The existence of a strongly polynomial time simplex algorithm ∗ Zi-zong Yan, Xiang-jun Li and Jinhai Guo † Abstract It is well known how to clarify whether there is a polynomial time simplex algorithm for linear programming (LP) is the most challenging open problem in optimization and discrete geometry. This paper gives a affirmative answer to this open question by the use of the parametric analysis technique that we recently proposed. We show that there is a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem. Keywords: Linear programming, parametric linear programming, com- plement slackness property, projection polyhedron, simplex algorithm, pivot rule, polynomial complexity AMS subject classifications. 90C05, 90C49, 90C60, 68Q25, 68W40 1 Introduction Linear programming (LP) is the problem of minimizing a linear objective function over a polyhedron P ⊂ Rn given by a system of m equalities and n nonnegative variables. As one of the fundamental problems in optimization, it has been a very successful undertaking in the field of polyhedral combinatorics in the last six decades. Part of this success relies on a very rich interplay between geometric and algebraic arXiv:2006.11466v8 [math.OC] 17 Sep 2021 properties of the faces of such a polyhedron and corresponding combinatorial struc- tures of the problem it encodes. The simplex algorithm, invented by Dantzig [8] solves LP problems by using pivot rules and procedures an optimal solution. These pivot rules are to iteratively improve the current feasible solutions by moving from one vertex of the polyhedron to an adjoint one according to some pivot rule, until no more improvement is possible and optimality is proven. All edges that connect these adjoint vertices of the polyhedron form a pivot path. The complexity of the simplex algorithm is then determined by the length of the path - the number of pivot steps. ∗Supported by the National Natural Science Foundation of China (11871118, 11771058). †School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China([email protected], [email protected] and [email protected]). 1 The simplex algorithm belongs to the '10 algorithms with the greatest influence on the development and practice of science and engineering in the 20th century', see [11]. The survey by Terlaky and Zhang [60] contributes the various pivot rules of the simplex algorithm and its variants, in which they categorized pivot rules into three types. The first alternative, which is called combinatorial pivot rule, is to take care of the sign of the variables and either costs or profits. The algorithms of this type known to the authors are that of Bland [4], Folkman and Lawrence [13], Fukuda [14] and etc. The second alternative, which is called parametric pivot rule, is closely related to parametric programming, more precisely to the shadow vertex algorithm [7, 21, 44] and to Dantig's self-dual parametric simplex algorithm [9]. The algorithms can be interpreted as Lemke's algorithm [34] for the corresponding lin- ear complementarity problem [36]. Algorithms of this third type, which are close connections to certain interior point methods (see Karmarker [31]), allow the iter- ative points to go inside the polytope. It is believed to be able to use more global information and therefore to avoid myopiness of the simplex method, for example, see Roos [48], Todd [61] and Tamura et al. [57]. It is a theoretically interesting and practically relevant question to recognize how a pivot rule [55] quickly leads to the optimal vertex. Connected to this are the diameter problem and the algorithm problem, for example, see [17]. The diameter problem, closely to the Hirsch conjecture and its variants, is whether there a short path to the optimal vertices. Unfortunately Santos' recent counter-example [51] disproves the Hirsch conjecture. On the subject for more information see the papers [6, 23, 40, 41] and the comments [12, 27, 35, 52, 59]. The algorithm problem is whether there is a very strong polynomial time algo- rithm for LP. Up to now, no pivoting rule yields such a algorithm depending only on the polynomiality of the number of constraints and the number of variables in the field of LP. In contrast to the excellent practical performance of the simplex algo- rithm, the worst-case time complexity of each analyzed pivot rule is known to grow exponentially, for example, see [2, 21, 22, 33, 42, 43, 44, 46, 49]. Other exponen- tial example was presented by Fukuda and Namiki [15] for linear complementarity problems. The simplex algorithm performs sufficiently well in practice, but theoretically the complexity of a pivot rule is not clear. To explain the large gap between