K. I. M. McKinnon

Solving stochastic ship fleet routing problems with inventory management using branch and price

This chapter describes a stochastic ship routing problem with inventory management. The problem involves finding a set of least cost routes for a fleet of ships transporting a single commodity when the demand for the commodity is uncertain. Storage at supply and consumption ports is limited and inventory levels are monitored in the model. Consumer demands are at a constant rate within each time period, and in the stochastic problem, the demand rate for a period is not known until the beginning of that period. The demand situation over the time periods is described by a scenario tree with corresponding probabilities. A decomposition formulation is given and it is solved using a Branch and Price framework. A master problem (set partitioning with extra inventory constraints) is built, and the subproblems, one for each ship, are solved by stochastic dynamic programming and yield the columns for the master problem. Each column corresponds to one possible tree of actions for one ship giving its schedule loading/unloading quantities for all demand scenarios. Computational results are given showing that medium sized problems can be solved successfully.

An aggregate stochastic dynamic programming model of multireservoir systems

We present a new method of determining an operating policy for a multireservoir system in which the operating policy for a reservoir is determined by solving a stochastic dynamic programming model consisting of that reservoir and a two-dimensional representation of the rest of the system. The method is practical for systems with many reservoirs because the time required to determine an operating policy only increases quadratically with the number of reservoirs in the system and because the operating policy for a reservoir is a function of few variables. We apply the method to examples of multireservoir systems with between 3 and 17 reservoirs and show that the operating policies determined are very close to optimal.

Local Solutions of the Optimal Power Flow Problem

The existence of locally optimal solutions to the AC optimal power flow problem (OPF) has been a question of interest for decades. This paper presents examples of local optima on a variety of test networks including modified versions of common networks. We show that local optima can occur because the feasible region is disconnected and/or because of nonlinearities in the constraints. Standard local optimization techniques are shown to converge to these local optima. The voltage bounds of all the examples in this paper are between ±5% and ±10% off-nominal. The examples with local optima are available in an online archive (http://www.maths.ed.ac.uk/optenergy/LocalOpt/) and can be used to test local or global optimization techniques for OPF. Finally we use our test examples to illustrate the behavior of a recent semi-definite programming approach that aims to find the global solution of OPF.

On Saddle Points of Augmented Lagrangians for Constrained Nonconvex Optimization

We present in this paper new results on the existence of saddle points of augmented Lagrangian functions for constrained nonconvex optimization. Four classes of augmented Lagrangian functions are considered: the essentially quadratic augmented Lagrangian, the exponential-type augmented Lagrangian, the modified barrier augmented Lagrangian, and the penalized exponential-type augmented Lagrangian. We first show that under second-order sufficiency conditions, all these augmented Lagrangian functions possess local saddle points. We then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions. The results obtained in this paper provide a theoretical foundation for the use of augmented Lagrangians in constrained global optimization. Our findings also give new insights to the role played by augmented Lagrangians in local duality theory of constrained nonconvex optimization.

Hyper-Sparsity in the Revised Simplex Method and How to Exploit it

The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems the result of one or more of these operations is usually sparse, a property we call hyper-sparsity. Analysis of the commonly-used techniques for implementing each step of the revised simplex method shows them to be inefficient when hyper-sparsity is present. Techniques to exploit hyper-sparsity are developed and their performance is compared with the standard techniques. For the subset of our test problems that exhibits hyper-sparsity, the average speedup in solution time is 5.2 when these techniques are used. For this problem set our implementation of the revised simplex method which exploits hyper-sparsity is shown to be competitive with the leading commercial solver and significantly faster than the leading public-domain solver.

A convexification method for a class of global optimization problems with applications to reliability optimization

A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.

Constructing integer programming models by the predicate calculus

A modelling language for Integer Programming (IP) based on the Predicate Calculus is described. This is particularly suitable for building models with logical conditions. Using this language a model is specified in terms of predicates. This is then converted automatically by a series of transformation rules into a normal form from which an IP model can be created. There is also some discussion of alternative IP formulations which can be incorporated into the system as options. Further practical considerations are discussed briefly concerning implementation language and incorporation into practical Mathematical Programming Systems.

Optimization-Based Islanding of Power Networks Using Piecewise Linear AC Power Flow

In this paper, a flexible optimization-based framework for intentional islanding is presented. The decision is made of which transmission lines to switch in order to split the network while minimizing disruption, the amount of load shed, or grouping coherent generators. The approach uses a piecewise linear model of AC power flow, which allows the voltage and reactive power to be considered directly when designing the islands. Demonstrations on standard test networks show that solution of the problem provides islands that are balanced in real and reactive power, satisfy AC power flow laws, and have a healthy voltage profile.

A Generic Global Optimization Algorithmfor the Chemical and Phase EquilibriumProblem

This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of ns substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing mb independent conserved quantities, where mb=ns when no reaction is possible and mb ≤ ne +1 when reaction is possible and ne is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the ‘reaction tangent-plane criterion’, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of mb(ns-1) dimensions) to a finite sequence of local optimization steps inK(ns-1) -space, K ≤ mb, and global optimization steps in (ns-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.

Controlling multi-reservoir systems

The paper extends the results of the form of the optimal policy for a hydroelectric reservoir problem from the one-reservoir case to multi-reservoir cases. The importance of these new results in practice is that they allow more efficient solution algorithms to be developed. Since multi-reservoir problems are extremely difficult to solve, such algorithms are of great value.