r) s Step 5 update We get ˆb = ˆb γa r, then ˆb(s) =γ Let B 1 =(l ij ) with i =1,,m and j =1,,m The inverse of the new base is determined as follows : ξ = 1 (a r ) s (a r ) 1 (a r ) 2 (a r ) (s 1) 1 (a r ) (s+1) (a r ) m then ξ(s) =ξ(s) 1 Extract the sth row of B 1 : L s =(l s1,l s2,,l sm ) Now we get : go to step 2 B 1 nouv = B 1 + ξ L s 32 Proof In the revised simplex, the inverse of the new base is determined by the product form of the inverse : B 1 next = EB 1
6 4646 S Sarr and Y Gningue E =(e 1,e 2,,e s 1,ξ,e s+1,,e m ) with ( ξ t = a 1r, a 2r,, a (s 1)r, a sr a sr a sr 1 a sr, a (s+1)r a sr I is an m-identity matrix Let s consider Q = E I then E = I + Q Bnext 1 = EB 1 =(I + Q)B 1 = B 1 + QB 1 We can note B 1 =(l ij ) with i =1,,m and j =1,,m QB 1 =(c ij ) with i =1,,m and j =1,,m Thus we get : c ij = a ir a sr l sj i s, a ) mr a sr c sj = ( 1 a sr 1 ) l sj Let L s =(l s1,l s2,,l sm ) We have the equality : QB 1 =(ξ e s )L s Therefore Bnext 1 = B 1 + QB 1 = B 1 +(ξ e s )L s Then Bnext 1 = B 1 +(ξ e s )L s 33 Algorithmic analysis To compute B 1 next we use the formula : B 1 next = B 1 + ξ L s with ξ = ξ e s Let D = ξ L s, we get now : Bnext 1 = B 1 + D D = ξ L s is a product of a vector column of dimension m and a row vector of dimension m, we have here m 2 multiplications Bnext 1 = B 1 + D is a sum of two matrices, we get m 2 additions Thus the new form product is O(m 2 ) 4 Numerical results In this section we present numerical results on randomly generated sparse LPs we use three different cases of density 10% and 5% and 25% The LPs that have been solved are under the form : max Z = CX subject to AX b X 0
7 A new product form of the inverse 4647 A is a matrix of dimension m n a ij [50, 400] C is a row vector of dimension n c j [0, 700] b is a vector column of dimension m b i [10, 100] The implemented algorithms are running in Matlab 65 We tested two algorithms : the revised simplex called revsimp, the algorithm in section 21 called mrs NbIter is the number of iterations and nnz is the nonzero elements of the matrix A 5 Conclusion The algorithm presented in this paper is based on the revised simplex method It uses an new inversion of matrix which complexity is quadratic This new inversion of matrix replaces the product form of the inverse The computations show that the operations are reduced considerably
8 4648 S Sarr and Y Gningue Table 1Results on sparse LPs of density 10% and dimension m m Table 2Results on sparse LPs of density 10% and dimension (n + 50) n Table 3Results on sparse LPs of density 10% and dimension n (n + 50)
9 A new product form of the inverse 4649 Table 4Results on sparse LPs of density 5% and dimension m m Table 5Results on sparse LPs of density 5% and dimension (n + 50) n Table 6Results on sparse LPs of density 5% and dimension n (n + 50)
10 4650 S Sarr and Y Gningue Table 7Results on sparse LPs of density 25% and dimension m m References [1] J Acher, J Gardelle, Programmation linéaire Dunod décision, 1978 [2] G Desbazeille,Exercices et problémes de recherche opérationnelle Bordas, Paris, 1976 [3] RFaure, BLemaire, C Picouleau, Précis de recherche opérationnelle 5 e édition Dunod (2000) [4] YGningue, La programmation linéaire avec applications aux problèmes de transport université Laurentienne, Ontario, Canada, (2000) [5] THamdy, Operations research an introduction sixth édition Prentice- Hall, Inc, (1997) [6] AKaufman, Méthodes et modéles de la recherche opérationnelle tome 1 Dunod, Paris (1962) [7] SS Morgan, A comparison of simplex method algorithms Thesis, university of Florida, (1997) [8] KPaparrizos, SNikolaos, GStephanides,A new primal dual simplex algorithm Computers and operations research, 2001 Received: April, 2012
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