Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology

Given any polytope $P$ and any generic linear functional ${\bf c}$, one obtains a directed graph $G(P,{\bf c})$ by taking the 1-skeleton of $P$ and orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. This paper raises the question of finding sufficient conditions on a polytope $P$ and generic cost vector ${\bf c}$ so that the graph $G(P, {\bf c} )$ will not have any directed paths which revisit any face of $P$ after departing from that face. This is in a sense equivalent to the question of finding conditions on $P$ and ${\bf c}$ under which the simplex method for linear programming will be efficient under all choices of pivot rules. Conditions on $P$ and ${\bf c}$ are given which provably yield a corollary of the desired face nonrevisiting property and which are conjectured to give the desired property itself. This conjecture is proven for 3-polytopes and for spindles having the two distinguished vertices as source and sink; this shows that known counterexamples to the Hirsch Conjecture will not provide counterexamples to this conjecture. A part of the proposed set of conditions is that $G(P, {\bf c} )$ be the Hasse diagram of a partially ordered set, which is equivalent to requiring non revisiting of 1-dimensional faces. This opens the door to the usage of poset-theoretic techniques. This work also leads to a result for simple polytopes in which $G(P, {\bf c})$ is the Hasse diagram of a lattice L that the order complex of each open interval in L is homotopy equivalent to a ball or a sphere of some dimension. Applications are given to the weak Bruhat order, the Tamari lattice, and more generally to the Cambrian lattices, using realizations of the Hasse diagrams of these posets as 1-skeleta of permutahedra, associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition substantially improved throughou

Curvature as a Complexity Bound in Interior-Point Methods

In this thesis, we investigate the curvature of interior paths as a component of complexity bounds for interior-point methods (IPMs) in Linear Optimization (LO). LO is an optimization paradigm, where both the objective and the constraints of the model are represented by linear relationships of the decision variables. Among the class ofalgorithms for LO, our focus is on IPMs which have been an extremely active research area in the last three decades. IPMs in optimization are unique in the sense that they enjoy the best iteration-complexity bounds which are polynomial in the size of the LO problem. The main objects of our interest in this thesis are two distinct curvature measures in the literature, the geometric and the Sonnevend curvature of the central path. The central path is a fundamental tool for the design and the study of IPMs and we will see both that the geometric and Sonnevend\u27s curvature of the central path are proven to be useful in approaching the iteration-complexity questions in IPMs. While the Sonnevend curvature of the central path has been rigorously shown to determine the iteration-complexity of certain IPMs, the role of the geometric curvature in the literature to explain the iteration-complexity is still not well-understood. The novel approach in this thesis is to explore whether or not there is a relationship between these two curvature concepts aiming to bring the geometric curvature into the picture. The structure of the thesis is as follows. In the first three chapters, we present the basic knowledge of path-following IPMs algorithms and review the literature on Sonnevend\u27s curvature and the geometric curvature of the central path. In Chapter 4, we analyze a certain class ofLO problems and show that the geometric and Sonnevend\u27s curvature for these problems display analogous behavior. In particular, the main result of this chapter states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when the number of inequalities is twice as big as the dimension. In Chapter 5, we study the redundant Klee-Minty (KM) construction and prove that the classical polynomial upper bound for IPMs is essentially tight for the Mizuno-Todd-Ye predictor-corrector algorithm. This chapter also provides a negative answer to an open problem about the Sonnevend curvature posed by Stoer et al. in 1993. Chapter 6 investigates a condition number relevant to the Sonnevend curvature and yields a strongly polynomial bound for that curvature in some special cases. Chapter 7 deals with another self-concordant barrier function, the volumetric barrier, and the volumetric path. That chapter investigates some of the basic properties of the volumetric path and shows that certain fundamental properties of the central path failto hold for the volumetric path. Chapter 8 concludes the thesis by providing some final remarks and pointing out future research directions

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes

We consider a family of linear optimization problems over the n-dimensional Klee—Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2n − 2 sharp turns. This fact sug-gests that any feasible path-following interior-point method will take at least O(2n) iterations to solve this problem, whereas in practice typically only a few iterations (e.g., 50) suffices to obtain a high-quality solution. Thus, the construction potentially exhibits the worst-case iteration-complexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n)

Linear programs and convex hulls over fields of puiseux fractions

We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful for computations in tropical geometry

The Complexity of Zadeh's Pivot Rule

The Simplex algorithm is one of the most important algorithms in discrete optimization, and is the most used algorithm for solving linear programs in practice. In the last 50 years, several pivot rules for this algorithm have been proposed and studied. For most deterministic pivot rules, exponential lower bounds were found, while a probabilistic pivot rule exists that guarantees termination in expected subexponential time. One deterministic pivot rule that is of special interest is Zadeh's pivot rule since it was the most promising candidate for a polynomial pivot rule for a long time. In 2011, Friedmann proved that this is not true by providing an example forcing the Simplex algorithm to perform at least a subexponential number of iterations in the worst case when using Zadeh's pivot rule. Still, it was not known whether Zadeh's pivot rule might achieve subexponential worst case running time. Next to analyzing Friedmann's construction in detail, we develop the first exponential lower bound for Zadeh's pivot rule. This closes a long-standing open problem by ruling out this pivot rule as a candidate for a deterministic, subexponential pivot rule in several areas of linear optimization and game theory

Advances in Interior Point Methods for Large-Scale Linear Programming

This research studies two computational techniques that improve the practical performance of existing implementations of interior point methods for linear programming. Both are based on the concept of symmetric neighbourhood as the driving tool for the analysis of the good performance of some practical algorithms. The symmetric neighbourhood adds explicit upper bounds on the complementarity pairs, besides the lower bound already present in the common N−1 neighbourhood. This allows the algorithm to keep under control the spread among complementarity pairs and reduce it with the barrier parameter μ. We show that a long-step feasible algorithm based on this neighbourhood is globally convergent and converges in O(nL) iterations. The use of the symmetric neighbourhood and the recent theoretical under- standing of the behaviour of Mehrotra’s corrector direction motivate the introduction of a weighting mechanism that can be applied to any corrector direction, whether originating from Mehrotra’s predictor–corrector algorithm or as part of the multiple centrality correctors technique. Such modification in the way a correction is applied aims to ensure that any computed search direction can positively contribute to a successful iteration by increasing the overall stepsize, thus avoid- ing the case that a corrector is rejected. The usefulness of the weighting strategy is documented through complete numerical experiments on various sets of publicly available test problems. The implementation within the hopdm interior point code shows remarkable time savings for large-scale linear programming problems. The second technique develops an efficient way of constructing a starting point for structured large-scale stochastic linear programs. We generate a computation- ally viable warm-start point by solving to low accuracy a stochastic problem of much smaller dimension. The reduced problem is the deterministic equivalent program corresponding to an event tree composed of a restricted number of scenarios. The solution to the reduced problem is then expanded to the size of the problem instance, and used to initialise the interior point algorithm. We present theoretical conditions that the warm-start iterate has to satisfy in order to be successful. We implemented this technique in both the hopdm and the oops frameworks, and its performance is verified through a series of tests on problem instances coming from various stochastic programming sources