Intl. Trans. in Op. Res. 10 (2003) 53–64 Performance evaluation of a family of criss-cross algorithms for linear programming Tibe´rius Bonates and Nelson Maculan Universidade Federal do Rio de Janeiro, Programa de Engenharia de Sistemas e Computac¸a˜o – COPPE-UFRJ, Caixa Postal 68511 21145-970 – Rio de Janeiro - RJ - Brasil E-mail: {tiberius,maculan}@cos.ufrj.br Received 11 April 2002; received in revised form 30 July 2002; accepted 8 August 2002 Abstract In this article we evaluate a family of criss-cross algorithms for linear programming problems, comparing the results obtained by these algorithms over a set of test problems with those obtained by the simplex algorithms implemented in the XPRESS1 commercial package. We describe the known criss-cross variants existing in the literature and introduce new versions obtained with a slight modification based on the original criss-cross algorithm. Moreover, we consider some computational details of our implementation and describe the set of test problems used. Keywords: Linear programming, simplex method, criss-cross algorithms 1. Introduction The first criss-cross algorithm was published in 1969 by Stanley Zionts (1969). At that time, there was great attention over the simplex method, that had proved to be successful in solving a number of real- life linear programming problems. However, there was a difficulty in applying the simplex method to some problems: to find a feasible initial basis for starting the algorithm. Although the algorithm itself was not able to find a feasible basis, it was shown to be possible to find such a basis for a specific problem applying the simplex algorithm to a modified version of this problem (Chva´tal, 1983). This procedure, known as the first phase of the simplex algorithm, returns a feasible basis from which the algorithm can be started. It has been widely used in linear programming solvers, shown to be effective in practice. We consider the following problem as the standard linear programming problem (LP). The algorithms will be described based on this definition: 1 XPRESS is a trademark of Dash Optimization, Inc. (http://www.dashoptimization.com/) # 2003 International Federation of Operational Research Societies. Published by Blackwell Publishing Ltd. 54 T. Bonates and N. Maculan / Intl. Trans. in Op. Res. 10 (2003) 53–64 (P) min c`x (1) subject to: Ax ¼ b (2) x > 0, (3) where A 2 Rm3n, with rank(A) ¼ m, x 2 Rn and real vectors c and b of appropriate dimensions. Unfortunately, to find a feasible initial basis for a linear programming problem is a hard computa- tional task (Bixby, 1992; Enge and Huhn, 1997). For some problems the first phase of the simplex algorithm can take most of the total time spent to solve a problem. In fact, the first phase procedure (i.e., to find a feasible basis) is theoretically equivalent to solving the problem (Enge and Huhn, 1997). The first criss-cross algorithm was an attempt to avoid the first phase procedure and solve the problem starting from any bases. While the simplex algorithm starts in a feasible vertex and walks along the edges of feasible space, the criss-cross algorithm of Zionts starts at any bases and walks through infeasible vertices, alternating primal and dual iterations, until it arrives at a feasible vertex. The primal iterations are made applying a primal simplex pivot rule to a primal feasible subproblem of the original problem, obtained by removing some infeasible rows of the problem. Similarly, the dual iterations are made applying a dual simplex pivot rule to a dual feasible subproblem, obtained by removing some infeasible columns of the problem. Once a (primal or dual) feasible vertex is acheived, the algorithm of Zionts reduces to the (primal or dual, respectively) simplex algorithm. This may seem to be the first phase for the simplex method, but some relevant differences can be realized: 1. Although we consider only a fraction of the problem in each iteration, there is no modification in the problem’s structure; 2. From its start the algorithm is directly searching for the optimal basis and not for a feasible basis. When we use a first phase procedure we change the objective function in some way to drive the algorithm in the search for a feasible basis. In fact, the main hope of Zionts’ algorithm is to find the optimal solution during such criss-cross iterations; 3. Reduction to a simplex-type algorithm is not specific as in two-phase simplex algorithms: the two- phase primal (respectively dual) simplex algorithm reduces to the standard primal (resp. dual) simplex algorithm when a feasible basis is found. In the criss-cross context, the choice of which algorithm must be used after feasibility is achieved depends on the nature of the