69 o N O PTIMA Mathematical Programming Society Newsletter JANUARY2003 69 Primal Integer Programming 2 mindsharpener 8 gallimaufry 10 OPTIMA69 JANUARY 2003 PAGE 2 Primal Integer Programming Robert Weismantel Faculty of Mathematics, University of Magdeburg Universitatsplatz 2, D-39106 Magdeburg, Germany [email protected] September 17, 2002 Primal integer programming is an area of Let us begin by recalling that all the problem specific cuts such as facet-defining cuts discrete optimization that -- in a broad sense -- algorithmic schemes applied to integer for the TSP in [18]. To the best of our includes the theory and algorithms related to programming problems without a-priori knowledge, a method of this type for Integer augmenting feasible integer solutions or knowledge about the structure of the side Programming has first been suggested by Ben- verifying optimality of feasible integer points. constraints resort to the power of linear Israel and Charnes [3]. They employed the fact This theory naturally provides an algorithmic programming duality. that for any feasible solution of an integer scheme that one might regard as a generic form Dual type algorithms start solving a linear program optimality can be proven by solving a of a primal integer programming algorithm: programming relaxation of the problem, subproblem arising from the non-basic columns First detect an integer point in a given domain typically with the dual simplex method. In the of a primal feasible simplex tableau. Various of discrete points. Then verify whether this course of the algorithm one maintains as an specializations and variants of a primal cutting point is an optimal solution with respect to a invariant both primal and dual feasibility of the plane algorithm were later given by Young [21, specific objective function, and if not, find solution of the relaxation. While the optimal 22] and Glover [6]. For an overview on this another feasible point in the domain that attains solution to the relaxation is not integral, one subject we refer to [11, 5]. Nemhauser and a better objective function value. continues to add cutting planes to the problem Wolsey [17] point out that because of poor The field of primal integer programming has formulation and reoptimizes. Within this computational experience, this line of research emerged in the 1960s beginning with the design framework, Gomory developed a method of has been very inactive, an exception being the of specific algorithms in combinatorial systematically generating valid inequalities work of Padberg and Hong [18], where strong optimization and Gomory's all integer directly from a given simplex tableau; see [7, 8]. primal-type cutting planes are used with some algorithms for the general integer programming A disadvantage of a dual-type method is, success. According to [17] the reason for the case. While the field stayed active in theory over however, that intermediate stopping does not poor computational performance of general the past 40 years, the algorithmic focus has been automatically yield a primal feasible integral primal cutting plane algorithms is that “it is nearly exclusively on combinatorial optimization solution. necessary to produce valid inequalities that problems. In fact, augmentation algorithms have In contrast to dual methods, primal type contain the one-dimensional faces (edges) on a been designed for and applied to a range of algorithms always preserve integrality of the path from the initial point to an optimal point”, specific linear integer programming problems: solution to the relaxation. They may be see also [13]. This is certainly one explanation augmenting path methods for solving maximum distinguished according to whether the why this line of research became inactive. flow problems or algorithms for solving the min- algorithm preserves dual feasibility or primal Recent theoretical advances regarding primal cost flow problem via augmentation along feasibility or none of those. Note that it is type algorithms and the theory of so-called test negative cycles are of this type. Other examples impossible to preserve integrality and both dual sets have also renewed the interest in primal include the greedy algorithm for solving the and primal feasibility. The only representative of cutting plane algorithms. Letchford and Lodi matroid optimization problem, alternating path the first family of primal type methods is an “all [15] propose a modern primal cutting plane algorithms for solving the weighted matching integer algorithm” that appeared in [9]. We are algorithm for 0/1 programs. The main problem, or methods for optimizing over the not aware of interesting computational ingredient is the subproblem of “primal intersection of two matroids. In fact, still many experiments with this method, however. separation”, see also [4, 16]. They give of the most recent combinatorial algorithms are An alternative to design a primal type computational results on small randomly of a primal nature; see for instance the algorithm is to preserve integrality of the generated 0/1 