Polynomial unconstrained binary optimisation – Part 2

The class of problems known as quadratic zeroone (binary) unconstrained optimisation has provided access to a vast array of combinatorial optimisation problems, allowing them to be expressed within the setting of a single unifying model. A gap exists, however, in addressing polynomial problems of degree greater than 2. To bridge this gap, we provide methods for efficiently executing core search processes for the general polynomial unconstrained binary (PUB) optimisation problem. A variety of search algorithms for quadratic optimisation can take advantage of our methods to be transformed directly into algorithms for problems where the objective functions involve arbitrary polynomials. Part 1 of this paper (Glover et al., 2011) provided fundamental results for carrying out the transformations and described coding and decoding procedures relevant for efficiently handling sparse problems, where many coefficients are 0, as typically arise in practical applications. In the present part 2 paper, we provide special algorithms and data structures for taking advantage of the basic results of part 1. We also disclose how our designs can be used to enhance existing quadratic optimisation algorithms

Approximate Approximation on a Quantum Annealer

Many problems of industrial interest are NP-complete, and quickly exhaust resources of computational devices with increasing input sizes. Quantum annealers (QA) are physical devices that aim at this class of problems by exploiting quantum mechanical properties of nature. However, they compete with efficient heuristics and probabilistic or randomised algorithms on classical machines that allow for finding approximate solutions to large NP-complete problems. While first implementations of QA have become commercially available, their practical benefits are far from fully explored. To the best of our knowledge, approximation techniques have not yet received substantial attention. In this paper, we explore how problems' approximate versions of varying degree can be systematically constructed for quantum annealer programs, and how this influences result quality or the handling of larger problem instances on given set of qubits. We illustrate various approximation techniques on both, simulations and real QA hardware, on different seminal problems, and interpret the results to contribute towards a better understanding of the real-world power and limitations of current-state and future quantum computing.Comment: Proceedings of the 17th ACM International Conference on Computing Frontiers (CF 2020

Polynomial unconstrained binary optimisation inspired by optical simulation

We propose an algorithm inspired by optical coherent Ising machines to solve the problem of polynomial unconstrained binary optimisation (PUBO). We benchmark the proposed algorithm against existing PUBO algorithms on the extended Sherrington-Kirkpatrick model and random third-degree polynomial pseudo-Boolean functions, and observe its superior performance. We also address instances of practically relevant computational problems such as protein folding and electronic structure calculations with problem sizes not accessible to existing quantum annealing devices. In particular, we successfully find the lowest-energy conformation of lattice protein molecules containing up to eleven amino-acids. The application of our algorithm to quantum chemistry sheds light on the shortcomings of approximating the electronic structure problem by a PUBO problem, which, in turn, puts into question the applicability of quantum annealers in this context.Comment: 10 pages, 6 figure

State-of-the-art in aerodynamic shape optimisation methods

Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners

Efficient Relaxations for Dense CRFs with Sparse Higher Order Potentials

Dense conditional random fields (CRFs) have become a popular framework for modelling several problems in computer vision such as stereo correspondence and multi-class semantic segmentation. By modelling long-range interactions, dense CRFs provide a labelling that captures finer detail than their sparse counterparts. Currently, the state-of-the-art algorithm performs mean-field inference using a filter-based method but fails to provide a strong theoretical guarantee on the quality of the solution. A question naturally arises as to whether it is possible to obtain a maximum a posteriori (MAP) estimate of a dense CRF using a principled method. Within this paper, we show that this is indeed possible. We will show that, by using a filter-based method, continuous relaxations of the MAP problem can be optimised efficiently using state-of-the-art algorithms. Specifically, we will solve a quadratic programming (QP) relaxation using the Frank-Wolfe algorithm and a linear programming (LP) relaxation by developing a proximal minimisation framework. By exploiting labelling consistency in the higher-order potentials and utilising the filter-based method, we are able to formulate the above algorithms such that each iteration has a complexity linear in the number of classes and random variables. The presented algorithms can be applied to any labelling problem using a dense CRF with sparse higher-order potentials. In this paper, we use semantic segmentation as an example application as it demonstrates the ability of the algorithm to scale to dense CRFs with large dimensions. We perform experiments on the Pascal dataset to indicate that the presented algorithms are able to attain lower energies than the mean-field inference method

Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach