Learning the kernel matrix with semidefinite programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space - classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

A new constraint programming approach for optimising a coal rail system

Because of the bottlenecking operations in a complex coal rail system, millions of dollars are costed by mining companies. To handle this issue, this paper investigates a real-world coal rail system and aims to optimise the coal railing operations under constraints of limited resources (e.g., limited number of locomotives and wagons). In the literature, most studies considered the train scheduling problem on a single-track railway network to be strongly NP-hard and thus developed metaheuristics as the main solution methods. In this paper, a new mathematical programming model is formulated and coded by optimization programming language based on a constraint programming (CP) approach. A new depth-first-search technique is developed and embedded inside the CP model to obtain the optimised coal railing timetable efficiently. Computational experiments demonstrate that high-quality solutions are obtainable in industry-scale applications. To provide insightful decisions, sensitivity analysis is conducted in terms of different scenarios and specific criteria. Keywords Train scheduling · Rail transportation · Coal mining · Constraint programming

A hybrid constructive heuristic and simulated annealing for railway crew scheduling

Railway crew scheduling problem is the process of allocating train services to the crew duties based on the published train timetable while satisfying operational and contractual requirements. The problem is restricted by many constraints and it belongs to the class of NP-hard. In this paper, we develop a mathematical model for railway crew scheduling with the aim of minimising the number of crew duties by reducing idle transition times. Duties are generated by arranging scheduled trips over a set of duties and sequentially ordering the set of trips within each of duties. The optimisation model includes the time period of relief opportunities within which a train crew can be relieved at any relief point. Existing models and algorithms usually only consider relieving a crew at the beginning of the interval of relief opportunities which may be impractical. This model involves a large number of decision variables and constraints, and therefore a hybrid constructive heuristic with the simulated annealing search algorithm is applied to yield an optimal or near-optimal schedule. The performance of the proposed algorithms is evaluated by applying computational experiments on randomly generated test instances. The results show that the proposed approaches obtain near-optimal solutions in a reasonable computational time for large-sized problems.

A flexible optimisation model for the railway crew scheduling problem

Addressing the Crew Scheduling Problem (CSP) in transportation systems can be too complex to capture all details. The designed models usually ignore or simplify features which are difficult to formulate. This paper proposes an alternative formulation using a Mixed Integer Programming (MIP) approach to the problem. The optimisation model integrates the two phases of pairing generation and pairing optimisation by simultaneously sequencing trips into feasible duties and minimising total elapsed time of any duty. Crew scheduling constraints in which the crew have to return to their home depot at the end of the shift are included in the model. The flexibility of this model comes in the inclusion of the time interval of relief opportunities, allowing the crew to be relieved during a finite time interval. This will enhance the robustness of the schedule and provide a better representation of real-world conditions.

A near Optimal Cane Rail Scheduler under Limited and Unlimited Capacity Constraints

Australia is the world’s third largest exporter of raw sugar after Brazil and Thailand, with around $2.0 billion in export earnings. Transport systems play a vital role in the raw sugar production process by transporting the sugarcane crop between farms and mills. In 2013, 87 per cent of sugarcane was transported to mills by cane railway. The total cost of sugarcane transport operations is very high. Over 35% of the total cost of sugarcane production in Australia is incurred in cane transport. A cane railway network mainly involves single track sections and multiple track sections used as passing loops or sidings. The cane railway system performs two main tasks: delivering empty bins from the mill to the sidings for filling by harvesters; and collecting the full bins of cane from the sidings and transporting them to the mill. A typical locomotive run involves an empty train (locomotive and empty bins) departing from the mill, traversing some track sections and delivering bins at specified sidings. The locomotive then, returns to the mill, traversing the same track sections in reverse order, collecting full bins along the way. In practice, a single track section can be occupied by only one train at a time, while more than one train can use a passing loop (parallel sections) at a time. The sugarcane transport system is a complex system that includes a large number of variables and elements. These elements work together to achieve the main system objectives of satisfying both mill and harvester requirements and improving the efficiency of the system in terms of low overall costs. These costs include delay, congestion, operating and maintenance costs. An effective cane rail scheduler will assist the traffic officers at the mill to keep a continuous supply of empty bins to harvesters and full bins to the mill with a minimum cost. This paper addresses the cane rail scheduling problem under rail siding capacity constraints where limited and unlimited siding capacities were investigated with different numbers of trains and different train speeds. The total operating time as a function of the number of trains, train shifts and a limited number of cane bins have been calculated for the different siding capacity constraints. A mathematical programming approach has been used to develop a new scheduler for the cane rail transport system under limited and unlimited constraints. The new scheduler aims to reduce the total costs associated with the cane rail transport system that are a function of the number of bins and total operating costs. The proposed metaheuristic techniques have been used to find near optimal solutions of the cane rail scheduling problem and provide different possible solutions to avoid being stuck in local optima. A numerical investigation and sensitivity analysis study is presented to demonstrate that high quality solutions for large scale cane rail scheduling problems are obtainable in a reasonable time. Keywords: Cane railway, mathematical programming, capacity, metaheuristics

Learning the Kernel Matrix with Semi-Definite Programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space -- classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semi-definite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -- using the labelled part of the data one can learn an embedding also for the unlabelled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method to learn the 2-norm soft margin parameter in support vector machines, solving another important open problem. Finally, the novel approach presented in the paper is supported by positive empirical results.

