• Aucun résultat trouvé

A modified algorithm for the strict feasibility problem

N/A
N/A
Protected

Academic year: 2022

Partager "A modified algorithm for the strict feasibility problem"

Copied!
5
0
0

Texte intégral

(1)

RAIRO Oper. Res.35(2001) 395-399

A MODIFIED ALGORITHM FOR THE STRICT FEASIBILITY PROBLEM

D. Benterki

1

and B. Merikhi

1

Communicated by Jean-Pierre Crouzeix

Abstract. In this note, we present a slight modification of an algo- rithm for the strict feasibility problem. This modification reduces the number of iterations.

Keywords:Strict feasibility, interior point methods, Ye–Lustig algorithm.

1. Introduction

This paper is concerned with the problem of findingx∈Rn such that

x >0 and Ax=b (F)

whereAis am×nreal matrix of rankm,b∈Rmand 0< m < n.

This problem, called a strict feasibility problem, occurs in many optimization problems in linear or quadratic programming. Such problems are of type

Minimizef(x) subject tox∈S ={x∈Rn : x≥0, Ax=b}· (P)

Let us denote bySethe following subset:

Se={x∈Rn : x >0, Ax=b}·

Received September, 2001.

1 epartement de Math´ematiques, Facult´e des Sciences, Universit´e Ferhat Abbas, S´etif, 19000 Alg´erie.

(2)

Interior point methods for solving (P) start from some arbitrary initial point x0 Se and build a sequence {xk} ⊂ Se expected to converge to some optimal solution x of (P). Thus, the first step in interior point methods consists in finding an initial pointx0 in a efficient way. To do that, it is usual to introduce an artificial variableλand to consider the linear programming problem

Minimizex, λ : Ax+λ(b−Aa0) =b, x≥0, λ0} (LF) where a0 is a arbitrary fixed point in the positive orthant of Rn. It is noticed that x is a solution of the feasibility problem (F) if and only if (x,0) is an optimal solution of (LF) withx>0.

In fact, numerical algorithms provide only approximate optimal solutions for an optimization problem. In our case, an approximate solution of (LF) can be obtainedvia a classical interior method as for instance the Ye–Lustig algorithm that we described below. But before, we precise the notation used in this algo- rithm:ε >0 corresponds to the precision of the approximation, r= 1

(n+1)(n+2), c= (0,0,· · ·,0,1)tRn+1,en+2= (1,1,· · ·,1)tRn+2, ˜x= (x, λ)tRn+1 and B = [A, b−Aa0] is a(n+ 1) matrix.

2. The original algorithm and its modification

Let us describe the original algorithm:

The Ye–Lustig algorithm[3]

a) Initialization: start with x0=a0, λ0= 1,x˜0= (x0, λ0)tand k= 0;

IfkAx0−bk ≤ε Stop: x0 is anε-approximate solution, If not go to b).

b)Ifλk ≤ε Stop: xk is anε-approximate solution, If not go to c).

c) Set Dk = diag(˜xk) and

Compute the projection pk of the vector (Dkc,−ctx˜k)t Rn+2 on the kernel of the(n+ 2) matrixBk= [BDk,−b],

Take yk+1= en+2n+2 −αkrkppkkk, whereαk is obtained by a line search, Take ˜xk+1= (xk+1, λk+1)t= (yn+2k+1)1Dkyk+1[n+ 1],

d) dok=k+ 1 and go back to b).

We propose a slight modification of this algorithm by modifying the stopping criteria.

(3)

The modified algorithm

a’) Initialization: start withx0=a0, λ0= 1,andk= 0;

IfkAx0−bk ≤ε Stop: x0 is anε-approximate solution.

If not compute the solutionu0of the linear system AAtu0=λ0(b−Aa0).

b’) Ifλk ≤ε Stop: xk is anε-approximate solution.

If not

Take uk =λku0,

Take zk =diag[(xk)]1Atuk,

– Ifmax|zk|i <1 Stop: xk+Atuk is anε-approximate solution.

If not go to c’)

c’) is identical to c) of the original algorithm.

d’) dok=k+ 1 and go back to b’).

The computation of the vector u0 occurs once only, it can be performed by a Cholesky method. It remains to prove the validity of the new stopping criteria max|zk|i <1. This is done in the following proposition.

Proposition 2.1. Ifmax|zk|i<1thenxk+Atuk>0and A(xk+Atuk) =b.

