Following The Central Trajectory Using The Monomial Method Rather Than Newton's Method


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1 Following The Central Trajectory Using The Monomial Method Rather Than Newton's Method YiChih Hsieh and Dennis L. Bricer Department of Industrial Engineering The University of Iowa Iowa City, IA February 1994 Abstract A new infeasible pathfollowing algorithm based on the monomial method, rather than Newton's method, is proposed to solve the convex quadratic programming problem. This algorithm generates a sequence of solutions which is exactly on the central trajectory. The different performances between the algorithms based on both Newton's and the monomial methods are illustrated by computational results. KEYWORDS: Convex Quadratic Programming, PathFollowing Algorithm, InteriorPoint Method, Monomial Method
2 1. Introduction We consider the linearly constrained convex quadratic programming problem (QP) where Q R n n 1 Min 2 xt Qx + c T x s.t. Ax b x 0 is a symmetric positive semidefinite matrix, A R m n, x, c R n, and b R m. QP problems have been widely studied, and many algorithms have been proposed to solve them. In 1979, Kozlov et al. [9] first proposed a polynomialtime algorithm for QP problems based on the ellipsoid method. With the advent of the interior point algorithm by Karmarar [7] for solving linear programming problems (LP), several algorithms based on the interior point method for solving LP and QP problems have been studied. Most of them are based on the Newton's method for solving the system of nonlinear KarushKuhn Tucer (KKT) equations. (See, for example, Goldfarb and Liu [6], Monteiro and Adler [12], and Mehrotra and Sun [10] for QP, and Renegar [14], Kojima et al. [8], Monteiro and Adler [11], and McShane et al. [13], etc., for LP.) Recently, a method called the "monomial method" has been used to solve a system of algebraic nonlinear equations (see, for example, Burns [2], [3], and [4]). It is well nown that Newton's method uses the linear part (firstorder) of the Taylor series expansion to approximate each nonlinear equation. In contrast to Newton's method, the monomial method is based on a system of approximating equations that are monomial in form. We may say that the monomial method is based on a different type of linearization. Although the monomial method is based on an alternative type of linearization, its performance seems very different from that of Newton's method. From Burns' examples (Burns [2], [3], and [4]), it appears that for some cases the performance of the monomial method seems better than that of Newton's method in many features. For example, the 1
3 monomial method converges much faster than Newton's method if given extreme starting points, and the monomial method reduces some computational error, e.g. floating point overflow. The main purpose of this paper is to demonstrate the different performances of the algorithms based on Newton's method and the monomial method for QP problems. The sections which follow are organized thus: After a brief description of the system of nonlinear KKT equations in Section 2, we outline the algorithm based on Newton's method in Section 3. In Section 4, the basic concept of the monomial method is presented and the algorithm based on this method is proposed. Computational results and brief conclusions are provided in the last Section. 2. Convex Quadratic Problem Consider the standard convex quadratic program: (QP) Its dual is: Min s.t. 1 2 xt Qx + c T x Ax y = b x, y 0 (QPD) Max 1 2 xt Qx + b T w s.t. Qx + A T w + s = c s,w 0 where x, s,c R n 1, y,w, b R m 1, Q R n n, and A R m n. We impose the following assumptions (A1) The matrix Q is positive semidefinite. (A2) The constraint matrix A has full row ran. (A3) The feasible region is nonempty and bounded. 2
