Dissertations / Theses on the topic 'KKT Conditions'

Esta tese é um estudo acerca da análise de convergência de vários métodos numéricos de primeira e de segunda ordem para resolver problemas de programação matemática e as condições de otimalidade associadas. Nossas principais ferramentas são as condições sequenciais de otimalidade. As condições sequenciais de otimalidade oferecem um quadro teórico para a análise de convergência para várias famílias de métodos de primeira ordem sob condições de qualificações fracas. Nesta tese, apresentamos, para cada condição sequencial de otimalidade, a condição de qualificação mínima associada e mostramos as relações com outras condições de qualificação conhecidas. Este fato tem implicações práticas, uma vez que enfraquece as hipóteses requeridas para a convergência de vários métodos numéricos cujos critérios de paradas estão associados às condições sequenciais de otimalidade. Ainda mais, esse tipo de resultado não pode ser melhorado usando outras condições de qualificações. Nós estendemos a noção de condições sequenciais de otimalidade de primeira ordem, para incorporar informações de segunda ordem. Apresentamos, segundo nosso conhecimento, a primeira condição sequencial de otimalidade de segunda ordem, adequada para a análise de convergência de vários métodos numéricos com convergência a pontos estacionários de segunda ordem, como por exemplo métodos baseados no Lagrangeano aumentado, regiões de confiança e SQP regularizado. Associada com a nova condição sequencial de segunda ordem, temos uma nova condição de qualificação, mais fraca que as outras condições de qualificações utilizadas para a análise de convergência para métodos numéricos de segunda ordem. Nós situamos essa nova condição de qualificação com respeito a outras condições de qualificação usadas em análise de convergência. Finalmente apresentamos outra razão pela qual a condição fraca necessária de segunda ordem é a condição de segunda ordem adequada quando lidarmos com a convergência de algoritmos práticos
This thesis deals with the convergence analysis for several rst-and-second-order numerical methods used to solve mathematical programming problems. Our main tools are the sequential optimality conditions. First-order sequential optimality conditions oer a framework to the study of the convergence analysis of several families of rst-order methods, under weak constraint qualications. In this thesis, we will introduce, for each sequential optimality condition the minimal constraint qualications associated with it and we will show their relationships with other constraint qualications. This fact has a practical aspect, since, we improve the convergence analysis of practical methods with stopping criteria associated with sequential optimality conditions. This results can not be improved by using another weak constraint qualications. We will extend the notion of rst-order sequential optimality conditions to incorporate secondorder information. We will introduce, to the best of our knowledge, the rst second-order sequential optimality condition, suitable to the study of the convergence analysis of several second-order methods including methods based on the augmented lagrangian, trust-region and regularized SQP. Associated with the second-order sequential optimality condition, we have a new constraint qualication weaker than all constraint qualications used for the convergence analysis of second-order methods. We show the relationships of this new constraint qualications with other constraint qualications used for algorithmic purposes. We will also present a new reason why the weak secondorder necessary condition is the natural second-order condition when we are dealing with practical numerical methods

Em um problema de natureza hierárquica, o nível mais influente toma certas decisões que afetam o comportamento dos níveis inferiores. Cada decisão do nível mais influente é considerada como fixa pelos níveis inferiores, que, com tais informações, tomam decisões que maximizam seus objetivos. Essas decisões podem influenciar os resultados obtidos pelo nível superior, que, por sua vez, também anseia pela decisão ótima. Em programação matemática, este problema é modelado como um problema de programação em níveis. Neste trabalho, consideramos uma classe particular de problemas de programação em níveis: os problemas de programação matemática em dois níveis. Estudamos uma técnica de resolução que consiste em substituir o problema do nível inferior por suas condições necessárias de primeira ordem, que podem ser formuladas de diversas maneiras, conforme as restrições de complementaridade são modificadas. O novo problema torna-se um problema de programação não linear e pode ser resolvido com algoritmos clássicos de otimização. Com o auxílio de condições de otimalidade de primeira e segunda ordem mostramos as relações entre o problema original e o problema reformulado. Aplicamos a técnica a problemas encontrados na literatura, analisamos o seu comportamento e apresentamos estratégias para eliminar certos inconvenientes encontrados.
In problems of hierarchical nature, the choices made by the most influential level - the so-called leader - affect the behavior of the lower levels. For each one of the leader\'s decisions there is a response from the lower levels, which maximizes the value of their respective objectives. These optimal choices, in return, may have influence in the results achieved by the leader, which also wants to make the optimal choices. In mathematical programming, this kind of problem is described as a multilevel programming problem. The present work considers a specific kind of multilevel problem: the bilevel mathematical problem. We study a resolution technique which consists in replacing the lower level problem by its necessary first order conditions, which can be formulated in various ways, as complementarity constraints occur and are modified. The new reformulated problem is a nonlinear programming problem which can be solved by classical optimization methods. Using first and second order optimality conditions, we show the relations between the original bilevel problem and the reformulated problem. We apply the described technique to solve a set of bilevel problems taken from the literature, analyse their behavior and discuss strategies to prevent undesirable difficulties that may arise.

