Text in Afikaans / In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. / Mathematical Sciences / D. Phil. (Operasionele Navorsing)

Text in Afikaans / In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. / Mathematical Sciences / D. Phil. (Operasionele Navorsing)

<p>Set-based methods have been leveraged in many engineering applications from robust control and global optimization, to probabilistic planning and estimation. While useful, these methods have most widely been applied to analysis over sets that are convex, due to their ease in both representation and calculation. The representation and analysis of nonconvex sets is inherently complex. When nonconvexity arises in design and control applications, the nonconvex set is often over-approximated by a convex set to provide conservative results. However, the level of conservatism may be large and difficult to quantify, often leading to trivial results and requiring repetitive analysis by the engineer. Nonconvexity is inherent and unavoidable in many applications, such as the analysis of hybrid systems and robust safety constraints. </p> <p>In this dissertation, I present a new nonconvex set representation named the hybrid zonotope. The hybrid zonotope builds upon a combination of recent advances in the compact representation of convex sets in the controls literature with methods leveraged in solving mixed-integer programming problems. It is shown that the hybrid zonotope is equivalent to the union of an exponential number of convex sets while using a linear number of continuous and binary variables in the set’s representation. I provide identities for, and derivations of, the set operations of hybrid zonotopes for linear mappings, Minkowski sums, generalized intersections, halfspace intersections, Cartesian products, unions, complements, point containment, set containment, support functions, and convex enclosures. I also provide methods for redundancy removal and order reduction to improve the compactness and computational efficiency of the represented sets. Therefore proving the hybrid zonotopes expressive power and applicability to many nonconvex set-theoretic methods. Beyond basic set operations, I specifically show how the exact forward and backward reachable sets of linear hybrid systems may be found using identities that are calculated algebraically and scale linearly. Numerical examples show the scalability of the proposed methods and how they may be used to verify the safety and performance of complex systems. These exact methods may also be used to evaluate the level of conservatism of the existing approximate methods provided in the literature.  </p>

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Abstract: This work is concerned with the development and the implementation of a global optimization method for solving nonlinear nonconvex problems with continuous or mixed integer variables, related to power systems analysis. The proposed method relaxes the problem under study into a linear outer approximation problem by using the concept of special ordered sets. The obtained problem is then successively refined by a branch-and-bound strategy. In this way, the convergence to a global optimum is guaranteed, provided the discrete variables or those appearing nonlinearly in the original problem are bounded. Our method, conceived to solve a specific kind of problem, has been developed in a general framework in such a way that it can be easily extended to solve a large class of problems. We first derive the method theoretically and next present numerical results, fixing some choices inherent to the method to make it as optimal as possible. / Résumé: Ce travail a pour objet la conception et l'implémentation d'une méthode d'optimisation globale pour la résolution de problèmes non linéaires et non convexes, continus ou avec variables mixtes (entières et continues), issus de l'analyse des réseaux électriques. La méthode proposée relâche le problème traité en un problème d'approximation externe linéaire en se basant sur le concept d ensembles spécialement ordonnés. Le problème obtenu est alors successivement raffiné grâce à une stratégie de branch-and-bound. La convergence vers un optimum global est ainsi assurée, pour autant que les variables discrètes ou apparaissant non linéairement dans le problème de départ soient bornées. Notre méthode, mise au point pour résoudre un type de problème bien particulier, a été conçue dans un cadre général permettant une extension aisée à la résolution d'une grande variété de problèmes. Nous développons tout d'abord la méthode théoriquement et présentons ensuite des résultats numériques dont le but est de fixer certains choix inhérents à la méthode afin de la rendre la plus optimale possible.