4 For x, y > 0 in (QP) and s, w > 0 in (QPD), we can apply the logarithmic barrier function technique, and obtain the nonlinear programming problems, (QP µ ) and (QPD µ ): (QP µ ) and Min s.t. n m 1 2 xt Qx + c T x µ log x j µ log y j Ax y = b x, y > 0 j =1 j =1 (QPD µ ) Max 1 2 xt Qx + b T w + µ log w j + µ log s j s.t. Qx + A T w + s = c w,s > 0 m j =1 n j =1 where µ > 0 is a barrier parameter. It is expected that the optimal solution of problem (QP µ ) would converge to the optimal solution of the original problem (QP) as µ 0. Convex programming theory further implies that the global solution, if one exists, is completely characterized by the KKT conditions as: Ax y = b x, y > 0 (primal feasibility) (2.1a) Qx + A T w + s = c s,w > 0 (dual feasibility) (2.1b) XSe n = µe n (complementary slacness) (2.1c) WYe m = µe m (complementary slacness) (2.1d) where X,S,W, and Y are diagonal matrices using the elements of vectors of x, s, w, and y as diagonal elements, respectively, and e i is the column matrix with i elements, each with value one. 3
5 3 Algorithm Based on Newton's Method Assume that (x,y,s,w ) >0 is a current solution of equation (2.1) for given µ >0. Applying Newton's method, we obtain a system of linear equations for the directions of translation. This system is given by A I 0 0 Q 0 I A T S 0 X 0 0 W 0 Y d x d y d s d w Ax y b Qx + A T w + s c = X S e n µe n W Y e m µe m (3.1) Note that (3.1) can be expressed as Ad x d y = t 1 Qd x + A T d w + d s = t 2 S d x + X d s = t 3 W d y + Y d w = t 4 where t 1 = b + y Ax where t 2 = Qx + c A T w s where t 3 = µ e n X S e n where t 4 = µ e m W Y e m (3.2a) (3.2b) (3.2c) (3.2d) Solving (3.2), one can derive the directions for iteration as [ ] 1 [(W t 1 + t 4 ) + W A(S + X Q) 1 (X t 2 t 3 )] (3.3a) d x = (S + X Q) 1 [ X A T d w (X t 2 t 3 )] (3.3b) d w = W A(S + X Q) 1 X A T + Y d y = Ad x t 1 (3.3c) and d s = X 1 (t 3 S d x ) (3.3d) Thus, a new solution can be obtained by choosing the appropriate step sizes α p for primal and α d for dual, such that x +1 = x + α p d x y +1 = y + α p d y s +1 = s + α d d s w +1 = w + α d d w (3.4a) (3.4b) (3.4c) (3.4d) where 4
6 α 1 p = min max 1, d x i i { αx i }, 1 max 1, d y i { i αy i } (3.5a) and α 1 d = min max 1, d w i i { αw i }, 1 max i 1, d { s i αs i } (0 < α <1). (3.5b) For each iteration, the barrier parameter µ is adjusted as follows: µ = σ (x ) T s + (y ) T w where 0 < σ <1 (3.6) n + m Therefore, we can introduce the algorithm based on Newton's method as : Algorithm based on Newton's method (Algorithm NM) : Step 1 : (Initialization) Set =0. Start with any initial solution (x 0,y 0,s 0,w 0 )>0. Choose three small values for ε 1, ε 2, and ε 3, and α,σ (0,1). Step 2 : (Intermediate computation) Compute µ by (3.6) and t 1,t 2,t 3 and t 4 by (3.2), respectively. Step 3 : (Checing optimality) If µ < ε 1, t 1 b +1 < ε, and t 2 2 Qx + c + 1 < ε 3 (3.7) then stop; the current solution is accepted as the optimal solution. Else proceed to the next step. Step 4 : (Finding the directions) Compute d w,d x,d y, and d s by (3.3). Step 5 : (Computing step sizes) Compute α p and α d by (3.5). Step 6 : (Finding the new solution) Compute x +1, y +1,s +1 and w +1 by (3.4) Set = +1 and go to step 2. 5
