Existence and uniqueness of best approximants, with numerical applications
Part I of the thesis deals with existence and uniqueness theorems. Strengthening a result due to J. Blatter, it is proved in chapter 3 that a normed linear space is complete if every closed, bounded, and convex set is proximinal. It is also shown, that in a semi-reflexive, locally convex, real linear metric space, every closed, bounded and convex set is proximinal. An example is constructed which proves that not every reflexive space is sequentially convex. In chapter 4, sequential and local uniform convexity are shown to be independent properties. It is proved that a sequentially convex space can be equivalently renormed with a locally uniformly convex norm. Various spaces are shown to be incapable of uniformly convex renorming. In chapter 5, a number of convexity properties and a class of convergence processes are generalized to metric spaces. It is shown that Clarkson's renorming technique can be extended to metrics and that each closed subset of a metric space can be made proximinal by introducing an equivalent metric. Chapter 6 provides a link between the abstract material of previous chapters and the numerical applications of part II. A unified theory is developed which comprises both discrete and continuous Chebyshev approximation. Part II of the thesis contains numerical applications to the approximation of functions, data analysis, mathematical modelling, and optimization. Chapter 7 deals with a modified exchange algorithm for Chebyshev approximation. In chapter 8, closed formulae for linear Chebyshev approximants are derived. A computer approximation is obtained which is subject to restrictions on the number of non-zero bits in its binary representation. In chapter 9, an algorithm is developed which determines the L^ solution set and selects a strictly best solution. Chapter 10 deals with the problem of balancing the input and output streams of mineral processing plants. A comparison is made of various existing methods and some new algorithms are suggested. In chapter 11, an integer programming algorithm is developed which allows the user to search for sub-optimal and alternative optimal solutions. Codings of the algorithms in chapters 7, 9, 10, and 11 are listed in the appendix of programs. A separate pocket at the end of the thesis contains two papers published in advance.