Este trabalho descreve uma abordagem do método primal-dual barreira logarítmica (MPDBL) associado ao método de Newton modificado para a resolução do problema de fluxo barreira logarítmica e nas condições de primeira ordem de Karush-Kuhn-Tucker (KKT). O sistema de equações resultantes das condições de estacionaridade, da função Lagrangiana, foi resolvido pelo método de Newton modificado. Na implementação computacional foram usadas as técnicas de esparsidade. Os resultados numéricos dos testes realizados em 5 sistemas (3, 14, 30, 57 e 118 barras) evidenciam o potencial desta metodologia na solução do problema de FPO.
This work describes an approach on primal-dual logarithmic barrier for solving the optimal power flow problem (OPF). The investigation was based on the logarithmic barrier function and Karush-Kuhn-Tucker (KKT) first-order necessary conditions. The equation system, obtained from the stationary conditions of the Lagrangian function, was solved using the Newton\'s modified method. The implementation was performed using sparsity techniques. The numerical results, carried out in five systems (3, 14,30, 57 and 118 bus), demonstrate the reliability of this approach in the solution OPF problem.

Este trabalho apresenta uma abordagem do método da função barreira logarítmica (MFBL) para a resolução do problema de fluxo de potência ótimo (FPO). A pesquisa fundamenta-se metodologicamente na função barreira logarítmica e nas condições de primeira ordem de Karush-Kuhn-Tucker (KKT). Para a solução do sistema de equações resultantes das condições de estacionaridade, da função Lagrangiana, utiliza-se o método de Newton. Na implementação computacional utiliza-se técnicas de esparsidade. Através dos resultados numéricos dos testes realizados em 5 sistemas (3, 8, 14, 30 e 118 barras) evidencia-se o potencial desta metodologia na solução do problema de FPO.
This work describes an approach on logarithmic barrier function method to solving the optimal power flow (OPF) problem. Search was based on the logarithmic barrier function and first order conditions of Karush-Kuhn-Tucker (KKT). To solve the equation system, obtained from the stationary conditions of the Lagrangian function, is used the Newton method. Implementation is performed using sparsity techniques. The numerical results, carried out in five systems (3, 8, 14, 30 and 118 bus), demonstrate the reliability of this approach in the solution OPF problem.

Die Arbeit abstrahiert bekannte Konzepte der endlichdimensionalen Optimierung im Hinblick auf deren Anwendung in Banachräumen. Hierfür werden zunächst grundlegende Elemente der Funktionalanalysis wie schwache Konvergenz, Dualräume und Reflexivität vorgestellt. Anschließend erfolgt eine kurze Einführung in die Thematik der Fréchet-Differenzierbarkeit und eine Abstraktion des Begriffs der partiellen Ordnungsrelation in normierten Räumen. Nach der Formulierung eines allgemeinen Existenzsatzes für globale Optimallösungen von abstrakten Optimierungsaufgaben werden notwendige Optimalitätsbedingungen vom Karush-Kuhn-Tucker-Typ hergeleitet. Abschließend wird eine hinreichende Optimalitätsbedingung vom Karush-Kuhn-Tucker-Typ unter verallgemeinerten Konvexitätsvoraussetzungen verifiziert.

The Kat River is an agricultural catchment that drains salt rich geology. Potential salinity impacts on ecological condition of the river were investigated. Monthly salt concentrations and flow discharges were monitored at ten sites along the Kat River below the Kat Dam. Monthly salt loads were computed to relate salinity to land use and ionic data used to assess the toxicity of major salts using the TIMS model. Concentration duration curves for sodium chloride were derived from flow concentration relationships, representing sodium chloride concentrations to which the aquatic ecosystem had been exposed. The ecological condition was assessed at nineteen sites using SASS5 biotic index over four seasons. Finally, the modelled instream salt concentrations and bioasessments were evaluated in terms of the modelled level of species protection afforded at different salt concentrations. Species Sensitivity Distributions (SSDs) were used for this exercise. There was a general downstream increase in salinity with the minimum concentrations recorded at the Fairbain tributary (84 mg/L) and maximum levels at the sewage outfall in Fort Beaufort (1222 mg/L). There was evidence that citrus irrigation upstream of Fort Beaufort increased salinisation. Sodium chloride, and to a lesser extent magnesium sulphate, were the dominant salts in the Kat River catchment, with the latter being more toxic. However these had little or no impact on the aquatic ecosystem. Flow-derived sodium chloride concentrations showed that both the Balfour and Blinkwater tributaries were in a fair/ poor condition. However with regard to ecological condition, it was demonstrated that the river is generally in a good state except for the Blinkwater River and the lower catchment. Degraded habitat condition at the Blinkwater was responsible for poor ecological condition. Integrating SSD derived classes, sodium chloride classes and ecological condition indicated that sodium chloride is a driver of ecological condition at the sewage treatment works and the subsequent site (only two of nineteen biomonitoring sites).

