A Generalized Homogeneous and SelfDual Algorithm. for Linear Programming. February 1994 (revised December 1994)


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1 A Generalized Homogeneous and SelfDual Algorithm for Linear Programming Xiaojie Xu Yinyu Ye y February 994 (revised December 994) Abstract: A generalized homogeneous and selfdual (HSD) infeasibleinteriorpoint algorithm for linear programming (LP) is proposed in this paper. The algorithm does not need to start from a big M initial point, while achieving O( p n (?) L)iteration complexity by following a certain central path on a central surface in a neighborhood N (), where can be any number between 0 and, n is the number of variables and L is the data length of the LP problem. In particular, an algorithm is developed, where the searching direction is obtained by solving a Newton equation system without infeasible residual terms on its right hand side. Key words: Linear programming, homogeneous and selfdual linear feasibility model, interiorpoint algorithm Institute of Systems Science, Academia Sinica, Beijing 00080, China, and currently visiting at Department of Management Sciences, The University of Iowa, Iowa City, Iowa 52242, USA. Research supported in part by NSF Grant DDM y Department of Management Sciences, The University of Iowa, Iowa City, Iowa 52242, USA. Research supported in part by NSF Grant DDM
2 Introduction Consider a linear programming (LP) problem in the standard form: (LP ) minimize c T x subject to Ax = b; x 0; where c 2 R n, A 2 R mn and b 2 R m are given, x 2 R n, and T denotes transpose. (LP) is said to be feasible if and only if its constraints are consistent; it is called unbounded if there is a sequence fx k g such that x k is feasible for all k but c T x k!?. (LP) has a solution if and only if it is feasible and bounded. The dual problem of (LP) can be written as (LD) maximize b T y subject to A T y c; where y 2 R m. We call z = c? A T y 2 R n dual slacks. Denote by F the set of all x and (y; z) that are feasible for the primal and dual, respectively. Denote by F 0 the set of points with (x; z) > 0 in F. Assuming that the LP problem has a feasible interior point, Megiddo [8] and Bayer and Lagarias [] dened the central path for a feasible LP problem as C() = (y; x; z) 2 F 0 : Xz = e; = xt z n where X = diag(x). As! 0, this path goes to a strictly complementary solution of LP. Based on following the central path, Kojima et al. [4] developed a primaldual interiorpoint algorithm in which the searching direction is generated by solving the following Newton equation system in iteration k: A d x = 0;?A T d y?d z = 0; Z k d x +X k d z = k e? X k z k ; where k = ((x k ) T z k )=n, X k = diag(x k ), Z k = diag(z k ), and is a scalar parameter. Kojima et al. [4] proved that their algorithm is O(nL)iteration bounded, where L is the data length of (LP) with integer numbers. Later, Kojima et al. [5] and Monteiro and Adler [2] gave an O( p nl)iteration bound for such a primaldual interiorpoint algorithm by restricting all iterates in a 2norm neighborhood of the central path, i.e. (y k ; x k ; z k ) 2 N () = (y; x; z) 2 F 0 : kxz? ek for some 2 (0; =2]. (Typically, = =4, or for a predictorcorrector algorithm = =2 in the predictor step, see [].) Throughout the paper, k:k represents 2norm. Unless a LP problem has a feasible interior point and such a point is given, an interiorpoint algorithm has to start from an infeasible point, or from an interior point feasible for an articial problem. In theory,
