A new infeasible interior-point algorithm for linear programming

A New Infeasible Interior-Point Algorithm ∗ for Linear Programming Miguel Argáez [email protected] Leticia Velázquez [email protected] Department of Mathematical Sciences The University of Texas at El Paso El Paso, Texas 79968-0514 ABSTRACT where c, x ∈ IRn , b ∈ IRm , A ∈ IRmxn , m < n, and full rank. The dual linear problem associated with problem (1) can be written In this paper we present an infeasible path-following interiorpoint algorithm for solving linear programs using a relaxed notion of the central path, called quasicentral path, as a central region. The algorithm starts from an infeasible point which satisfies that the norm of the dual condition is less than the norm of the primal condition. We use weighted sets as proximity measures of the quasicentral path, and a new merit function for making progress toward this central region. We test the algorithm on a set of NETLIB problems obtaining promising numerical results. maximize bT y subject to AT y + z = c z ≥ 0, where z ∈ IRn and y ∈ IRm . A point (x, y, z) is said to be an interior point for problems (1) and (2) if (x, z) > 0. For µ > 0, a point on the central path satisfies the perturbed Karush-Kuhn-Tucker (KKT) conditions associated with problems (1) and (2) are 1 0 Ax − b Fµ (x, y, z) = @AT y + z − cA = 0, XZe − µe Categories and Subject Descriptors G.4 [Mathematics of Computing]: Mathematical Software—efficiency; G.1.6 [Numerical Analysis]: Optimization—linear programming (x, z) ≥ 0, General Terms where X = diag(x), Z = diag(z), and e = (1, . . . , 1)T ∈ IRn . The Newton direction for this system is 1 10 1 0 0 b − Ax A 0 0 ∆x @ 0 AT I A @∆y A = @c − AT y − z A . (3) ∆z µe − XZe Z 0 X Algorithms, Experimentation Keywords Interior-point methods, primal-dual, Newton’s method, merit function 1. (2) Now, the perturbed KKT conditions promote global convergence and avoids the procedure for converging to spurious solutions. The central path has the property that runs through the strictly feasible set INTRODUCTION We consider the primal linear problem in the standard form minimize cT x subject to Ax = b x ≥ 0, F o = {(x, y, z) ∈ IR2n+m : Ax = b, AT y+z = c, (x, z) > 0}, (1) keeps iterates at an adequate distance from the non-optimal borders, and ends at a particular solution called the analytic center. Due to this property, most of the primal-dual interiorpoint algorithms are based on following explicitly or implicitly the central path as a guide for obtaining a solution of the primal and dual problems. Contrary to this central region, we use a relaxed notion of the central path called quasicentral path introduced in nonlinear programming by [1, 2], and analyzed in [3]. The use of the quasicentral path, as opposed to the central path, yields a definite advantage. Specifically, for problems where the strictly feasible set F o is empty. ∗This work was supported by the US Department of the Army under Grant DAAD19-01-1-0741. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. TAPIA’03, October 15–18, 2003, Atlanta, Georgia, USA. Copyright 2003 ACM 1-58113-790-7/03/0010 ...$5.00. 12 2. PATH-FOLLOWING STRATEGY Table 1: Numerical Results for Algorithm 3.1 We present a general description of a path-following strategy as follows: For a fixed µ > 0, working from an interior point (x, y, z), apply a linesearch Newton’s method to the perturbed KKT conditions until an iterate arrives at a specified measure of nearness of the central region. If an optimal solution is not found, decrease µ and the process is repeated. A solution is found as µ approaches zero. The quasicentral path, introduced by [1, 2], is defined as the collection of points (x, z) > 0 parameterized by µ > 0 such that Ax − b = 0, XZe − µe = 0. Problem afiro blend adlittle sc205 sc50a sc50b scsd1 scsd6 scagr7 sctap1 and The quasicentral path is equivalent to the region of strictly feasible points for the primal problem. Now, let us define the errors for the primal and dual equations as e1p = b − Ax1 = (1 − α1 ) eop , and e1d = c − AT y1 − z1 = (1 − α1 ) eod . Iteratively, we obtain (1 − αj ) eop , and j=1 k Y (1 − αj ) eod . j=1 The proof follows from the last two equations, and the fact that e0d  ≤ e0p . ✷ This property suggests that the quasicentral path can be used as a central region to reach a solution of the problem. Instead of following the quasicentral path exactly, which requires high computational