feasible solution encountered (primal or dual). For a detailed description of the original criss-cross algorithm we refer to Zionts (1969), in which the author introduces his method. Some years later, another paper was published by Zionts (1972) concerning the empirical behavior of his algorithm. The author made a comparison between an implementation of the criss-cross algorithm and similar implementations of the standard primal and dual simplex algorithms. According to Zionts the criss-cross algorithm was shown to be very efficient in solving the test problems, presenting better results than the standard primal and dual simplex methods. In general, the criss-cross algorithm required a smaller number of iterations and solved the problems faster than the standard simplex algorithms. T. Bonates and N. Maculan / Intl. Trans. in Op. Res. 10 (2003) 53–64 55 Despite the good results related by Zionts, the need for a more rigorous evaluation of the empirical performance of the criss-cross algorithm is evident. The test problems used by Zionts were problems with very small dimension (at most 25 rows or columns). Moreover, a visible weakness of Zionts’ algorithm is its convergence proof. The convergence guarantee was based on the insertion of a regularization constraint, followed by a reduction to one of the simplex algorithms, if a large number of criss-cross iterations were made without finding a feasible basis. At that time, it was not clear if one could propose a finite criss-cross pivot rule, i.e., a criss-cross pivot rule that could solve the problem (or report it as infeasible or unbounded) in a finite number of iterations, without invoking one of the simplex algorithms. 2. The criss-cross variants Terlaky (1985) proposed a finite version of the original criss-cross algorithm of Zionts to the setting of oriented matroid programming. Fukuda and Matsui (1991) proved that Terlaky’s algorithm was also finite in the linear programming case. Since Terlaky’s algorithm could be started from an arbitrary bases and could visit infeasible vertices of the problem, it is considered the first finite criss-cross variant, i.e., the first algorithm that could solve a linear program in one single phase based on Zionts’ idea of following an external path outside the feasible space during the solution process. In fact, the same algorithm was developed almost at the same time by three different researchers in different contexts (Bonates, 2001). But, for simplicity, it is commonly cited as Terlaky’s algorithm or as the general criss-cross scheme. The algorithm consists of a least index pivot rule applied to the problem. Among the infeasible (primal or dual) variables, the one with least index is chosen to drive the pivoting operation. For this reason it is also called the least index criss-cross algorithm (LICC). We need to make some definitions before introducing Terlaky’s algorithm. These definitions will also hold for the entire text. Let B be a basis for (P), N be the non-basis set and AB, AN , cB , cN , xB , xN the submatrices and subvectors of A, c and x indexed by B and N , respectively. Let us also define ` ` À1 ~ À1 ~ À1 ~ c~ ¼ (cN À cB AB AN ), A ¼ AB AN and b ¼ AB b. Whenever we refer to bi, c~j and a~ij as compo- nents of b~, c~ and A~, we are in fact, referring to those components of b~, c~ and A~ that are associated with the variables of B and N indexed by i, j and (i, j), respectively. According to these conventions, we can describe Terlaky’s algorithm as follows in Figure 1. The pivot ( p, q) statement means a pivoting operation that makes the p-th variable enter the basis and the q-th variable leave. Unlike the original criss-cross method, Terlaky’s algorithm works in a single phase, which means that the algorithm follows the same pivot rule from its start until its end. Therefore, it does not reduce to one of the simplex algorithms when a feasible solution is reached. Fukuda and Matsui (1991) also showed that the pivot rule used by Terlaky could be relaxed, showing that it was possible to devise new pivot rules for the criss-cross algorithm. In fact, some years later, two new variants of the criss-cross algorithm were proposed by Zhang (1997). Zhang showed that the criss- cross method under a last in first out/last out first in (LIFO/LOFI) rule and under a most-often- selected-variable (MOSV) rule was still finite. The variants of Zhang were very similar to Terlaky’s algorithm: there was no need for a first phase procedure and they could follow a path through infeasible points before solving the problem. 56 T. Bonates and N. Maculan / Intl. Trans. in Op. Res. 10 (2003) 53–64 Fig. 1. Least index criss-cross algorithm. Fig. 2. Most-often-selected-variable criss-cross algorithm.