practical experience and the disappointing worstcase, the tools of average case analysis and smoothed analysis have been devised, and to conquer the worst case bounds, research has turned to randomized pivot rules. For the average case analysis, a polynomial upper bounded was achieved [6]. In contrast, so far, smoothed analysis has only been done for the shadow-vertex pivot rule introduced by Gaas and Saaty [18]. Under reasonable probabilistic assumptions its expected number of pivot steps is polynomial [56]. It is worth noting that none of the existing results exclude the possibility of (randomized) pivot rules being the desired (expected) polynomial- time pivot rules. For more information on the randomized pivot rules reference the papers [16, 17, 29, 38, 44, 63] and etc. Two important advances have been made in polynomial time solvability for the 2 ellipsoid method developed by Khachain [32] and the interior-point method initiated by Karmarkar [31] since the 1970s. However, the run times complexity of such two algorithms is only qualified as weak polynomial. Actually, constructing a strongly polynomial time pivot rule is the most challenging open problem in optimization and discrete geometry [23, 40, 54, 62]. Our main goal of this paper is to give an existence result of the above open problem by the use of the parametric analysis technique that we recently proposed in [64]. We show that there exists a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem. The organization of the rest of the paper is as follows. In Section 2, we recall the strong duality theorem of LP. In Section 3, we review the main results of the paper [64]: we define the set-valued mappings between two projection polyhedrons for parametric LP problems and establish the relationship between perturbing the objective function data (OFD) and perturbing the right-hand side (RHS) for a LP problem without using dual. We then investigate the projected behavior of the set-valued mappings in Section 4. As a application, we present the existence of a strongly polynomial time pivot rule for a LP problem in the final section. 2 Preliminaries Consider the following linear programming (LP) problem pair in the standard forms: min hc; xi s:t: Ax = b; (1) x ≥ 0 and max hb; wi s:t: AT w + y = c; (2) y ≥ 0; where c 2 Rn; b 2 Rm and A 2 Rm×n are given. As usual, the first LP problem (1) is called the primal problem and the second problem (2) is called the dual problem, and the vectors c and b are called the cost and the profit vectors, respectively. By P = fx 2 RnjAx = b; x ≥ 0g and D = f(w; y) 2 Rm × RnjAT w + y = c; y ≥ 0g, we denote the feasible sets of the primal and dual problems, respectively. If the problems (1) and (2) are feasible and if b = Ad, then hd; c − yi = hd; AT wi = hAd; wi = hb; wi: Therefore, the dual problem (2) can be equivalently expressed as the form max hd; c − yi s:t: AT w + y = c; (3) y ≥ 0: 3 In following statement, we always use the problem (3) instead of the dual problem (2). The strong duality theorem provides a sufficient and necessary condition for optimality, for example, see [8, 20, 43, 53]. Theorem 2.1. (Strong duality) If the primal-dual problem pair (1) and (3) are feasible, then the two problems are solvable and share the same objective value. If the primal and dual programs have optimal solutions and the duality gap is zero, then the Karush-Kuhn-Tucker (KKT) conditions for the primal-dual LP problem (1) and (3) pair are Ax = b; x ≥ 0; (4a) AT w + y = c; y ≥ 0; (4b) hx; yi = 0; (4c) in which the last equality is called the complement slackness property. Conversely, if (x∗; y∗) 2 Rn × Rn is a pair of solutions of the system (4a)-(4c), then (x∗; y∗) is a pair of optimal solutions of the primal-dual problem pair (1) and (3). The feasible region of a LP problem is a polyhedron. In particular, a polyhedron is called a polytope if it is bounded. A nontrivial face F of a polyhedron is the intersection of the polyhedron with a supporting hyperplane, in which F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a k-face of the polyhedron. The empty set and the polyhedron itself are regarded as trivial faces. 0-faces of the polyhedron are called vertices and 1-faces are called edges. Two different vertices x1 and x2 are neighbors if x1 and x2 are the endpoints of an edge of the polyhedron. For material on convex polyhedron and for many references see Ziegler's book [65]. 3 Parametric KKT conditions Describing a pivot path of the simplex algorithm is a difficult task. The ingredient in our construction is to investigate the relationship between two projection polyhe- drons associated with a pair of almost primal-dual problems.