Purpose:Omeprazole is a proton pump inhibitor (PPI) used in the treatment of gastrointestinal conditions, such as gastrointestinal esophageal reflux disease (GERD). Omeprazole is often prepared as an oral suspension to accommodate certain patients. Historically, oral suspensions of omeprazole were prepared using pharmaceutical compounding with sodium bicarbonate, but a kit for preparation of omeprazole oral suspension is available, FIRST® - Omeprazole. The purpose of this project is to compare the stability of the active pharmaceutical ingredient (API), omeprazole, in the FIRST® kit product to a traditionally compounded omeprazole suspension, when stored in refrigerated unit-dosed syringes. Methods: Five 100-mL batches of compounded omeprazole oral suspension (2 mg/mL) and five 300-mL kits of FIRST® - Omeprazole were prepared by a licensed pharmacist, and aliquoted into 5-mL doses in clear luer-lock plastic oral syringes, and stored at refrigerated temperature (2-8oC). Omeprazole concentration was assessed in each batch/kit on the day of preparation. Triplicate syringes from each batch/kit (n = 15 per test group per day) were removed after 7 days, 14 days, 21 days, and 30 days of refrigerated storage. Samples were diluted to assay concentration (1 mg/mL) in ion-free water and filtered using a 0.22-micron microcentrifuge filter tube. Samples were analyzed for omeprazole recovery using a validated high-performance liquid chromatography with ultraviolet detection (HPLC-UV) method. Recovery was quantitatively assessed by comparing sample peak area to a freshly prepared calibration curve (1 – 0.125 mg/mL) using United States Pharmacopoeia (USP) reference standard on each day of sampling. Refrigerator temperatures were recorded daily using a digital thermometer. Results:Stability was defined as recovery of 90 - 110% of initial concentration of API. For the FIRST® - Omeprazole samples, the chemical potency remained within this window for the entire study period of 30 days. The compounded omeprazole suspension demonstrated a less than 90% average recovery at the day 21 sample. Furthermore, a statistically significant difference in the initial concentration was detected on the day of compounding (p = 0.0244), with the compounded omeprazole starting at 1.89 ± 0.10 mg/mL and the FIRST® - Omeprazole at 1.98 ± 0.04 mg/mL. After 30 days, the compounded omeprazole suspension had an 89.13% average API recovery (standard deviation; ± 5.17%) and the FIRST® - Omeprazole 97.20% API recovery (± 3.59%). Conclusion:Both traditionally compounded omeprazole suspension (2mg/mL) and FIRST® - Omeprazole suspension (2mg/mL) may be stored in clear luer-lock oral syringes under refrigeration for 14 days, and retain potency between 90 to 110% based on initial concentration. Furthermore, the FIRST® - Omeprazole suspension can be stored for the duration of the product’s beyond-use date of 30 days and retain potency between 90 to 110% of initial concentration or label claim. Finally, the data suggest that API concentration in FIRST® - Omeprazole suspension is more consistent from batch to batch than traditionally compounded omeprazole suspension.

This project explores the question of how to support women living with endometriosis in the transition into live with a chronic condition while maintaining the best possible quality of life. Design process was planned out and exercises in four steps - preparations and background research (outlining well-being perspective, desk research, research online via social media), design research (multiple surveys, interviews, reflections online, co-creation sessions, probing, interviews following probing with co-creation sessions, ideation sessions, analysis), explorations (creation of directions, analysis, initial concepts creation, feedback, synthesis, design proposal, feedback and evaluation sessions, design improvements) and final design proposal. Activities were aiming to repeatedly open up the scope of the project and then through the analysis of each phase narrow it down again following diamond shape. One important aspect of the process was to engage women living with endometriosis as well as their loved ones throughout the whole process and design together instead of designing for them. Of course, experts were consulted as well to make sure project is following standards of medical care, but the primary focus on the users was important since the product was shaped rather as a lifestyle change facilitator as well as data collection tool rather than a strictly medical solution. The outcome of the project is a flexible system VEA serving as a transition kit for women diagnosed with endometriosis, but also early detection screening tool and awareness platform for those who haven’t been diagnosed yet. Even though this might seem complicated I’ll try to explain why only this kind of a multifaceted approach can fully support women in their journey toward their best possible quality of life. VEA allows users to access Habit Nooks supporting them in introducing changes while guided by experts and supported by the community. Physical tracker allowing to track symptoms together with the Data Dashboard allow users to explore data patterns and make informed decisions. The system gives also an opportunity to get in touch with experts via calls or structured long-term programs. Beside the design proposal project was closed with a series of reflection touching on the bias in a medical field and how it might affect healthcare design, as well as designer’s experience of working on a project with a personal relevance.

In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

This paper presents the findings of part of a research project entitled "Population Mobility and Settlement Patterns in the Eastern Cape, 1950 to 1990", which was funded by the Human Sciences Research Council. The part of the project with which this paper is concerned, is the study of resettlement in the Border/Ciskei area of the (new) Eastern Cape Province. It involves two main foci: a) the Whittlesea district of the former Ciskei, where research was done in the resettlement area of Sada (where findings are compared with research done there in 1981) and Dongwe; and b) the Fort Beaufort area, where we looked at the two 'black spot' communities of Upisdraai and Gqugesi which were uprooted and moved to the Fort Beaufort township of Bhofolo in the 1960s, and at the establishment of black citrus farmers in the Kat River Valley in the late 1980s, on previously White owned farms which were bought out by the (then) Ciskei government. In the Conclusion, some important differences are suggested between resettlement in the Eastern Cape and in QwaQwa, one of the areas of South Africa that has been most severely affected by resettlement. Ways in which the South African material may be seen in terms of prevailing models for the analysis of resettlement, and may provide an input for the modification of these approaches, are briefly considered.
Digitised by Rhodes University Library on behalf of the Institute of Social and Economic Research (ISER)