3 a big M interior point suces for establishing complexity result. However, such a big M approach is not practical at all. Furthermore, a robust algorithm has to be able to correct possible error accumulated from computations even starting from a feasible interior point. Algorithms, which allow to start from a nonbig M initial point in both theory and practice, are called infeasible interiorpoint algorithms and reported perform very well in practice (see [6], [7], [9], [0], [4], [6], and [9]). Unlike for feasible algorithms (in which a feasible interior point is given as the initial point), the besttodate O( p nl)iteration complexity for infeasible interiorpoint algorithms was not established until Ye et al. [8] proposed an homogeneous and selfdual (HSD) algorithm. Recently Mizuno et al. [] studied the trajectories followed by many primaldual infeasible interiorpoint algorithms. For given (y 0 ; x 0 > 0; z 0 > 0), they dened two dimensional central surface fq(; ) : 0; 0g with < T x z Q(; ) = (y; x > 0; z > 0) : Xz = e; = : n r P A r0 = P A r D rd 0 ; where rp 0 = b? Ax 0 and rd 0 = c? A T y 0? z 0 ; r P = b? Ax and r D = c? A T y k? z k are primal and dual residuals respectively. If the LP problem possesses a solution, many primaldual infeasible interiorpoint algorithms (e.g., Kojima et al. [6], Lustig [7], Mehrotra [9]) follow some paths on this central surface and approach optimality and feasibility simultaneously: for t! 0 : (t)! 0; (t)! 0: Mizuno et al. [] also discussed in detail the boundary behavior of central surface for primaldual type infeasible interiorpoint algorithms. Very recently, Xu et al. [7] proposed a simplied version of the HSD Algorithm of Ye et al. [8]. The algorithm deals with a homogeneous and selfdual linear feasibility model (HLF ) Ax?b = 0;?A T y +c 0; b T y?c T x 0; () y free x 0 0: Denote by z the slack vector for the second (inequality) constraint and by the slack scalar for the third (inequality) constraint. Then, the problem is to nd a strictly complementary point such that x T z = 0 and = 0: 2
4 The kth iteration of the HSD algorithm solves the following system of linear equations for direction (d y ; d x ; d ; d z ; d ) and A d x?b d = r k P ;?A T d y +c d?d z =? r k D ; b T d y?c T d x?d = r k G ; (2) X k d z + Z k d x = k e? X k z k ; k d + k d = k? k k ; (3) where > 0, are scalar parameters, and k = ((x k ) T z k + k k )=(n + ); (4) r k P = b k? Ax k ; r k D = c k? A T y k? z k ; r k G = c T x k? b T y k + k : (5) Xu et al. [7] showed that if we set =? in each iteration, then the algorithm becomes the HSD algorithm of Ye et al. [8], which follows a path fq( 0 t; 0 t) : 0 t g on the central surface. More precisely, Xu et al. [7] set ) k 0 k ( 0 O( 0 0 ): The limit points of these paths are strictly complementary point for (HLF), according to Mizuno et al. []. If setting <? or >? at each iteration, then the algorithm generates iterates converging to the allzero solution or diverging, respectively. In this paper by introducing a simple update, we generalize the HSD algorithm of Xu et al. [7] so that a strictly complementary solution is obtained even when 6=?. In section 3, we prove that the generalized algorithm achieves O( p n (?) L)iteration complexity by following certain central path on the central surface in a neighborhood N (), where can be any number 2 (0; ). By setting = 0, we get an interesting algorithm in which the searching direction is obtained by solving a Newton equation system without infeasible residual terms on its right hand side, as rst proposed by de Ghellinck and Vial [2] and later by Nesterov [3]. This approach obviously saves the computation of these residual terms. 2 Generalized HSD algorithms Generic HSD algorithm Given initial point y 0 ; x 0 > 0; 0 > 0; z 0 > 0; 0 > 0, k 0. While ( stopping criteria not satised ) do. Let r k P = b k? Ax k ; r k D = c k? A T y k? z k ; r k G = ct x k? b T y k + k : 3
