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# A lower bound on the number of iterations of long-step primal-dual linear programming algorithms

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## Abstract

Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at leastn 1/3 iterations to improve the initial duality gap by a constant factor. He also showed that this lower bound holds for some polynomial variants of primal-dual interior-point methods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to long-step primal-dual variants that restrict the iterates to a wider neighborhood. This neigh-borhood seems the least restrictive one to guarantee polynomiality for primal-dual path-following methods, and the variants are also even closer to what is implemented in practice.
... Each main iteration consists of one so-called feasibility step, a μ-update, and a few centering steps, respectively. First, we find new iterates x f , y f and s f that satisfy equations (4) and (5), with ν replaced by ν + . As we will see, by taking θ small enough, this can be realized by one feasibility step, as discussed subsequently. ...
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In Roos [Roos, C., 2006, A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM Journal on Optimization, 16(4), 1110–1136.] presented a new primal-dual infeasible interior-point algorithm that uses full-Newton steps and whose iteration bound coincides with the best-known bound for infeasible interior-point algorithms. Each iteration consists of a step that restores the feasibility for an intermediate problem (the so-called feasibility step) and a few (usual) centering steps. No more than O(nlog(n/ϵ)) iterations are required for getting an ϵ-solution of the problem at hand, which coincides with the best-known bound for infeasible interior-point algorithms. In this article, we introduce a different feasibility step and show that the same complexity result can be obtained with a relatively simpler analysis.
... Sonnevend et al. [13] showed that a variant of MTY predictor-corrector algorithm requires Ω(n 1 3 ) iterations to reduce the duality gap by log n for certain LO problems. A similar result has been obtained by Todd et al. [15] for the primal-dual affine scaling algorithm and has been later extended by Todd and Ye [16] for long step primal-dual IPMs; they showed that these algorithms take Ω(n 1 3 ) iterations to reduce the duality gap by a constant. ...
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It is an open question whether there is an interior-point algorithm for linear optimization problems with a lower iteration-complexity than the classical bound O(√ n log(µ1 µ0)). This paper provides a negative answer to that question for a variant of the Mizuno-Todd-Ye predictor-corrector algorithm. In fact, we prove that for any > 0, there is a redundant Klee-Minty cube for which the aforementioned algorithm requires n (1 2 − iterations to reduce the barrier parameter by at least a constant. This is provably the first case of an adaptive step interior-point algorithm, where the classical iteration-complexity upper bound is shown to be tight.
Chapter
In the first part of this chapter we cover basic LP facts, including LP duality, special LP structure, as well as two main tools for solving LP: simplex and interior point methods (IPMs).
Chapter
It is an open question whether there is an interior-point algorithm for linear optimization problems with a lower iteration-complexity than the classical bound $$\mathcal {O}(\sqrt{n} \log (\frac{\mu _1}{\mu _0}))$$. This paper provides a negative answer to that question for a variant of the Mizuno-Todd-Ye predictor-corrector algorithm. In fact, we prove that for any $$\varepsilon >0$$, there is a redundant Klee-Minty cube for which the aforementioned algorithm requires $$n^{( \frac{1}{2}-\varepsilon )}$$ iterations to reduce the barrier parameter by at least a constant. This is provably the first case of an adaptive step interior-point algorithm where the classical iteration-complexity upper bound is shown to be tight.
Article
We prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with $3r+1$ inequalities in dimension $2r$ for which the number of iterations performed is in $\Omega(2^r)$. The total curvature of the central path of these linear programs is also exponential in $r$, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature, in a general setting.