multidimensional knapsack combinatorial methods for minimizing a solution to the relaxation and simultaneously problems with up to twenty five variables. Their submodular function [12] [19] or the primal feasibility. In order to achieve this, one computational experiments reveal that the combinatorial algorithm for the independent typically starts with a primal feasible simplex proposed algorithm is superior to the original path matching problem [20]. tableau. Iteratively, pivot steps are performed on algorithm of Young. However, the results do not The attempt, however, to solve unstructured entries equal to one, only. If such a pivot provide evidence that primal cutting plane integer programs with a primal strategy had lost element cannot be detected by the standard dual algorithms can become superior to dual cutting its popularity for at least two decades. Which simplex rule, a special cut row with this property plane methods. Only future research and developments did lead to this situation? What is generated and then chosen for pivoting. When experiments can clarify this situation. I have are the obstacles for a primal strategy in the at any time the reduced cost coefficients of the doubts, however, that primal cutting plane general integer case? What is the state of the art? tableau point into the “right direction”, algorithms will become an alternative to the What are the next steps to be taken? These are optimality is proved by the simplex criterion. “standard method” of today. One additional some of the questions that I try to address in the The cuts one makes use of in this framework problem for primal cutting plane algorithms following. may be Gomory’s rounding cuts as in [21], or remains: To this day there is no primal cutting OPTIMA69 JANUARY 2003 PAGE 3 algorithm for mixed integer programs available, reformulation” of a tableau according to [10]. reduced cost coefficient yield a model in which despite the fact that mixed integer programming The substitution is guided by the concept of the linear programming relaxation becomes models are the most important models of irreducibility of lattice points. In contrast to significantly stronger than the original linear discrete optimization. generating a basis of a lattice one generates programming relaxation. Therefore, the effect of It is, however, important to remark that even iteratively a minimal set of lattice points in the reformulating a tableau with the help of if primal cutting plane algorithms will not work positive orthant that permits a representation of irreducible lattice points is not only interesting in practice, this will not imply that a primal any feasible solution as a non-negative integer per se, but rather suggests a primal-dual augmentation approach will not work. An combination of the selected subset. This approach for integer programming: perform alternative to primal cutting plane algorithms is distinguishes the latter approach from the Integral Basis Method steps in addition to the Integral Basis Method [10]. It uses a reformulation method of Aardal et al [1]. We cutting planes and branching. Each of the three completely different idea for solving the refer to [10] for the theoretic foundations and operations, i.e., substitution of variables, cutting, augmentation subproblem. After turning a computational results with the Integral Basis and branching, has an impact on the model and feasible solution into a basic feasible solution of Method. We also remark that the method can be on the other operations. This is -- in my opinion an appropriate simplex tableau, one manipulates extended to mixed integer programming -- an area of research to advance our current the set of non-basic columns until either problems, see [14]. A predecessor of the Integral ability to solve integer programming models. If optimality of the given solution is proved or an Basis Method for the special case of set we apply in addition to branching and cutting “applicable” improving direction, i.e., an partitioning problems was invented by Balas and the operation of substituting variables, then we augmentation vector, is detected. In each Padberg about 25 years ago [2]. cannot lose performance, but gain a lot. This is manipulation step, one substitutes one column Computational results with the Integral Basis where I see one future direction of research in by new columns in a way that no feasible Method demonstrate that a couple of iterative integer programming algorithms. solutions are lost; this is known as a “proper substitutions of non-basic columns with a “bad” References [1] Karen Aardal, Robert E. Bixby, [7] Ralph E. Gomory. Outline of [11] T. C. Hu. Integer Programming [17] George L. Nemhauser and Cor A. J. Hurkens, Arjen K. an algorithm for integer and Network Flows. Addison – Laurence A. Wolsey. Integer Lenstra, and Job W. Smeltink. solutions to linear programs. Wesley, Reading, Mass., 1969. and Combinatorial Market split and basis Bulletin of the American Optimization. Wiley, reduction: towards a solution Mathematical Society, 64:275-- [12] S.