In conventional wireless channel models, there is no control on the gains of different subchannels. In such channels, the transmitted signal undergoes attenuation and phase shift and is subject to multi-path propagation effects. We herein refer to such channels as passive channels. In this dissertation, we study the problem of joint power allocation and channel design for a parallel channel which conveys information from a source to a destination through multiple orthogonal subchannels. In such a link, the power over each subchannel can be adjusted not only at the source but also at each subchannel. We refer to this link as an active parallel channel. For such a channel, we study the problem of sum-rate maximization under the assumption that the source power as well as the energy of the active channel are constrained. This problem is investigated for equal and unequal noise power at different subchannels. For equal noise power over different subchannels, although the sum-rate maximization problem is not convex, we propose a closed-form solution to this maximization problem. An interesting aspect of this solution is that it requires only a subset of the subchannels to be active and the remaining subchannels should be switched off. This is in contrast with passive parallel channels with equal subchannel signal-tonoise- ratios (SNRs), where water-filling solution to the sum-rate maximization under a total source power constraint leads to an equal power allocation among all subchannels. Furthermore, we prove that the number of active channels depends on the product of the source and channel powers. We also prove that if the total power available to the source and to the channel is limited, then in order to maximize the sum-rate via optimal power allocation to the source and to the active channel, half viii ix of the total available power should be allocated to the source and the remaining half should be allocated to the active channel. We extend our analysis to the case where the noise powers are unequal over different subchannels. we show that the sum-rate maximization problem is not convex. Nevertheless, with the aid of Karush-Kuhn-Tucker (KKT) conditions, we propose a computationally efficient algorithm for optimal source and channel power allocation. To this end, first, we obtain the feasible number of active subchannels. Then, we show that the optimal solution can be obtained by comparing a finite number of points in the feasible set and by choosing the best point which yields the best sum-rate performance. The worst-case computational complexity of this solution is linear in terms of number of subchannels.
UOIT

碩士
淡江大學
管理科學研究所碩士班
99
This research applies bi-level programming to make a decision of the subsidy to recycling industries for waste printers in Taiwan. The upper-level decision unit is design for the social welfare system; the lower-level unit is modeled as the recycling industries. The different and conflict of these two objectives can be simulated by the bi-level programming model. This research solves the model through KKT conditions and 0-1 variable transformation, fuzzy approach and intuitionistic fuzzy approach, and compares the results of these approaches with current subsidy for obtaining a better managerial control. 　　Recycling Fund Management Board (RFMB), Environment Protection Administration of ROC Government was founded being responsible for resource recycling. The RFMB levy the fee from responsible manufactures or importers, and give the recycling and treatment fee to the recycling industries. The recycling industries also give the incentives to the populace who come to recycle their wastes. The upper-level, it aims to increase the recycling rate, reduce the recycling and treatment fees of waste printers, and reduce the administrative cost. The goal of the lower-level is to increase their profits. For comparing the managerial control, this research applies KKT conditions and 0-1 variable transformation which are upper-level model; the fuzzy approach and intuitionistic fuzzy approach which is lower-level model. 　　As a result, the profits of recycling industries arise when the resources value increase, and the profit are depending on the recycling rate. Besides, the correlation between the incentives and recycling rate are positive. Higher incentives get higher recycling rate, then the profit of recycling industries increased and the environment pollution can be reduced. Thus, this research can improve recycling rate and the circumstance of social welfare system. Lastly, 0-1 programming has larger value for the recycling and treatment fee, but the incentives are lower than fuzzy approach and intuitionistic fuzzy approach. It causes the lower recycling industries profit and recycling rate. The results can be an option for setting the subsidy on waste printers for RFMB.

This dissertation studies several aspects of the interface between information theory and signal processing. Several new and existing results in information theory are researched from the perspective of signal processing. Similarly, some fundamental results in signal processing and statistics are studied from the information theoretic viewpoint. The first part of this dissertation focuses on illustrating the equivalence between Stein's identity and De Bruijn's identity, and providing two extensions of De Bruijn's identity. First, it is shown that Stein's identity is equivalent to De Bruijn's identity in additive noise channels with specific conditions. Second, for arbitrary but fixed input and noise distributions, and an additive noise channel model, the first derivative of the differential entropy is expressed as a function of the posterior mean, and the second derivative of the differential entropy is expressed in terms of a function of Fisher information. Several applications over a number of fields, such as statistical estimation theory, signal processing and information theory, are presented to support the usefulness of the results developed in Section 2. The second part of this dissertation focuses on three contributions. First, a connection between the result, proposed by Stoica and Babu, and the recent information theoretic results, the worst additive noise lemma and the isoperimetric inequality for entropies, is illustrated. Second, information theoretic and estimation theoretic justifications for the fact that the Gaussian assumption leads to the largest Cramer-Rao lower bound (CRLB) is presented. Third, a slight extension of this result to the more general framework of correlated observations is shown. The third part of this dissertation concentrates on deriving an alternative proof for an extremal entropy inequality (EEI), originally proposed by Liu and Viswanath. Compared with the proofs, presented by Liu and Viswanath, the proposed alternative proof is simpler, more direct, and more information-theoretic. An additional application for the extremal inequality is also provided. Moreover, this section illustrates not only the usefulness of the EEI but also a novel method to approach applications such as the capacity of the vector Gaussian broadcast channel, the lower bound of the achievable rate for distributed source coding with a single quadratic distortion constraint, and the secrecy capacity of the Gaussian wire-tap channel. Finally, a unifying variational and novel approach for proving fundamental information theoretic inequalities is proposed. Fundamental information theory results such as the maximization of differential entropy, minimization of Fisher information (Cramer-Rao inequality), worst additive noise lemma, entropy power inequality (EPI), and EEI are interpreted as functional problems and proved within the framework of calculus of variations. Several extensions and applications of the proposed results are briefly mentioned.