Article
In this thesis, we present new results on the complexity of classical linear programming on the one hand, and of tropical linear programming and mean payoff games on the other hand.Our contributions lie in the study of the interplay between these two problems provided by the dequantization process. This process tranforms classical linear programs into linear programs over tropical semirings, such as the $\R \cup\{ -\infty\}$ endowed with $\max$ as addition and $+$ as muliplication. Concerning classical linear programming, our first contribution is a tropicalization of the simplex method. More precisely, we describe an implementation of the simplex method that, under genericity conditions, solves a linear program over an ordered field. Our implementation uses only the restricted information provided by the valuation map, which corresponds to the orders of magnitude'' of the input. Using this approach, we exhibit a class of classical linear programs over the real numbers on which the simplex method, with any pivoting rule, performs a number of iterations which is polynomial in the input size of the problem. In particular, this implies that the corresponding polyhedra have a diameter which is polynomial in the input size.Our second contribution concerns interior point methods for classical linear programming. We disprove the continuous analog of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with $3r + 4$ inequalities indimension $2r + 2$ where the central path has a total curvature which is exponential in $r$. We also point out suprising features of the tropicalization of the central path. For example it has a purely geometric description, while the classical central path depends on the algebraic representation of a linear program. Moreover, the tropical central path may lie on the boundary ofthe tropicalization of the feasible set, and may even coincide with a path of the tropical simplex method.Concerning tropical linear programming and mean payoff games, our main result is a new procedure to solve these problems based on the tropicalization of the simplex method. The latter readily applies to tropical linear programs satisfying genericity conditions. In order to solve arbitrary problems, we devise a new perturbation scheme. Our key tool is to use tropical semirings based on additive groups of vectors ordered lexicographically.Then, we transfer complexity results from classical to tropical linear programming. We show that the existence of a polynomial-time pivoting rule for the classical simplex method, satisfying additional assumptions, would provide a polynomial algorithm for tropical linear programming and thus for mean payoff games. By transferring the analysis of the shadow-vertex rule of Adler, Karp and Shamir, we also obtain the first algorithm that solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a symmetry property. We establish tropical counterparts of the notions of basic points and edges of a polyhedron. This yields a geometric interpretation of the tropicalization of the simplex method. As in the classical case, the tropical algorithm pivots on the graph of an arrangement of hyperplanes associated to a tropical polyhedron. This interpretation is based on a geometric connection between the cells of an arrangement of classical hyperplanes and their tropicalization.Building up on this geometric interpretation, we present algorithmic refinements of the tropical pivoting operation. We show that pivoting along an edge of a tropical polyhedron defined by $m$ inequalities in dimension $n$ can be done in time $O(n(m+n))$, a complexity similar to the classical pivoting operation. We also show that the computation of reduced costs can be done tropically in time $O(n(m+n))$.
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We disprove a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with $3r+4$ inequalities in dimension $2r+2$ where the central path has a total curvature in $\Omega(2^r)$. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. The lower bound for the classical curvature is obtained by developing a combinatorial concept of a tropical angle.
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The Klee-Minty cube is a well-known worst case example for which the simplex method takes an exponential number of iterations as the algorithm visits all the 2 n vertices of the n-dimensional cube. While such behavior is excluded by polynomial interior point meth-ods, we show that, by adding an exponential number of redundant inequalities, the central path can be bent along the edges of the Klee-Minty cube. More precisely, for an arbitrarily small δ, the central path takes 2 n − 2 turns as it passes through the δ-neighborhood of all the vertices of the Klee-Minty cube in the same order as the simplex method does.
Chapter
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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 suggests 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).
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This paper proposes an interior point algorithm for a positive semi-definite linear complementarity problem: find an (x, y)∈ℝ2n such thaty=Mx+q, (x,y)⩾0 andx T y=0. The algorithm reduces the potential function$$f(x,y) = (n + \sqrt n )\log x^T y - \sum\limits_{i = 1}^n {\log x_i y_i }$$ by at least 0.2 in each iteration requiring O(n 3) arithmetic operations. If it starts from an interior feasible solution with the potential function value bounded by$$O(\sqrt n L)$$, it generates, in at most$$O(\sqrt n L)$$ iterations, an approximate solution with the potential function value$$- O(\sqrt n L)$$, from which we can compute an exact solution in O(n 3) arithmetic operations. The algorithm is closely related with the central path following algorithm recently given by the authors. We also suggest a unified model for both potential reduction and path following algorithms for positive semi-definite linear complementarity problems.
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We propose an approach based on interior-point algorithms for linear programming (LP). We show that the algorithm solves a class of LP problems in strongly polynomial time, O(nlogn)-iteration, where each iteration solves a system of linear equations with n variables. The recent statistical data of the solutions of the NETLIB LP problems seem to indicate that virtually all of these problems are in this class. Then, we show that some random LP problems, with high probability (probability converges to one as n approaches infinity), are in this class. These random LP problems include recent Todd’s probabilistic models with the standard Gauss distribution.
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