Topics in optimisation

© 2018 Dr. David Paul Jerome KirszenblatThis thesis addresses four problems in continuous and discrete optimisation. The first problem is about using column generation – an advanced technique in mixed integer programming – to design schedules for students in the Royal Australian Navy who are learning to fly helicopters. Selected results were published in the proceedings of the International Congress on Modelling and Simulation. The second problem is about shortest curvature-constrained paths in the plane, i.e., Dubins curves, which find applications in, for example, path planning for unmanned aerial vehicles. We use geometric arguments to produce a classification of length minimising curvature- constrained paths. The results were published in Communications in Analysis and Geometry. The third problem is about the construction of minimal curvature-constrained networks with applications in the design of underground mines. This problem is a novel combination of two classical problems in optimisation, namely, the Steiner problem and the Dubins problem. We give two algorithms for the construction of “Dubins trees” in the plane and in 3D space. The results were selected for publication in a special issue of Journal of Global Optimization on “Distance Geometry: Theory and Applications”. The fourth problem is the polynomial Hirsch conjecture. The Hirsch conjecture, posed in 1957, states that the least number edges required to join any pair of vertices of a d-dimensional polyhedron with n facets, i.e., its diameter, is bounded from above by n − d. The conjecture is of relevance to the efficiency of the simplex method, a path following method for the solution of linear programming problems. The version of the conjecture for polytopes was disproved by Santos in 2010. However, the question of whether there exists an upper bound for the diameter that is polynomial in d and n remains open. We propose two approaches for tackling this problem. First, we provide an algorithm that allows us to enumerate polytopes and verify a variant of the Hirsch conjecture, called the d-step conjecture, in low dimensions. Second, we provide a method for constructing higher dimensional polyhedra that satisfy the Hirsch bound from arbitrary polyhedra. The method allows us to take an arbitrary linear programming problem and construct an equivalent linear programming problem for which the best possible sequence of pivots to an optimal solution is small