cost, proximity measures of the quasicentral path are defined. For µ > 0 and γ ∈ (0, 1), we say that a point (x > 0, z > 0) ∈ IRn+n is sufficiently close to the quasicentral path, if it is contained in the following neighborhood: The Algorithm 3.1, written in MATLAB, was run on a set of ten NETLIB problems using a Sun Ultra 10 machine. We report the number of Newton iterations it took for solving each problem on Table I. The perturbation parameter µ is updated by µ = 10−2 ∗ ( ep 2 + (XZ)−1/2 (XZe − µe)2 ) when the iterate (x, z) is inside the proximity measure set. We set the initial value of y = 0, and choose the initial point (x, z) using a standard procedure. Now if b−Ax > c−z, we find a positive scalar ζ such that c − z ≤ b − A(ζx), and let the initial point be (ζx, z). To make progress to the quasicentral path, we use the following function as a merit function: n X 1 Ax − b2 + (xi zi − µ ln(xi zi )). 2 i=1 The Newton step is a descent direction for this function at any point that is not in the quasicentral path. 3. e0p  2.94e3 2.06e3 2.80e4 2.38e3 1.13e3 1.47e3 8.91e2 1.51e3 2.00e4 1.51e6 4. NUMERICAL EXPERIMENTATION N (γµ) = {Ax − b2 + (XZ)−1/2 (XZe − µe)2 ≤ γµ}. Φµ (x, z) = e0d  3.07e1 4.00e1 1.93e4 4.46e1 2.21e1 2.19e1 8.12e2 1.38e3 1.43e4 4.86e3 (1) Given an interior point (x, z), ǫ, µ > 0, and τ, γ ∈ (0, 1). (2) Initialize ed = c − z, ep = b − Ax, and ec = µe − XZe where ed  ≤ ep . (3) Repeat Steps (3)(a)-(3)(f) until step (3)(f) is satisfied. (a) Newton direction. Solve system (3) for (∆x, ∆y, ∆z). (b) Maintain x and z positive. Calculate α̃ = min(1, τ α̂) where α̂ = min(X −1 −1 . ∆x,Z −1 ∆z) (c) Sufficient decrease. Find α = ( 21 )t α̃ where t is the smallest positive integer such that Φµ (x + α∆x, z + α∆z) ≤ Φµ (x, z) + 10−4 α∗ ∇Φµ (x, z)T (∆x, ∆z). (d) Update (x, z) = (x, z) + α(∆x, ∆z), ed = (1 − α)ed , and ep = (1 − α)ep . (e) Proximity to the quasicentral path. if ( ep 2 + (XZ)−1/2 (XZe − µe)2 ) ≤ γµ, then go to step (3)(f). else set ec = µe − XZe, and go to step (3)(a). (f) Stopping criteria. ` ´ 2ep  xT z if max(1,b,c) + max(1,|c ≤ ǫ, STOP. T x|) else update µ, set ec = µe − XZe, and go to step (3)(a). (4) Termination. Return x. Proof. By applying a damped Newton’s method to primal and dual residuals at an infeasible starting point, then ekd = c − AT yk − zk = No. of Iterations 9 16 18 15 11 10 12 15 15 23 Algorithm 3.1 Proposition 1. If the initial point is such that e0d  ≤ then the dual error, ekd , is zero if the primal error, ekp , is zero. e0p , k Y n 51 114 137 317 77 76 760 1350 185 660 as a central region. As a measure of nearness to the quasicentral path and to make progress towards this region, we use the set N (γµ) and merit function Φµ (x, z), respectively. ekp = b − Axk , and ekd = c − AT yk − zk . ekp = b − Axk = m 27 74 55 205 49 48 77 147 129 300 INFEASIBLE ALGORITHM 5. CONCLUSIONS We present an infeasible path-following algorithm for computing a solution that uses the primal-dual interior-point framework proposed by [4]. The quasicentral path is used We present an infeasible primal-dual interior-point algorithm for linear programming featuring a new central region 13 [2] M. Argáez and R. A. Tapia. On the global convergence of a modified augmented lagrangian linesearch interior-point newton method for nonlinear programming. J. Optim. Theor. Appl., 114(1):1–25, August 2002. [3] M. Argáez, R. A. Tapia, and L. Velázquez. Numerical comparisons of path–following strategies for a primal–dual interior–point method for nonlinear programming. J. Optim. Theor. Appl., 114(2):255–272, August 2002. [4] M. Kojima, S. Mizuno, and A. Yoshise. A primal-dual interior point method for linear programming. Progress in Mathematical Programming: Interior-Point and Related Methods, pages 29–47, 1989. called quasicentral path, a new merit function, and weighted proximity measures. The dual variable y does not play any role, at least explicity, in our procedure. The numerical results indicate the quasicentral path is enough to guide the iterates to a solution of the problem. Further numerical and theoretical research is needed to establish the role that the quasicentral path plays for solving linear programming problems. 6. ACKNOWLEDGMENTS The authors thank the referees for their valuable comments. 7. REFERENCES [1] M. Argáez. Exact and Inexact Newton Linesearch Interior-Point Algorithms for Nonlinear Programming Problems. PhD thesis, Rice University, Houston, TX, 1997. 14