Optimistic linear programming gives logarithmic regret for irreducible MDPs

We present an algorithm called Optimistic Linear Programming (OLP) for learning to optimize average reward in an irreducible but otherwise unknown Markov decision process (MDP). OLP uses its experience so far to estimate the MDP. It chooses actions by optimistically maximizing estimated future rewards over a set of next-state transition probabilities that are close to the estimates, a computation that corresponds to solving linear programs. We show that the total expected reward obtained by OLP up to time T is within C(P) log T of the reward obtained by the optimal policy, where C(P) is an explicit, MDP-dependent constant. OLP is closely related to an algorithm proposed by Burnetas and Katehakis with four key differences: OLP is simpler, it does not require knowledge of the supports of transition probabilities, the proof of the regret bound is simpler, but our regret bound is a constant factor larger than the regret of their algorithm. OLP is also similar in flavor to an algorithm recently proposed by Auer and Ortner. But OLP is simpler and its regret bound has a better dependence on the size of the MDP.

Mathematical modelling of tumour-induced angiogenesis

The growth of solid tumours beyond a critical size is dependent upon angiogenesis, the formation of new blood vessels from an existing vasculature. Tumours may remain dormant at microscopic sizes for some years before switching to a mode in which growth of a supportive vasculature is initiated. The new blood vessels supply nutrients, oxygen, and access to routes by which tumour cells may travel to other sites within the host (metastasize). In recent decades an abundance of biological research has focused on tumour-induced angiogenesis in the hope that treatments targeted at the vasculature may result in a stabilisation or regression of the disease: a tantalizing prospect. The complex and fascinating process of angiogenesis has also attracted the interest of researchers in the field of mathematical biology, a discipline that is, for mathematics, relatively new. The challenge in mathematical biology is to produce a model that captures the essential elements and critical dependencies of a biological system. Such a model may ultimately be used as a predictive tool. In this thesis we examine a number of aspects of tumour-induced angiogenesis, focusing on growth of the neovasculature external to the tumour. Firstly we present a one-dimensional continuum model of tumour-induced angiogenesis in which elements of the immune system or other tumour-cytotoxins are delivered via the newly formed vessels. This model, based on observations from experiments by Judah Folkman et al., is able to show regression of the tumour for some parameter regimes. The modelling highlights a number of interesting aspects of the process that may be characterised further in the laboratory. The next model we present examines the initiation positions of blood vessel sprouts on an existing vessel, in a two-dimensional domain. This model hypothesises that a simple feedback inhibition mechanism may be used to describe the spacing of these sprouts with the inhibitor being produced by breakdown of the existing vessel's basement membrane. Finally, we have developed a stochastic model of blood vessel growth and anastomosis in three dimensions. The model has been implemented in C++, includes an openGL interface, and uses a novel algorithm for calculating proximity of the line segments representing a growing vessel. This choice of programming language and graphics interface allows for near-simultaneous calculation and visualisation of blood vessel networks using a contemporary personal computer. In addition the visualised results may be transformed interactively, and drop-down menus facilitate changes in the parameter values. Visualisation of results is of vital importance in the communication of mathematical information to a wide audience, and we aim to incorporate this philosophy in the thesis. As biological research further uncovers the intriguing processes involved in tumourinduced angiogenesis, we conclude with a comment from mathematical biologist Jim Murray, Mathematical biology is : : : the most exciting modern application of mathematics.

Mathematical modelling and experimental validation of solar assisted heat pump system

Singapore is located at the equator, with abundant supply of solar radiation, relatively high ambient temperature and relative humidity throughout the year. The meteorological conditions of Singapore are favourable for efficient operation of solar energy based systems. Solar assisted heat pump systems are built on the roof-top of National University of Singapore’s Faculty of Engineering. The objectives of this study include the design and performance evaluation of a solar assisted heat-pump system for water desalination, water heating and drying of clothes. Using MATLAB programming language, a 2-dimensional simulation model has been developed to conduct parametric studies on the system. The system shows good prospect to be implemented in both industrial and residential applications and would give new opportunities in replacing conventional energy sources with green renewable energy.

Computation of Mathematical Models for Complex Industrial Processes

Designed for undergraduate and postgraduate students, academic researchers and industrial practitioners, this book provides comprehensive case studies on numerical computing of industrial processes and step-by-step procedures for conducting industrial computing. It assumes minimal knowledge in numerical computing and computer programming, making it easy to read, understand and follow. Topics discussed include fundamentals of industrial computing, finite difference methods, the Wavelet-Collocation Method, the Wavelet-Galerkin Method, High Resolution Methods, and comparative studies of various methods. These are discussed using examples of carefully selected models from real processes of industrial significance. The step-by-step procedures in all these case studies can be easily applied to other industrial processes without a need for major changes and thus provide readers with useful frameworks for the applications of engineering computing in fundamental research problems and practical development scenarios.

Linear programming for large-scale Markov decision problems

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose a technique based on stochastic convex optimization and give bounds that show that the performance of our algorithm approaches the best achievable by any policy in the comparison class. Most importantly, this result depends on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithm in a queuing application.

Mathematics, programming, and STEM

Learning mathematics is a complex and dynamic process. In this paper, the authors adopt a semiotic framework (Yeh & Nason, 2004) and highlight programming as one of the main aspects of the semiosis or meaning-making for the learning of mathematics. During a 10-week teaching experiment, mathematical meaning-making was enriched when primary students wrote Logo programs to create 3D virtual worlds. The analysis of results found deep learning in mathematics, as well as in technology and engineering areas. This prompted a rethinking about the nature of learning mathematics and a need to employ and examine a more holistic learning approach for the learning in science, technology, engineering, and mathematics (STEM) areas.