5 2. Solve (2) and (3) for d y ; d x ; d ; d z ; d. 3. Let x = d x + (?? )x k ; y = d y + (?? )y k ; z = d z + (?? )z k ; (6) = d + (?? ) k ; = d + (?? ) k : 4. Choose a step size k > 0 and update x k+ = x k + k x > 0; y k+ = y k + k y ; z k+ = z k + k z > 0; (7) k+ = k + k > 0; k+ = k + k > 0: 5. k k +. Note that we have X k z + Z k x = X k d z + Z k d x + (?? )(X k z k + Z k x k ) = k e? X k z k + 2(?? )X k z k = k e? (2 + 2? )X k z k ; (8) k + k = k? (2 + 2? ) k k : Similar to the proofs in Xu et al. [7], we rst establish the following lemmas: Lemma. The direction resulting from (6) satises ( x ) T z + = 0: (9) Proof. Xu et al. [7] established following result for the solution of system (2) and (3): d T x d z + d d = (?? )(n + ): Thus ( x ) T z + = d T x d z + d d + (?? )((x k ) T d z + (z k ) T d x + k d + k d )+ (?? ) 2 ((x k ) T z k + k k ) = [(?? ) + (?? )(? ) + (?? ) 2 ](n + ) = 0: Q.E.D. 4
6 Lemma 2. The generic algorithm generates f k g and f k g satisfying 0 = [(x 0 ) T z ] = (n + ); k+ = ( + k (?? 2)) k (0) and 0 = ; k+ = ( + k (?? 2)) k () such that r k P = k r 0 P ; rk D = k r 0 D ; rk G = k r 0 G : Proof. By (9) and (8), we have k+ = [(x k+ ) T z k+ + ( k+ ) T k+ ] = (n + ) By (2), (6) and (7), we also have = f [(x k ) T z k + k k ] + k [(x k ) T z + ( x ) T z k + k + k ] g = (n + ) = [ + k (? 2? 2 + )] k = [ + k (?? 2)] k : Similarly, we have this relation for r k+ D r k+ P = [( k + k )b? A(x k + k x )] = [r k P + k ( b? A x )] = [ + k (?? 2)]r k P : and rk+ G as well. From Lemma 2, for any choice of and, our algorithm ensures k and k having a nice xed ratio Q.E.D. k = k = 0 = 0 : The nonnegativity of (x k+ ; k+ ; z k+ ; k+ ) results in k+ 0, which implies that the step size k must satisfy + k (?? 2) 0: Therefore, letting + 2? > 0; (2) yields 0 + k (?? 2) < : According to Mizuno et al. [], we have the following corollary. 5
7 Corollary 3. If the generic algorithm generates f(y k ; x k ; k ; z k ; k )g satisfying k! 0 and min[min(x k i zk i ); k k ] k i for certain > 0, then every limit point of the sequence is a strictly complementary solution of (HLF). p 3 n O( (?) L)iteration HSD algorithms For (HLF) the two dimensional central surface and its neighborhood are dened as r >< Q(; ) = (y; x > 0; > 0; z > 0; > 0) Xz P r P 0 >= A = e; B D C A = B r D C A >: r G rg 0 >; r >< N () = (y; x > 0; > 0; z > 0; > 0) : Xz P r P 0 >= A? ek ; B D C A = B r D C A >: r G rg 0 >; for some 2 (0; ), respectively. Theorem 4. For a given 0 < < and (y k ; x k ; k ; z k ; k ) 2 N (), if k minf 2 + 2? ; 2(? ) [( + 2? ) p n + = + (2 + 2? )] 2 g (3) then 0 Xk+ z k+ k+ k+ A? k+ ek k+ : Proof. To simplify the notation, we use x and z to represent ( x ) and ( z ). Therefore kek = p n +. Note that this notation is only employed in the proof of this theorem. As usual, the capital expression denotes the diagonal matrix of a vector. Thus x = diag( x ), z = diag( z ). Consider kx k+ z k+? k+ ek = kx k z k + k (X k z + Z k x ) + ( k ) 2 x z? [ + k (?? 2)] k ek kx k z k + k (X k z + Z k x )? [ + k (?? 2)] k ek + ( k ) 2 k x z k = j? k (2 + 2? )j kx k z k? k ek + ( k ) 2 k x z k: 6
