This paper investigates four methods of one variable function minimising that are used in the optimisation process („Golden cut”, „DSK”, „Pauel”, „Half distribution”), executes the comparison of their efficiency and generalises the results of this research.

Nous étudions la minimisation du périmètre et de l'énergie de Willmore en présence de contraintes ainsi que le flot, défini par les mouvements minimisants, de l'énergie de Willmore. Les problèmes d'optimisation géométriques et les flots que nous considérons reposent sur une propriété de semi-continuité inférieure que nous pouvons assurer en prenant l'enveloppe semi-continue inférieurement des énergies incluant les contraintes.Dans la première partie de la thèse, nous étudions trois problèmes d'optimisation. Le premier concerne le périmètre avec une contrainte de connexité. Le second est un problème de reconstruction de domaine à partir de sections planaires. Cette reconstruction est basée sur la minimisation du périmètre ou de l'énergie de Willmore avec des contraintes d'inclusion-exclusion. Nous développons un modèle de champ de phase pour implémenter numériquement la reconstruction en 2D et 3D à partir de contraintes d'inclusion-exclusion variées. Le troisième problème est l'étude des propriétés des courbes fermées, confinées dans un ouvert borné du plan, minimisant l'énergie élastique (Willmore).La deuxième partie étudie le flot de l'énergie de Willmore par les mouvements minimisants. Le flot pour une surface régulière est difficile à analyser, entre autre car il peut développer des singularités en temps fini. L'enveloppe semi-continue inférieurement et les mouvements minimisants permettent de définir un flot en temps long pour des surfaces moins régulières. Ce flot est étudié dans deux situations : pour la somme de l'énergie de Willmore et du périmètre dans le plan et pour l'énergie de Willmore des fonctions radiales à profil décroissant en toute dimension / We study the minimisation with constraints of the perimeter and of the Willmore energie and the flow of the Willmore energie, defined by minimising movements. The geometric optimisation problems and flows we handle rely on a lower semicontinuous property that we enforce by taking the lower semicontinuous envelop of the energies including the constraints.In the first part of the thesis, we consider three optimisation problems. The first one deals with the perimeter and connectedness constraints in the plan. The second one is a reconstruction problem of a volumetric domain from planar slices. This reconstruction is based on the minimisation of the perimeter or of the Willmore energy with inclusion-exclusion constraints. A phase field numerical approach and experiments are implemented. The third problem is the study of closed curves confined in an open bounded subset of the plane that minimise the bending energy (or Willmore energy). The second part of the thesis studies the Willmore flow defined by minimising movements. The flow of a regular surface is complex to analyse and may develop singularities in finite time. We use the lower semicontinuous envelop of the Willmore energy and the minimising movement to define a long time flow for surfaces with less regularity. This flow is studied within two contexts: for the sum of the Willmore energy and the perimeter in the plane and for the Willmore energy of radial functions with a non-increasing profile in any dimension

The purpose of the study is to investigate optimal asset allocation as a means of minimising the investment risk, drawdown risk and longevity risk associated with an investment linked living annuity. The three risk elements were tested for various categories of retirees investing the full retirement savings amount in a living annuity. At first the paper examines the South African public's current pre-retirement savings habits, propensity to save and knowledge on the financial industry. The literature concludes that very few people are saving adequately for retirement, thus leaving a gap between required retirement savings capital and accumulated retirement savings capital. As a consequence, retirees have to take on more risk, usually in the form of equity exposure, (only available in an investment linked living annuity) or delaying retirement, to try and breach the gap. Secondly the paper examines the constructs in developing an optimal asset allocation. An analysis of the constructs includes risk versus return relationships for retirees, various unit trust sectors and portfolios within the South African financial market, the investment horizon also stated as the life expectancy of a retiree and withdrawal strategies applied by investors or retirees. The practical data and theory from the literature study formed the basis of the empirical study where different retirement savings balances were tested at various drawdown rates and asset allocations in an investment linked living annuity. The study concluded that retirees have to consider, among other factors, the required standard of living (stated as a net replacement ratio), the need to withdraw one third of the retirement capital and life expectancy before investing in an investment linked living annuity. These factors will have the biggest influence on the risks associated with an investment linked living annuity. Furthermore, the study concluded that an optimal asset allocation would be able to support a retiree during the post-retirement phase. A well diversified portfolio with a minimum of 50% allocation towards equity and property assets seems to be optimal. / Thesis (M.B.A.)--North-West University, Vaal Triangle Campus, 2010.

The purpose of the study is to investigate optimal asset allocation as a means of minimising the investment risk, drawdown risk and longevity risk associated with an investment linked living annuity. The three risk elements were tested for various categories of retirees investing the full retirement savings amount in a living annuity. At first the paper examines the South African public's current pre-retirement savings habits, propensity to save and knowledge on the financial industry. The literature concludes that very few people are saving adequately for retirement, thus leaving a gap between required retirement savings capital and accumulated retirement savings capital. As a consequence, retirees have to take on more risk, usually in the form of equity exposure, (only available in an investment linked living annuity) or delaying retirement, to try and breach the gap. Secondly the paper examines the constructs in developing an optimal asset allocation. An analysis of the constructs includes risk versus return relationships for retirees, various unit trust sectors and portfolios within the South African financial market, the investment horizon also stated as the life expectancy of a retiree and withdrawal strategies applied by investors or retirees. The practical data and theory from the literature study formed the basis of the empirical study where different retirement savings balances were tested at various drawdown rates and asset allocations in an investment linked living annuity. The study concluded that retirees have to consider, among other factors, the required standard of living (stated as a net replacement ratio), the need to withdraw one third of the retirement capital and life expectancy before investing in an investment linked living annuity. These factors will have the biggest influence on the risks associated with an investment linked living annuity. Furthermore, the study concluded that an optimal asset allocation would be able to support a retiree during the post-retirement phase. A well diversified portfolio with a minimum of 50% allocation towards equity and property assets seems to be optimal. / Thesis (M.B.A.)--North-West University, Vaal Triangle Campus, 2010.