arXiv subjects

Henning Seidler

Publications and source records attributed to Henning Seidler.

4 recordsLinked to original sources

arXiv2021

Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM Exponentials

We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the number of terms of the input, and linear in the distance to the boundary of the cone. We also provide experimental comparisons regarding the implementation of the two optimization algorithms.

cs.SC↗

arXiv2021

Improved Lower Bounds for Global Polynomial Optimisation

We present a branch-and-bound algorithm to improve the lower bounds obtained by SONC/SAGE. The running time is fixed-parameter tractable in the number of variables. Furthermore, we describe a new heuristic to obtain a candidate for the global minimum of a multivariate polynomial, based on its SONC decomposition. Applying this approach to thousands of test cases, we mostly obtain small duality gaps. In particular, we optimally solve the global minimisation problem in about 70% of the investigated cases.

math.OC↗

arXiv2018

An Experimental Comparison of SONC and SOS Certificates for Unconstrained Optimization

Finding the minimum of a multivariate real polynomial is a well-known hard problem with various applications. We present a polynomial time algorithm to approximate such lower bounds via sums of nonnegative circuit polynomials (SONC). As a main result, we carry out the first large-scale comparison of SONC, using this algorithm and different geometric programming (GP) solvers, with the classical sums of squares (SOS) approach, using several of the most common semidefinite programming (SDP) solvers. SONC yields bounds competitive to SOS in several cases, but using significantly less time and memory. In particular, SONC/GP can handle much larger problem instances than SOS/SDP.

math.OC↗

arXiv2017

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs.

cs.CC↗