Proof. 1) Notice thatdiag(xk)zk =Atuk, thenxk+Atuk =xkdiag(xk)zk= diag(xk)(en−zk)>0 becausexk>0 and|zk|i<1 for all i.

2) Since (xk, λk)tis a feasible solution of (LF) thenA(xk+Atuk) =Axk+AAtuk= b−λk(b−Aa0) +λkAAtu0=b−λk(b−Aa0) +λkλ0(b−Aa0) =b.

This modification brings a significant improvement in the number of iterations with only a very small increasing in the cost per iteration. We illustrate that by a few examples.

3. Examples

In this examplesεhas been taken equal to 103 or 106 according to the case.

3.1. Some examples

The following examples are taken form the literature see for instance [1, 4]. In

(4)

example taille Nbr of iterations Nbr of iterations m×n Ye–Lustig modified algorithm

1 3×5 4 1

2 3×6 3 1

3 5×11 5 4

4 6×12 3 1

5 11×25 4 3

6 16×27 6 5

7 11×28 7 6

3.2. Cube example

n= 2m,A[i, j] = 0 if i6=j or (i+ 1)6=j A[i, i] =A[i, i+m] = 1, b[i] = 2, fori, j= 1· · ·m.

Dimension Nbr of iterations Nbr of iterations m×n Ye–Lustig Modified algorithm

50×100 3 1

100×200 3 1

150×300 3 1

200×400 3 1

3.3. Hilbert example

n= 2m,A[i, j] = i+j1 , A[i, i+m] = 1, b[i] =Pm

j=1 1

i+j, fori, j= 1· · ·m.

Dimension Nbr of iterations Nbr of iterations m×n Ye–Lustig Modified algorithm

50×100 3 1

100×200 3 1

150×300 3 1

200×400 3 1

(5)

References

[1] D. Benterki,Etude des performances de l’algorithme de Karmarkar pour la programmation´ lin´eaire. Th`ese de Magister, D´epartement de Math´ematiques, Universit´e de Annaba, Alg´erie (1992).

[2] J.C. Culioli,Introduction `a l’optimisation. ´Edition Marketing, Ellipses, Paris (1994).

[3] I.J. Lustig,A pratical approach to Karmarkar’s algorithm. Technical report sol 85-5, De- partment of Operations Research Stanford University, Stanford, California.

[4] A. Keraghel,Etude adaptative et comparative des principales variantes dans l’algorithme´ de Karmarkar, Th`ese de Doctorat de math´ematiques appliqu´ees. Universit´e Joseph Fourier, Grenoble, France (1989).

[5] D.F. Shanno and R.E. Marsten, A reduced-gradient variant of Karmarkar’s algorithm and null-space projections.J. Optim. Theory Appl.57(1988) 383-397.

[6] S.J. Wright,Primal-dual interior point method. SIAM, Philadelphia, PA (1997).

to access this journal online:

www.edpsciences.org

Références

Documents relatifs

Hence the idea: If we saw servers or nodes of the graph as a node in an electrical circuit, and the &#34;distances&#34; between vertices as a resistor values in a circuit, then more

Ranking is itself information that also reflects the political, social, and cultural values of the society that search engine companies operate within...’ (p.148).. Since 2011,

Zawadzki: Der Kommentar Cyrills von Alexandrien zum 1.. Zawadzki: Der Kommentar Cyrills von Alexandrien

running CPU time in seconds; rp: denotes the ratio y c + where y + is the capacity level where the algorithm detects that the periodicity level y ∗ is reached; vrs: number of

• Stopping condition redesigned (integrated with the exploration/exploitation strategy and based on the interest of the search and diversity of new solutions). ACOv3f: algorithm

To determine the role of the regulation of brain LDLR by PCSK9 and because apoE is the main apolipoprotein in brain that can bind LDLR, we quantifi ed the protein levels of

Our first experiment (4.2), shows that although undirected graphical models with empirical risk minimization (EGMs) are trained specifically for certain inference tasks,

لولأا لصفلا 19 ➢ :نولدتعلما نويسويرلأا نويسويرلأا ناك ،ىمادق ينينايجروأ نولدتعلما ةيرصيق فقسأ برتعي ثيح سويباسوأ ،مله اميعز مهف صي اتلا ىلع نور و بلآا ينب زييم