Random walks on polytopes and an affine interior point method for linear programming

Abstract

Let K be a polytope in R n defined by m linear inequalities. We give a new Markov Chain algorithm to draw a nearly uniform sample from K. The underlying Markov Chain is the first to have a mixing time that is strongly polynomial when started from a "central" point x 0 . If s is the supremum over all chords pq passing through x 0 of |p−x0| |q−x0| and is an upper bound on the desired total variation distance from the uniform, it is sufficient to take O mn n log(sm) + log 1 steps of the random walk. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately. More precisely, suppose Q = {z Bz ≤ 1} contains a point z such that c T z ≥ d and r := sup z∈Q Bz + 1, where B is an m × n matrix. Then, after τ = O mn n ln mr + ln 1 δ steps, the random walk is at a point x τ for which c T x τ ≥ d(1 − ) with probability greater than 1 − δ. The fact that this algorithm has a run-time that is provably polynomial is notable since the analogous deterministic affine algorithm analyzed by Dikin has no known polynomial guarantees.

Efficient Inverse Maintenance and Faster Algorithms for Linear Programming

“…Here consider a typical case where the convex set is a polytope that is explicitly given as { x ∈ R d : A x ≥ b} for A ∈ R n×d . The current fastest algorithm for this setting is Hit-and-Run [22] and Dikin walk [9]. Given an initial random point in the set, Hit-and-Run takes O * (d 3 ) iterations and each iteration takes time O * (nnz(A)) while Dikin walk takes O * (nd) iterations and each iteration takes time O * (nd ω−1 ) where the O * notation omits the dependence on error parameters and logarithmic terms.…”

“…Solving a sequence of linear systems is the computational bottleneck in many state-of-the-art optimization algorithms, including interior point methods for linear programming [12,29,30,19], the Dikin walk for sampling a point in a polytope [9], and multiplicative weight algorithms for grouped least squares problem [1], etc. In full generality, any particular iteration of these algorithms could require solving an arbitrary positive definite (PD) linear system.…”

Modelling chance and necessity in natural systems

“…Several algorithms exists to sample polytopes; a simple and nevertheless efficient method is the Hit and Run algorithm (Kannan and Narayanan, 2012). The sample is the result of a walk inside the polytope.…”

“…In this paper, we use a variant of this algorithm, the Dikin algorithm (Kannan and Narayanan, 2012;Sachdeva and Vishnoi, 2016;Chen et al, 2018), which allows to deal with this issue, and results in a Metropolis-Hasting walk. It is a Monte-Carlo Markov chain algorithm and its mixing properties can be theoretically studied (Chen et al, 2018).…”

Pangolin: Incremental Hybrid Fuzzing with Polyhedral Path Abstraction

“…Specifically, in addition to finding a single feasible solution of the outer branch through constraint solvers, we also target at obtaining the bounded ranges of the related input variables for the target path constraint. We describe such bounded ranges by linearizing the path constraint as a polyhedron, denoted as the "polyhedral path abstraction", for guiding both the mutation and the constraint 1 i n t main ( ) { 2 u n s i g n e d v , w, x , y , z = i n p u t ( ) ; • The polyhedral path abstraction of a path constraint enables us to convert the problem of mutating the seeds into the problem of sampling over a polyhedron, which has been well studied and has many efficient solutions [10], [11], [12], [13], [14], [15]. In this work, we adopt a state-of-the-art technique, the Dikin walk algorithm [15], which can achieve the polynomial time complexity, to efficiently generate a large number of inputs while still respecting the target path constraints.…”