7 4. Algorithm Based on the Monomial Method 4.1 Basic Concepts of the Monomial Method Consider the following general class of N nonlinear equations with N unnowns of the form: T q s ˆ ˆ iq i =1 c iq N x ˆ a ijq = 0, q = 1,2,..., N. (4.1) j j =1 where s ˆ iq { 1,+1} refer to signs of the terms, ˆ c iq >0 are the coefficients, a ijq, which are real numbers without restriction in sign, are the exponents, ˆ x j >0 are the variables, T q is the total number of terms in equation q. We define u iq = ˆ c iq Let T q + = i ˆ s iq = 1 N a x ˆ ijq j j =1, so that (4.1) can be rewritten as T q s ˆ u iq iq = 0, q = 1,2,..., N. (4.2) i =1 { } and T q = { i s ˆ iq = 1} for q = 1,2,..., N. Hence, (4.2) can be further expressed as : u iq = 0, q = 1,2,..., N. u iq + i T q or equivalently, Note that u iq >0. We further define u iq + u iq = 1, q = 1,2,..., N. (4.3) = u iq P q i T + q and = u iq Q q (4.4) 6
8 where P q = and Q q = u iq, u iq i T q + u iq = u iq ˆ x = x, P q = P q ˆ x = x, and Q q = Q q x ˆ. = x Property 4.1: u iq u iq + + ( ) and uiq ( ) δ iq u iq, q =1,2,..., N. with equalities if and only if u iq is a constant for q = 1,2,..., N. Using this property, we can approximate (4.3) as δ ( u iq ) iq + ( u iq ) =1 (4.5) or equivalently N x ˆ D jq =1 (4.6) j H q j =1 where c ˆ iq i T H q = q +( ) ˆ c iq i T q ( ) and D jq = a ijq i T q + Transforming the variables according to x ˆ j = e z j, we have N D jq j =1 a ijq (4.7) z j = log H q, q = 1,2,..., N. (4.8) Thus, solving the linear equation (4.8) for z j and we can find the new iterate as x ˆ = e z j. 4.2 Algorithm Based on the Monomial Method Applying the monomial method to the system of KKT equations (2.1), we have the following system of equations: 7
9 A 1x A 1y 0 0 A 2x 0 A 2s A 2 w I 0 I 0 0 I 0 I z x +1 z y +1 z s +1 z w +1 = ξ x ξ y ξ s ξ w (4.9) where A 1x R m n, A 1y R m m, A 2x, A 2s R n n, A 2w R n m, ξ x,ξ w R m 1, and ξ y,ξ s R n 1. Note that the dimension for the matrix of lefthandside of (4.9) is (2n + 2m) (2n + 2m), that is, there are (2n + 2m) variables in the system of linear equations Property 4.2: The elements of matrices ξ s and ξ w are all log( 1 µ ) where µ = σ (x ) T s + (y ) T w n + m, 0 < σ <1. This property implies that the sequence of solutions is exactly on the central trajectory. Equation (4.9) may be solved as follows: From (4.9), we get By (4.10c), By (4.10d), A 2x A 1x z x +1 + z x +1 + A 2 s A 1y z s +1 + z y +1 = ξ x A 2w z +1 x + z s = ξ s z +1 y + z w = ξ w z +1 s = ξ +1 s z x z +1 w = ξ +1 w z y z +1 w = ξ y (4.10a) (4.10b) (4.10c) (4.10d) (4.11) (4.12) Substituting (4.11) and (4.12) to (4.10b), we have z +1 x + (ξ s z +1 x ) + A 2x A 2 s A 2w (ξ w z +1 y ) = ξ y which implies Hence, if ( A 2x ( A 2x A 2s ) z x +1 A 2w A 2s ) has full ran, we further have z +1 y = ξ y ξ s A 2s A 2w ξ w (4.13) 8
10 z +1 x ( A 2x A 2s ) 1 A 2w z +1 y = ( A 2 x Multiplying (4.14) by A 1x, it produces A 1x z +1 x A 1x ( A 2x A 2s ) 1 A 2w z +1 y = A 1x ( Subtracting (4.10a) from (4.15), we obtain A 1x ( A A ) 1 A + A [ ] z +1 2x 2s 2w 1y y = A 1x ( That is z +1 y = A 2 x A 2s A 2x A ( A A ) 1 A + A [ ] 1 ξ 1x 2x 2s 2w 1y x A 1x ( By (4.14), we have z +1 x = ( A 2x A 2s ) 1 A 2 s A ( 2w ξ w ) (4.14) ) 1 ξ y ξ s A 2s ) 1 A 2s ξ y A ξ 2s s A ( ξ 2w w ) (4.15) A A ( 2 s ξ 2w w ) ξ x (4.16) ) 1 ξ y ξ s [ A A ) 1 ( ξ 2x 2 s y A 2s ξ s A ξ )] (4.17) 2 w w ξ y A ξ 2s s A ξ 2w w + A +1 ( 2w z y ) (4.18) Thus, after computing (4.17), (4.18), (4.11), and (4.12), respectively, we may find the new iterate as x +1 y +1 s +1 w +1 = +1 z x e +1 z y e +1 z s e +1 z w e (4.19) Algorithm based on the monomial method (Algorithm MM) : Step 1: (Initialization) Set = 0. Start with any initial solution (x 0,y 0,s 0,w 0 )>0, and choose three small values for ε 1, ε 2, and ε 3. Step 2: (Checing optimality) Compute µ by (3.6) and t 1,t 2 by (3.2), respectively. If (3.7) is satisfied then stop; the current solution is accepted as the optimal solution. Else proceed to the next step. Step 3: (Evaluating weights) Compute the weights of each term and equation for iteration by (4.4). Step 4: (Intermediate computation) 9