博士
國立清華大學
材料科學工程學系
98
A novel and disposable microchip (named as K-kit) with electron-transparent SiO2 nano-membranes was developed using microelectromechanical system techniques and used as a specimen kit for in situ imaging of living organisms in an aqueous condition by transmission electron microscopy (TEM) without equipment modification. The resolving power of living bacterial cells negatively stained with phosphotungstic acid was theoretically calculated to be 2.2 nm in K-kit at incident current density of 30 pA/cm2. Experimentally, this K-kit which can enclose aqueous specimens enabled the successful TEM observation of living Escherichia coli cells, the 8-18 nm type 3 fimbriae of living Klebsiella pneumoniae, and the tellurite reduction process in K. pneumoniae in-situ. The survival ratio of K. pneumoniae sealed in the K-kit for 12 h exceeded 80% before TEM imaging. Besides, the viability of bacterial cells sealed in the K-kits during TEM electron irradiation was examined. The K. pneumoniae (gram-negative bacteria) and Saccharomyces cerevisiae (yeast cells) can stay alive in K-kit after continuous TEM imaging for up to 14 s and 42 s, respectively. Utilizing the measurement by electron energy-loss spectroscopy, the total electron energy of 329.1 eV nm-3 and 405.3 eV nm-3 dissipated in K. pneumoniae　and S. cerevisiae were calculated, respectively. By the comparison of their surface structures, S. cerevisiae cells exhibit more dense-packed and covalently linked cell walls than the loosely attached capsular polysaccharides on the outer membrane of K. pneumoniae. It is possible that K. pneumoniae is more susceptible to electron beam irradiation compared to S. cerevisiae. Furthermore, the bio-reaction, tellurite reduction in K. pneumonia, was in situ monitored by TEM for 12.5 h through the use of the K-kit. The different tellurite reduction profiles in cells grown in aerobic and anaerobic environments can be also clearly revealed. These results demonstrate that the K-kit developed in this study can be useful for observing living organisms and in situ monitoring bio-reaction.

In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.

