In this thesis, we study several interior point continuous trajectories for linearly constrained convex programming (CP) and convex semidefinite programming (SDP). The continuous trajectories are characterized as the solution trajectories of corresponding ordinary differential equation (ODE) systems. All our ODE systems are closely related to interior point methods.. First, we propose and analyze three continuous trajectories, which are the solutions of three ODE systems for linearly constrained convex programming. The three ODE systems are formulated based on an variant of the affine scaling direction, the central path, and the affine scaling direction in interior point methods. The resulting solutions of the first two ODE systems are called generalized affine scaling trajectory and generalized central path, respectively. Under some mild conditions, the properties of the continuous trajectories, the optimality and convergence of the continuous trajectories are all obtained. Furthermore, we show that for the example of Gilbert et al. [Math. Program., { 103}, 63-94 (2005)], where the central path does not converge, our generalized central path converges to an optimal solution of the same example in the limit.. Then we analyze two primal dual continuous trajectories for convex programming. The two continuous trajectories are derived from the primal-dual path-following method and the primal-dual affine scaling method, respectively. Theoretical properties of the two interior point continuous trajectories are fully studied. The optimality and convergence of both interior point continuous trajectories are obtained for any interior feasible point under some mild conditions. In particular, with proper choice of some parameters, the convergence for both continuous trajectories does not require the strict complementarity or the analyticity of the objective function.. For convex semidefinite programming, four interior continuous trajectories defined by matrix differential equations are proposed and analyzed. Optimality and convergence of the continuous trajectories are also obtained under some mild conditions. We also propose a strategy to guarantee the optimality of the affine scaling algorithm for convex SDP.

The submitted work consists of a collection of publications arising from research carried out at Rhodes University (1970-1980) and at Heriot-Watt University (1980-1992). The theme of this research is the process of program development, i.e. the process of creating a computer program to solve some particular problem. The papers presented cover a number of different topics which relate to this process, viz. (a) Programming methodology programming. (b) Properties of programming languages. aspects of structured. (c) Formal specification of programming languages. (d) Compiler techniques. (e) Declarative programming languages. (f) Program development aids. (g) Automatic program generation. (h) Databases. (i) Algorithms and applications.

No description available.

The conditions required for a solution of general non-linear programming problems of the form min{f(x): x є X, g(x) ≤ 0, h(x)=0}; where f is called the objective function, g the inequality constraint and. h the equality constraint, are presented in this thesis. The following cases are studied: (1) X, a finite dimensional space; f, a real valued function; and g and h finite dimensional vector functions. (2) X, an infinite dimensional space; f, a real valued function; and g and h either finite or infinite dimensional vector functions. An application of this type of problem to optimal control will be given and the recent developments in this area will be discussed. / Science, Faculty of / Mathematics, Department of / Graduate

Problems of minimizing a convex function or maximizing a concave function over a convex set are called convex programming problems. Duality principles relate two problems, one a minimization problem, the other a maximization problem, in such a way that a solution to one implies a solution to the other and that the minimum value of one is equal to the maximum value of the other. When the functions are linear and the constraint sets are polyhedral, the problems are called linear programming problems. Their duality is well-known. Certain duality results of linear programming can be extended to convex programming by means of the theory of conjugate convex functions introduced by Fenchel ([1], [2]). In this thesis the theory of conjugate functions is generalized and applied to convex programming problems. In particular a duality theorem is given for a class of convex programming problems. This theorem is compared with a duality theorem for convex programming problems given by Dorn [3]. / Science, Faculty of / Mathematics, Department of / Graduate

Reservoir operation policies are often formulated deterministically on the basis of critical flow hydrology. However, if a dynamic river daily flow forecast system is available for the whole season, the forecast information should be fully utilized in reservoir regulation. Given such a forecast system, two approaches to determining optimal daily operation policies for a single purpose flood control reservoir are suggested. Both approaches use stochastic dynamic programming: one involves the minimization of the expected value of flood damages, and the other involves minimizing the probability of occurrence of an undesirable event, which is a flood damage exceeding a certain amount. The probabilistic approach not only offers a set of alternative optimal daily operation policies, but also indicates the probabilities of being able to achieve the objectives, and thus it forms a basis for comparing and evaluating the alternative objectives. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Exception handling is a crucial aspect of practical programming, particularly in systems allowing logical concurrency such as multi-process distributed systems. First, a survey of existing exception handling mechanisms in operating systems is performed, which shows a diversity of implementations, depending on the process model and the method of inter-process communication. The thesis then develops a model for designing software which exploits the different mechanisms for handling normal and exceptional events. The model is applicable in many multi-process programming environments, and not only preserves modularity, but also enhances efficiency and reliability, while often increasing concurrency. To derive such a model, exceptions in multi-process software are classified primarily according to the program level at which they are detected and handled. Server-to-client exceptions are of particular interest because of their ubiquity; these are exceptions detected by a server and handled by a client. The model treats systems programs as event driven, and proposes dividing the events into normal or exceptional, according to the cost and mechanisms for handling them. Techniques are described for designing software according to three criteria: minimising the average run-time, minimising the exception processing time, and incrementally increasing the program's functionality. Many examples are given which illustrate the use of the general model. Program paradigms in several languages and in several systems are introduced to model features which are system dependent, through illustrative examples for asynchronous i/o multiplexing, and for exception notification from a server to its client or clients. Finally, some programs which have been implemented according to the rules of the model are described and compared with their more conventional counterparts. These programs illustrate the practicality and usefulness of the model for diverse systems and concurrent environments. / Science, Faculty of / Computer Science, Department of / Graduate

The implementation of a unified facility for program development and maintenance is described. The prototype (named "Eighty-One") provides for the entry, editing and compilation of program text within a unified system in which source text is immediately transformed into an intermediate tree structured representation for storage. The implications of the unification of these facilities under this representation are discussed from the viewpoints of user convenience and system efficiency. Technical problems posed by the implementation are also discussed. Finally, some comments are made upon the nature of the user interface to systems of this type. / Science, Faculty of / Computer Science, Department of / Graduate

In the thesis, geometric programming is considered as a numerical optimization technique. The problem of minimizing the integral square error of a system characterized by a second order plant with proportional- integral-derivative (PID) controller is investigated. Constraints are imposed upon the state of the system In order to obtain feasible solutions and conditions that are amenable to the geometric programming technique. The application of geometric programming requires the use of approximation procedures to eliminate untenable conditions in the objective and constraint functions. The techniques utilized render solutions that are easily obtainable, usually amounting to solving a set of linear equations and requiring no differentiation of terms. In addition, there is rapid convergence to an optimum. The accuracy of the results is dependent upon the validity of the approximations. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Finding the convex hull of a finite set of points is important not only for practical applications but also for theoretical reasons: a number of geometrical problems, such as constructing Voronoi diagrams or intersecting hyperspheres, can be reduced to the convex hull problem, and a fast convex hull algorithm yields fast algorithms for these other problems. This thesis deals with the problem of constructing the convex hull of a finite point set in R . Mathematical properties of convex hulls are developed, in particular, their facial structure, their representation, bounds on the number of faces, and the concept of duality. The main result of this thesis is an O(nlogn + n[(d+1)/2]) algorithm for the construction of the convex hull of n points in Rd. It is shown that this algorithm is worst case optimal for even d≥2. / Science, Faculty of / Computer Science, Department of / Graduate