11 Compute A 1x, A 1y,,,,ξ x,ξ y,ξ s, and ξ w by (4.7). A 2x A 2s A 2w Step 5: (Solving nonlinear equations) z +1 y, z x +1 z +1 w and z +1 s by (4.17), (4.18), (4.11), and (4.12), respectively. Step 6: (Finding the new solution) Compute x +1, y +1,s +1 and w +1 by (4.19) Set = +1, and go to Step Computational Results and Conclusions 5.1 Computational Results The first test problem is an example due to Bazaraa and Shetty [1] as shown below: Min 2 x x 2 2 2x 1 x 2 4x 1 6x 2 s.t. x 1 x 2 2 x 1 5x 2 5 x 1, x 2 0 This is a very simple example, but will demonstrate the different performances of these two algorithms, Algorithms NM and MM. For ease of comparison of these algorithms, we employ the following procedure: 1. We try 15,000 starting points, in which x 0 = (x 0 1, x 0 2 ) Integer, where x 0 1 [1,150], and x 0 2 [1,100]. 2. y 0 = s 0 = w 0 = (1,1) T for each starting point. 3. The tolerances are ε 1 = ε 2 = ε 3 = σ +1 = if µ +1 µ µ if µ +1 µ µ > 0.5 Figures 1 and 2 illustrate the number of iterations required for these two algorithms to satisfy the convergence criterion. That is, each of the 15,000 starting points is shaded according to the number of iterations required to converge. One can see that Figure 2, 10
12 based on the monomial method, is more regular than Figure 1, based on Newton's method. It should also be noted that the average number of iterations to converge in Figure 1 is (ranging from 17 to 27 iterations), which is larger than that of Figure 2, namely (ranging from 16 to 21 iterations). In addition, we have tested several different sizes of convex quadratic problems. These are separable problems, based upon Calamai's procedure for generating test problems (Calamai et al. [5]), in the form of (SQP) Min F(x) = f l (x 1l, x 2l ) s.t. M l =1 a 11l x 1l + a 12l x 2l α l a 21l x 1l + a 22l x 2l α l x 1l x 2l 30 x 1l, x 2l 0, l { 1,2,..., M} (5.1) where a 11l,a 12 l,a 21l,a 22 l and α l are randomly generated such that assumptions (A2) and { } ρ 1l,ρ 2l 0,1 (A3) are satisfied. f l (x 1l, x 2l ) = 1 2 ρ 1l (x 1l t 1l )2 +ρ 2l (x 2l t 2l ) 2 { } and t 1l,t 2l R n. Note that, for this type of separable convex quadratic problems, the optimal solutions can be specified, and may be either extreme points, interior points, or boundary points. Based on (5.1), we test different sizes of test problems, M =1, 2, 4, 8, 16, 18, 20, and 22. The number of variables and constraints are 2M and 3M, respectively. For each test problem, the optimal solutions for the subproblems may be either extreme points, interior points, or boundary points. We also impose the following conditions. 1. For each problem size, we generate 10 random test problems, and for each of these 10 test problems we try 5 random starting points ((x 0,y 0,s 0,w 0 )) selected from the interval [1,100]. 11
13 2. For each combination of test problem and starting point, algorithms NM and MM were applied. 3. The convergence tolerances are ε 1 = ε 2 = ε 3 = The results shown in Table 1 were obtained using the HP 715/50 worstation. From Table 1, one can see that the average number of iterations for the monomial method is less than those of Newton's method. The cpu time for the monomial method is less than that of Newton's method when the problem size, M, is larger (for instance, when M = 16, 18, 20, and 22). It should be pointed out that Algorithm MM needs more arithmetic operations for each iteration owing to the computation of weights in step 3. However, because for the larger problems the number of iterations required by the Algorithm MM is reduced by approximately one iteration, the