English with Englsih, Northern Sotho and Tsonga summaries
Since the inception of the homestead food garden programme in Gauteng Province, more than 3 000 individuals in the City of Tshwane Region 7 have benefited from the programme. It is not known whether the gardens were actually established after beneficiaries had received the tools from government, or whether the gardens still exist and produce food. The aim of the study was to investigate the contribution of the homestead food garden programme to household food security in Region 7 in the City of Tshwane Metropolitan Municipality. The study population consisted of beneficiaries of the homestead food garden programme from 2013 to 2016 in various wards in the region. The objectives of the study were to determine the socio-demographic characteristics of the beneficiaries; to assess the current status of the homestead food gardens and other support programmes in which the beneficiaries were engaged; to assess the contribution of the homestead food garden programme to household food security (availability, accessibility, utilisation and stability); to ascertain the factors influencing food availability in the households of beneficiaries; and to identify the constraints and benefits of homestead food production. The study used a quantitative research approach involving a survey design. A semistructured questionnaire was used for data collection through face-to-face interviews in the homes of participants. The random sampling technique was employed to acquire a proportionate sample of 258 participants. Data were analysed using the Statistical Package for Social Sciences (SPSS) version 24. The analysis of quantitative data included frequencies, percentages, mean and standard deviation, standard error of mean, a binomial test and the Ordered Logistics Regression (OLR) model. Themes and codes were used to analyse data from open-ended questions (qualitative data) and convert them to frequencies and percentages. The results showed that 73% of women participated and96% participants were Black Africans. Youth participation was 19%. The participants who could read and write because they had received primary, secondary and tertiary education were 84%. The average family size of the respondents was five, ranging between 1 and 47 household members. 83% of the participants relied on social grants (child, old-age and disability grants) from government as their main source of income. 83.3% participants had backyard (homestead) gardens for their households and 26% of participants consumed vegetables from their homestead food gardens. The household expenditure of 74% of participants included the cost of vegetables, which they had to buy regularly. The food availability results showed that 51% and 50% of the participants cultivated their backyard gardens during summer and spring respectively. This implied that there were adequate vegetables in the households of most participants during those seasons. It was discovered that 60% of the participants experienced anxiety and uncertainty because they were worried over the previous four weeks that their households would not have enough vegetables. The coping strategy mostly adopted by 69% of participants was to purchase cheap vegetables when there was lack of vegetables from the gardens. 55% participants consumed vegetables frequently because they either had enough harvests from their backyard gardens or had money to buy vegetables. The factors that positively influenced vegetable availability were participants’ age; the number of support programmes available to them; the existence of backyard gardens; and access to water. Some of the major challenges that systematically hindered homestead food gardens were insects; pests and diseases; crop damage by livestock, birds and rats; a lack of water for irrigation; a lack of production inputs (seeds, chemicals, fertilisers and others); and insufficient space to plant or establish a garden. The study found that the major benefits of the homestead food gardens were the consumption of fresh, healthy vegetables from the backyard garden and saving money by not having to buy vegetables. Based on the results of the current study, it is suggested that youth participation should be bolstered to ensure the future of homestead food gardens as the majority of the participants were older than 35 years. It is also suggested that other support measures should be taken to ensure the continuity of the homestead food garden programme, such as training programmes and access to water, fencing and production inputs (seeds and fertilisers) for homestead food gardeners.
