Quadratic programming with constant norm with parallel applications
This thesis is concerned with the problem of minimizing a quadratic function defined on an ellipsoid, and subject to the set of linear inequality constraints. An original method is proposed, a method which generates the descent flow on the spiral, the curve on the surface of the constraining ellipsoid, characterized locally by the steepest descent of the objective function. The spiral is constructed discretely in a number of points which are the solutions of the sequence of certain artificially constrained subproblems. The properties of the spiral are studied extensively to show that the associated descent flow can be divided into three categories of the regular, separated and multiple descent flow. Particular attention is devoted to the case of the separated descent flow, which involves the discontinuity of the spiral and may pose certain difficulties in the search for the solution. The important and the original part of the thesis is also the issue of all local minima of the equality constrained version of the problem. The identification of all such minima is vital to the main problem and the corresponding method. The method which may find many practical applications, has been inspired by the opportunities given by a massively parallel computer, namely the Distributed Array Processor, although the method can find the applications in other parallel environments. The algorithm has been implemented in DAP Fortran-Plus, and a series of numerical examples is presented, some of them of practical importance in structural engineering.