total cpu time is less than that of Algorithm NM. 5.2 Conclusions We have proposed a pathfollowing algorithm based on the monomial method rather than Newton's method for the convex quadratic problems. From the limited computational results which we have presented, one may see that the Algorithm MM seems better than Algorithm NM in various features. For example, from Figures 1 and 2, we find that the former is more regular than the latter. From Table 1, one can see that the total number of iterations and cpu time to converge are better for Algorithm MM when the problem size is larger. Further study is required in order to draw more definite conclusions; the authors are now investigating the global convergence and complexity of this new algorithm, as well as performing more complete computational testing. 12
14 REFERENCE [1] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming : Theory and Algorithms, John Wiley and Sons, (1979). [2] S. A. Burns and A. Locascio, "A MonomialBased Method for Solving Systems of NonLinear Algebraic Equations", International Journal for Numerical Methods in Engineering, Vol. 31, pp , (1991). [3] S. A. Burns, "The Monomial Method and Asymptotic Properties of Algebraic Systems", to appear in: International Journal for Numerical Methods in Engineering, (1993). [4] S. A. Burns, "The Monomial Method: Extensions, Variations, and Performance Issues", to appear in: International Journal for Numerical Methods in Engineering, (1993). [5] P. H. Calamai, L. N. Vicente, and J. J. Judice, "New techniques for generating Quadratic Programming Test Problems", Mathematical Programming, Vol. 61, pp , (1993). [6] D. Goldfarb and S. Liu, "An O(n 3 L) Primal Interior Point Algorithm for Convex Quadratic Programming", Mathematical Programming, Vol. 49, pp , (1991). [7] N. Karmarar, "A New Polynomial Time Algorithm for Linear Programming", Combinatorica, Vol. 4, pp , (1984). [8] M. Kojima, N. Megiddo, and S. Mizuno, "Theoretical Convergence of LargeStep PrimalDual Interior Point Algorithms for Linear Programming", Mathematical Programming, Vol. 59., pp. 121, (1993). [9] M. K. Kozlov, S. P. Tarasov, and L. G. Khachian, "Polynomial Solvability of Convex Quadratic Programming", Dolady Aademiia Nau USSR, pp , (1979). 13
15 [10] S. Mehrotra and J. Sun, "An Interior Point Algorithm for Solving Smooth Convex Programs Based on Newton's Method", Contemporary Mathematics, Vol. 114, pp , (1990). [11] R. D. C. Monteiro and I. Adler, "Interior Path Following PrimalDual Algorithms. Part I: Linear Programming", Mathematical Programming, Vol. 44, pp , (1989). [12] R. D. C. Monteiro and I. Adler, "Interior Path Following PrimalDual Algorithms. Part II: Convex Quadratic Programming", Mathematical Programming, Vol. 44, pp , (1989). [13] K. McShane, C. Monna, and D. Shanno, "An Implementation of a PrimalDual Interior Point Method for Linear Programming, ORSA Journal on Computing, Vol. 1, No. 2, pp , (1989). [14] J. Renegar, "A PolynomialTime Algorithm Based on Newton's Method for Linear Programming", Mathematical Programming, Vol. 40, pp , (1988). 14
16 Figure 1. Iteration counts for various starting points using Newton's method for Bazaraa's example. 15
17 Figure 2. Iteration counts for various starting points using monomial method for Bazaraa's example. 16
18 Number of subproblems M=1 M=2 M=4 M=8 M=16 M=18 M=20 M=22 Newton's iterations (cpu time) (0.3512) (0.6912) (1.4610) (6.4020) ( ) ( ) ( ) ( ) Monomial iterations (cpu time) (0.5300) (1.2440) (2.5482) (8.7306) ( ) ( ) ( ) ( ) Table 1. Average numbers of iterations and cpu time for algorithms based on Newton's and monomial methods. 17
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