Go tloga mola go thongwago lenaneo la dirapana tša dijo la motsana ka Profenseng ya Gauteng, batho ba go feta ba 3 000 ka Seleteng sa 7 sa Tshwane ba holegile ka go fiwa dithulusi tša serapana, seabe sa tšweletšo le tlhahlo go thoma dirapana tša ka mafuri tša malapa a bona. Ga go tsebege ge eba dirapana dihlomilwe ka nnete morago ga gore baholegi ba amogele dithulusi go tšwa mmušong, goba ge eba dirapana di sa le gona le gore di tšweletša dijo. Dinepo tša dinyakišišo e be e le go nyakišiša seabe sa lenaneo la dirapana tša dijo la motsana go tšhireletšo ya dijo tša lapa Seleteng sa 7 ka Mmasepaleng wa Metropothene wa Toropo ya Tshwane. Bakgathema ba dinyakišišo ba bopilwe ke baholegi ba lenaneo la serapana sa dijo sa motsana go tloga ka 2013 go fihla ka 2016 ka diwateng tša go fapana tša selete. Maikemišetšo a dinyakišišo e be e le laetša dipharologantšho tša batho le leago tša baholegi; go sekaseka maemo a bjale a dirapana tša dijo tša motsana le mananeo a mangwe a thekgo ao baholegi ba bego ba le karolo ya ona; go sekaseka seabe sa lenaneo la serapana sa dijo sa motsana go tšhireletšo ya dijo (khwetšagalo, phihlelelo, tšhomišo le tielelo); go laetša mabaka a go huetša khwetšagalo ya dijo ka malapeng a baholegi; le go hlaola mathata le dikholego tša tšweletšo ya dijo tša motsana. Dinyakišišo di šomišitše mokgwa wa dinyakišišo tša khwalithethifi wa go akaretša popo ya lenaneopotšišo. Lenaneopotšišo le beakantšwego seripa le go šomišetšwa kgoboketšo ya datha ka dipoledišano tša go dirwa go lebelelanwe mahlong ka malapeng a bakgathatema. Thekniki ya tiro ya sampole ya go se kgethe e šomišeditšwe go hwetša sampole ya go lekanetšwa ya bakgathema ba 258. Datha di sekasekilwe ka go šomiša tlhalošo ya 4 ya Statistical Package for Social Sciences (SPSS). Tshekatsheko ya datha ya khwalithethifi e akareditše dikelo, diphesente, palogare, phapano ya tekanetšo, phošo ya tekanetšo ya palogare, teko ya phapano le mmotlolo wa ponelopele ya diphetogo. Dikgwekgwe le Dikhoute di šomišitšwe go sekaseka datha go tšwa go dipotšišo tše di bulegilego (datha ya khwalithethifi) le go di fetolela go dikelo le diphesente. Dipoelo di laeditše gore phesente ya godimo (73%) ya basadi e kgathile tema. Bontši (96%) e bile baarabi e bile Maafrika a Bathobaso. Kgathotema ya baswa e be e le ka fase ga 19%. Bontši (84%) bja baholegi bo be bo kgona go bala le go ngwala gobane ba amogetše thuto ya poraemari, sekontari le ya godimo. Bogolo bja lapa bja palogare ya baarabi e be e le hlano, go tloga gare ga tee le 47 ya maloko a lapa. Karolo ye kgolo (83%) ya baarabi ba tshephile dithušo tša leago (dithušo tša bana, botšofadi le bogolofadi) go tšwa mmušong bjalo ka mothopo wa bona wo mogolo wa letseno gomme bontši (83.3%) bo be bo na le dirapana (motsana) tša ka mafuri tša malapa a bona. Go tloga go ponego ya tšhireletšo ya dijo, dipoelo di laeditše gore 26% ya baarabi ba ja merogo ya go tšwa ka dirapaneng tša dijo tša motsana wa bona. Ditshenyegalelo tša lapa tša bontši (74%) bja baarabi di akareditše tshenyegelo ya merogo, yeo ba bego ba swanela ke go e reka kgafetša. Dipoelo tša khwetšagalo ya dijo e laeditše gore 51% le 50% ya baarabi ba lemile dirapana tša ka mafuri a bona nakong ya selemo le seruthwane ka tatelano, seo se rago gore go bile le merogo ye e lekanego la malapeng a baarabi ba bantši dihleng tšeo. Ka go realo go utulotšwe gore, mabapi le phihlelelo ya dijo, bontši (60%) bja baarabi ba itemogetše tlalelo le pelaelo bjalo ka ge ba laeditše gore ba be ba hlobaela mo dibekeng tše nne tša go feta gore malapa a bona a ka se be le merogo ye e lekanego. Leano la go laola le le amogetšwego gagolo ke bontši (69%) bja baarabi go kgonthišiša tielelo ya dijo e be e le go reka merogo ka theko ya fase ge go sena merogo go tšwa ka dirapaneng. Mabapi le tielelo ya dijo, bontši (55%) bja bakgathatema ba be ba e ja merogo kgafetša gobane ba bunne tše di lekanego go tšwa dirapaneng tša ka mafuri a bona goba ba bile le tšhelete ya reka merogo. Mabaka ao a hueditšego khwetšagalo ya merogo gabotse e bile mengwaga ya baarabi; palo ya mananeo a thekgo ao ba ka a fihlelelago; go ba gona ga dirapana tša ka mafuri; le phihlelelo ya meetse. Tše dingwe tša ditlhohlo tše kgolo tšeo di tshwentšego ka botlalo dirapana tša dijo ka motsaneng e bile dikhunkhwane; disenyi le malwetši, tshenyo ya dibjalo ka leruo, dinonyana le magotlo; tlhokego ya meetse a go nošetša; tlhokego ya ditshepetšo tša tšweletšo (dipeu, dikhemikale, manyora le tše dingwe); le sekgoba se lekanego go bjala le go hloma serapana. Dinyakišišo di hweditše gore dikholego tše kgolo tša dirapana tša dijo tša motsana e be e le go ja merogo ye meswa, ye mebotse go tšwa ka serapaneng sa ka mafuri le go boloka tšhelete ka go se reke merogo. Go ya ka dipoelo tša dinyakišišo tša bjale, go šišintšwe gore kgathotema ya baswa e swanetše go thekgwa go kgonthišiša bokamoso bja dirapana tša dijo tša motsana ka ge bontši bja bakgathatema ba be ba le bogolo bja mengwaga ya ka godimo ga 35. Gape go šišintšwe gore dikelo tša thekgo tše dingwe di swanela go tšewa go kgonthišiša tšwelopele ya lenaneo la serapana sa dijo sa motsana, go swana le mananeo a tlhahlo le phihlelelo ya meetse, legora le production inputs (dipeu le manyora) ya boradirapana tša dijo tša motsana.
Ku sukela loko ku sunguriwe nongonoko wa swirhapa swa swakudya eka Xifundzhankulu xa Gauteng, vanhu vo tlula nhlayo ya 3 000 eTshwane eka Region 7 va vuyeriwile hi ku pfunetiwa hi mathulusi ya swirhapa, swipfuneto swo rima na vuleteri ku sungula swirhapa lomu majaratini ya miti ya vona. A swi tiveki loko swirhapa swi sunguriwe endzhaku ka loko vavuyeriwa va amukerile mathulusi eka mfumo, kumbe leswo xana swirhapa leswi swa ha ya emahlweni no rima swakudya. Xikongomelo xa ndzavisiso lowu wa tidyondzo a ku ri ku endla vulavisisi hi ndlela leyi nongonoko wa swirhapa swa swakudya wu pfunetaku hi yona eka ku sirheleleka hi swakudya eka Region 7 eka Masipala wa Dorobankulu ra City of Tshwane Metropolitan Municipality. Vanhu lava a ku dyondziwa hi vona a va katsa vavuyeriwa va nongonoko wa swirhapa swa swakudya lomu makaya ku sukela hi lembe ra 2013 ku fikela hi 2016 eka tiwadi