8 Using T x z = 0, we have k x z k = k[(x k )? Z k ] =2 x [X k (Z k )? ] =2 z ek 2 k[(xk )? Z k ] =2 x e + [X k (Z k )? ] =2 z ek 2 = 2 k(xk Z k )?=2 (Z k x + X k z )k 2 2 mini xizi kzk x + X k z k 2 = 2 mini xizi kk e? (2 + 2? )X k z k k 2 2 mini xizi [k( + 2? )k ek + k(2 + 2? )(X k z k? k e)k] 2 ( k ) 2 2 mini x k i zk [( + 2? ) p n + + (2 + 2? )] 2 : i By min i x k i zk i (? )k ; we have kx k+ z k+? k+ ek j? k (2 + 2? )j k + (k ) 2 ( k ) 2 [( + 2? ) p n + + (2 + 2? )] 2 2 mini x k i zk i = fj? k (2 + 2? )j + (k ) 2 2(?) [( + 2? )p n + = + (2 + 2? )] 2 g k : Again, Lemma 2 tells us k+ = [? k ( + 2? )] k : (4) Therefore, kx k+ z k+? k+ ek k+ if or j? k (2 + 2? )j + (k ) 2 2(? ) [( + 2? )p n + = + (2 + 2? )] 2 [? k ( + 2? )]; 2(? ) [( + 2? )p n + = + (2 + 2? )] 2 ( k ) 2 [? k ( + 2? )]? j? k (2 + 2? )j: If we further assume then it becomes k Thus we have proved the theorem.? k (2 + 2? ) 0; 2(? ) [( + 2? ) p n + = + (2 + 2? )] 2 : Q.E.D. Using the simple continuarity argument ([]), we see from Theorem 4 that, as long as the step size k satises (3), the resulting point is still in the neighborhood of the central path (y k+ ; x k+ ; k+ ; z k+ ; k+ ) 2 N (). 7
9 Let us now consider the following optimization problem for a given 0 < < minimize k+ = k subject to k ; ; satisfy (2); (3): (5) Setting step size according to (3), we obtain k+ = k =? k ( + 2? ) =? minf 2+2? ; Letting! = + 2?, we can rewrite problem (5) as (0 < < ) minimize? minf +! ; subject to! > 0; > 0: By setting! =!, problem (5) further becomes 2(?) [(+2?) p n+=+(2+2?)] 2 g ( + 2? ): 2(?) (! p n+=++!) 2 g! minimize? minf! +! ; subject to! > 0: 2(?)! [( p n+=+)!+] 2 g (6) It is easy to verify that the problem minimize? 2(?)! [( p n+=+)!+] 2 has the optimal value with the optimizer?! =? 2( p n + + ) p n + = + : This implies that and satisfy Therefore, the optimal solution of (6) is clearly bounded by maxf? p n ;? = ( p n + = + )( + 2? ): (7)? 2( p n + + ) g k+ = k?? 2( p n + + ) : (8) Above analysis points out that for a given 0 < <, using (3), the best reduction rate for that the algorithm can achieve is? O( p n ). This results in the O( p nl)iteration complexity. It also implies that a better complexity in worst case is very hard to achieve if the 2norm neighborhood is used. By setting and according to (7), the reduction rate is? p n++2 when is near 0, or? near, respectively. Therefore, the algorithm achieves O( p n (?) L)iteration complexity.? 2( p n++) when is A simple choice (y 0 = 0; x 0 = e; 0 = ; z 0 = e; 0 = ) ensures that the initial point is Q(; ) on the central surface. In summary, we have the following theorem. 8
10 Theorem 5. Let (LP) have integer data with a total bit length L. Then, (HLF) has integer data with a bit length O(L). Furthermore, let 0 < < and (y 0 ; x 0 ; 0 ; z 0 ; 0 ) 2 N () (for instance (y 0 ; x 0 ; 0 ; z 0 ; 0 ) = (0; e; ; e; )) and set = ( p n + = + )( + 2? ) > 0 p k n + + = minf ( p n + + 2) ;? g 2 > 0: Then, the generalized HSD algorithm generates a strictly complementary optimal solution of (HLF) in O( p n (?) L) iterations. As showed in Goldman and Tucker [3][5], Ye et al. [8], we have the following corollary. Corollary 6. The algorithm specied in Theorem 5 obtains a strictly complementary optimal solution of (LP) and (LD) or detects infeasibility of either (LP) or (LD) in O( p n (?) L) iterations. 4 An HSD algorithm In this section, we consider a special case of the generalized HSD algorithms. Let = 0. The modied Newton equation system becomes A d x?b d = 0;?A T d y +c d?d z = 0; b T d y?c T d x?d = 0; (9) Z k d x +X k d z = k e? X k z k ; k d + k d = k? k k : We observe that r P ; r D ; r G disappear in (9). From (2), we have to set >. Thus, we can avoid computing these residuals. Designing algorithms basing on the homogeneous and selfdual linear feasibility model (HLF) seems to have exploited the special properties of linear programming better than on original model. We observe that the introduction of the homogeneous variable and the updating of solutions (6) plays an important role in this algorithm. It makes the feasible and infeasible interiorpoint algorithms with no dierence at all. From Lemma 2, clearly, a large is desired for this new algorithm, since is xed at zero now. In practice, we can make use of a predictorcorrector strategy to choose a very large in each iteration, similar to Xu et al. [7] where they used the strategy to choose a very small. 9