to hlayanyana ta region. Swikongomelo swa ndzavisiso lowu i ku vona muxaka wa vanhu lava nga vuyeriwa; ku kambela xiyimo xa sweswi xa swirhapa swa swakudya na minongonoko yin'wana ya nseketelo laha vavuyeriwa va nga na xiavo eka yona; ku kambela xiavo xa nongonoko wa swirhapa swa swakudya eka ku sirheleleka hi ku kuma swakudya eka mindyangu (ku kumeka, ku fikelela, ku tirhisa, na ku tshamiseka); ku kumisisa mintiyiso kumbe swilo leswi swi nga na nkucetelo eka ku kumeka ka swakudya lomu makaya ya lava vuyeriwaka; na ku vona leswi swi nga swihingakanyi eka mimbuyelo ya ku rima swakudya. Ndzavisiso wu tirhise fambiselo leri vuriwaka quantitative research ku nga ku hlayela leswi a swi katsa dizayini ya survey. Ku tirhisiwe endlelo ra nongonoko wa swivutiso leswi nga tsariwa ku nga semi-structured questionnare ku hlengeleta data hi ku endla ti-interview ta xikandza-na-xikandza emakaya ya lava a va teka xiavo. Ku tirhisiwe thekniki ya random sampling ku kuma mpimanyeto wa proporionate sample eka vanhu va 258 lava a va teka xiavo eka ndzavisiso. Data yi ve yi hlahluviwa hi ku tirhisa Statistical Package for Social Sciences (SPSS) version 24. Nhlahluvo wa quantitative data a wu katsa vungani bya ku humelela ku nga frequences, tiphesente, mean, standard deviation, standard error of mean, binomial test na modlele wa Ordered Logistics Regresion (OLR). Eku hlahluveni ku tirhisiwe tithemes na tikhodi ku kambela data eka swivutiso leswi a swi ri na tinhlamulo to hambana-hambana ku nga open-ended questions (qualitative data) na ku swi hundzuluxa ku va swikombisa leswo swi endleke kangani (frequencies) na tiphesente ta ku endleka. Vuyelo byi kombise tiphesente ta le henhla eka vamanana hi (73%) lava a va ri na xiavo. Vunyingi bya vanhu lava a va hlamula swivutiso a ku ri Vanhu va Vantima va Ma-Afrika. Xiavo xa lavantshwa a xi ri ehansi ka 19%. Vunyingi bya vavuyeriwa (84%) a va kota ku hlaya no tsala hikuva a va kumile dyondzo ya prayimari, sekondari na ya le tikholichi na tiyunivhesiti. Mpimanyeto wa nhlayo ya vanhu va mindyangu a ku ri vanhu va ntlhanu, ku sukela eka wun'we ku fikela eka swirho swa 47 eka ndyangu. Ntsengo wa le henhla wo ringana (83%) wa vahlamula swivutiso i vanhu lava va hanyaka hi mpfuneto wa mali ya mfumo eka vanhu (ku nga xipfuneto xo wundla vana, mudende wa vadyuhari, na mpfuneto eka vo lamala) ku suka eka mfumo tani hi mali leyi nghenaka leyi va tshembeleke eka yona ngopfu kasi vunyingi bya vona (83.3%) va na swirhapa (lomu makaya) emitini ya vona. Mayelana na ku sirheleleka hi swakudya, vuyelo byi kombise leswo 26% wa vahlamuri va swivutiso va dya miroho yo huma eka swirhapa swa swakudya swa vona. Mpimanyeto wa mali lowu tirhisiwaka hi mindyangyu yo tala ku nga (74%) wa vahlamuri va swivutiso va katse ntsengo wa nxavo wa miroho, leyi va yi tirhisaka ku xava hi mikarhi na mikarhi. Vuyelo bya ku kumeka ka swakudya wu kombe leswo 51% wa vahlamuri va swivutiso va rima swakudya swa vona lomu swirhapeni swa le makaya ya vona hi nkarhi wa ximumu na hi nkarhi wa ximun'wana, leswi swu vulaka leswo a ku ri na miroho yo ringanela eka miti eka vahlamuri votala va swivutiso hi mikarhi ya tisizini leti. Ku ve ku kumeka leswaku mayelana na ku kumeka ka swakudya, vunyingi lebyi nga (60%) wa vahlamuri va swivutiso va karhateka na ku va va nga ri na ku tiyiseka hikuva va kombise leswo va xaniseka eka mavhiki ya mune lama nga hundza leswo mindyangu ya vona yi nga ka yi nga vi na miroho yo ringanela. Leswi vunyingi byi swi endleka ku ringeta ku hanya eka xiyimo lexi, vunyingi byi nga (69%) wa vahlamuri va swivutiso va vule leswo va xava miroho ya ntsengo wa nxavo wa le hansi loko ku nga ri na miroho eswirhapeni swa vona. Mayelana na ku tshamiseka hi swakudya, vunyingi bya vahlamuri va swivutiso (55%) va dya miroho hakanyingi hikuva va ri na ntshovelo wo ringanela eka swirhapa swa miroho ya vona kumbe hikuva va ri na mali yo xava miroho. Swilo leswi nga na nkucetelo lowunene hi ku kumeka ka miroho, ku ve malembe ya vukulu bya vahlamuri va swivutiso; nhlayo ya minongonoko ya nseketelo eka vona; vukona bya swirhapa swa miroho lomu makaya; na ku kumeka ka mati. Swin'wana swa mintlhontlho leyikulu leyi nga nkavanyeto eka swirhapha swa swakudya i switsotswana na vuvabyi ya swimila; ku onhiwa ka swibyariwa hi swifuwo; swinyenyane na makondlo; ku pfumaleka ka mati yo cheleta; ku pfumaleka ka swipfuneto swo rima (ku nga timbewu, tikhemikali, swinonisi na swin'wana swo tano); ku ka vanhu va nga ri na ndhawu yo ringanela ku byala kumbe ku endla swirhapa. Ndzavisiso wa dyondo wu kume leswo vunyingi bya vavuyeriwa eka swirhapa swa swakudya ku ve ku kota ku va na miroho ya furexe, miroho leyi nga na rihanyu eka swirhapa swa vona lomu makaya no hlayisa mali hikuva va nga xavi miroho. Hi ku landza vuyelo bya ndzavisiso wa dyondo wa sweswi, ku ringanyetiwa leswaku ku fanele ku khutaziwa ku va na xiavo ka vantshwa ku tiyisa vumundzuku bya swirhapa swa miroho, hikuva vunyingi bya lava va nga na xiavo a ku ri vanhu va malembe ya le henhla ka 35 hi vukulu. Ku pimanyetiwa na leswo ku fanele ku va na tindlela tin'wana to seketela ku tiyisa leswo nongonoko wa swirhapa swa swakudya wu ya emahlweni, ku endliwa ka swilo swo fana na minongonoko ya vuleteri, ku biyela swirhapa hi mifensi na swipfuneto swo byala (swo fana na timbewu na swinonisi eka swirhapa swa swakudya lomu makaya.
Agriculture and  Animal Health
M. Sc. (Agriculture)