11 References [] D. A. Bayer and J. C. Lagarias, \The nonlinear geometry of linear programming: I. Ane and projective scaling trajectories, II. Legendre transform coordinates and central trajectories," Transactions of the American Mathematical Society 34 (989) [2] G. de Ghellinck and J.P. Vial \A polynomial Newton method for linear programming," Algorithmica (986) [3] A. J. Goldman and A. W. Tucker, \Polyhedral convex cones," in: H. W. Kuhn and A. W. Tucker eds., Linear Inequalities and Related Systems (Princeton University Press, Princeton, NJ, 956) [4] M. Kojima, S. Mizuno, and A. Yoshise, \A primaldual interior point algorithm for linear programming," in: N. Megiddo, ed., Progress in Mathematical Programming, Interior Point and Related Methods (SpringerVerlag, New York, 989) [5] M. Kojima, S. Mizuno, and A. Yoshise, \A polynomialtime algorithm for a class of linear complementarity problems," Mathematical Programming 44 (989) 26. [6] M. Kojima, N. Megiddo, and S. Mizuno, \A primaldual infeasibleinteriorpoint algorithm for linear programming," Mathematical Programming 6 (993) [7] I. J., Lustig, R. E. Marsten, and D. F. Shanno, \Computational experience with a primaldual interior point method for linear programming" Linear Algebra and Its Applications 52 (99) [8] N. Megiddo, \Pathways to the optimal set in linear programming," in: N. Megiddo, ed., Progress in Mathematical Programming, Interior Point and Related Methods (SpringerVerlag, New York, 988) [9] S. Mehrotra \On the implementation of a (primaldual) interior point method," SIAM Journal on Optimization 2 (992) [0] S. Mizuno, \Polynomiality of infeasible interior point algorithms for linear programming," Mathematical Programming 67 (994) [] S. Mizuno, M. J. Todd, and Y. Ye, \A surface of analytic centers and infeasibleinteriorpoint algorithms for linear programming," Technical Report, School of Operations Research and Industrial Engineering, Cornell University, (Ithaca, New York, 992), to appear in Mathematics of Operations Research. [2] R. C. Monteiro and I. Adler, \Interior path following primaldual algorithms, part I: linear programming," Mathematical Programming 44 (989)
12 [3] Yu. Nesterov, \Longstep strategies in interiorpoint potential reduction methods," Central Economical and Mathematical Institute, Russian Academy of Science, (Moscow, Russia, 993). [4] F. A. Potra, \An infeasible interiorpoint predictorcorrector algorithm for linear programming," Report No. 26, Department of Mathematics, University of Iowa (Iowa City, IA, 992), to appear in SIAM Journal on Optimization. [5] A. W. Tucker, \Dual systems of homogeneous linear relations," in: H. W. Kuhn and A. W. Tucker eds., Linear Inequalities and Related Systems (Princeton University Press, Princeton, NJ, 956) 38. [6] S. J. Wright, \A pathfollowing infeasibleinteriorpoint algorithm for linear complementarity problems," Optimization Methods and Software 2 (993) [7] X. Xu, P. F. Hung, and Y. Ye, \A simplied homogeneous and selfdual linear programming Algorithm and Its Implementation," College of Business Administration, The University of Iowa (Iowa City, IA, 993). [8] Y. Ye, M. J. Todd, and S. Mizuno, \An O( p nl)iteration homogeneous and selfdual linear programming algorithm," Mathematics of Operations Research 9 (994) [9] Y. Zhang, \On the convergence of a class of infeasible interiorpoint methods for the horizontal linear complementarity problem," SIAM Journal on Optimization 4 (994)
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