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La modélisation multi-échelle multi-physique de la solidification d'alliages métalliques demande de combiner des phénomènes à l'échelle macroscopique du produit et microscopiques à l'échelle des structures de solidification. Dans cette thèse, l'empilement aléatoire des grains équiaxes avec des morphologies typiques de solidification est étudié. Nous mettons tout d'abord en évidence les paramètres hydrodynamiques adimensionnels qui régissent l'empilement de grains équiaxes : le nombre de Stokes, St, le nombre d'Archimède, Ar, et le rapport entre le temps caractéristique de la croissance et le temps caractéristique du mouvement, Γ. Un dispositif expérimental a été conçu par similitude hydrodynamique avec le phénomène réel de l'empilement de la solidification afin d'étudier l'influence de la géométrie des grains équiaxes et l'influence des conditions hydrodynamiques sur la fraction d'empilement. En outre, un outil numérique basé sur le méthode des éléments discrets a été développé pour compléter le travail expérimental de détermination de : la fraction d'empilement locale, le nombre de particules voisines en contact et l'orientation des particules. Des fractions d'empilement entre environ 0,53 et 0,67 ont été mesurées et calculées pour les grains sphériques non-cohésifs, alors que des valeurs allant jusqu'à environ 0,30 sont trouvées pour les grains dendritiques non-cohésifs. Enfin, nous étudions la dynamique de l'empilement, qui est la transition d'un régime de sédimentation à l'équilibre mécanique. L'évolution des variables comme la fraction locale de solide, le nombre de particules voisines en contact et l'orientation du grain en fonction du temps est présentée / Solidification multiphase multiscale modeling of metal alloys is based on the combination of the phenomena at the macroscopic scale of the product and at the microscopic scale of the solidification structures. In this thesis, the random packing of the typical equiaxed grain morphologies in metal alloy solidification is investigated. Firstly, we highlight the hydrodynamic dimensionless parameters governing the grain packing in the melt: the Stokes number, St, the Archimedes number, Ar, and the growth-to-motion ratio, Γ. Subsequently, an experimental setup is designed by hydrodynamic similarity with the actual solidification packing phenomenon in order to investigate the influence of the equiaxed grain geometry and the hydrodynamic conditions on the average solid packing fraction. Additionally, a numerical Discrete Element Method tool is developed to complement the experimental work by accessing to those granular variables which result difficult to be experimentally obtained such as the local packing fraction, the contacting neighbors and the particle orientation. Packing fractions between approximately 0.53 and 0.67 are measured and computed for the spherical noncohesive grains, for different hydrodynamic, frictional and polydispersity conditions, whereas values down to approximately 0.30 are found for noncohesive dendrite envelopes. Finally, we investigate the packing dynamics, which is the transition from a sedimentation regime to the mechanical equilibrium (packing). The evolution of the local solid fraction, contacting neighbors, mechanical contacts and grain orientation are given

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-06-16T17:52:10Z No. of bitstreams: 1 camiladefreitasfernandes.pdf: 740367 bytes, checksum: fac5ab7dcb039b31d587151b9a53fab1 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T14:25:13Z (GMT) No. of bitstreams: 1 camiladefreitasfernandes.pdf: 740367 bytes, checksum: fac5ab7dcb039b31d587151b9a53fab1 (MD5) / Made available in DSpace on 2016-07-13T14:25:13Z (GMT). No. of bitstreams: 1 camiladefreitasfernandes.pdf: 740367 bytes, checksum: fac5ab7dcb039b31d587151b9a53fab1 (MD5) Previous issue date: 2016-04-08 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho estudamos um algoritmo para solução de problemas de otimização irrestrita com funções não necessariamente convexas ou diferenciáveis, denominado Nonsmooth Feasible Direction Nonconvex Algorithm - NFDNA, e fazemos uma aplicação deste algoritmo que consistiu em utilizá-lo como subrotina de um outro algoritmo chamado Interior Epigraph Direction (IED) method. O IED, desenvolvido para resolver problemas de otimização não convexa, não diferenciável mas com restrições, utiliza Dualidade Lagrangeana que requer a minimização da função Lagrangeana. A eficiência do IED depende fortemente de tal minimização. Como aplicação, substituímos a rotina fminsearch do Matlab, utilizada originalmente pelo IED, pelo NFDNA. Mostramos através da solução de problemas teste que a performance do IED foi mais eficiente com a utilização do NFDNA. / In this work we study an algorithm for solving unsconstrained, not necessarily convex or differentiable optimization problems called Nonsmooth Feasible Direction Nonconvex Algorithm - NFDNA. We also employ this algorithm as a subroutine of the Interior Epigraph Directions (IED) method. The IED method, devised for solving constrained, nonconvex and nonsmooth optimization problems uses Lagrangean Duality which requires the minimization of the Lagrangean function. The effectiveness of the IED depends strongly on the Lagrangean function minimization. As an application, we replace the Matlab routine fminsearch, originally used by IED, with NFDNA. We show through the solution of test problems that the IED performance is more efficient by employing NFDNA.

Abstract This thesis examines radio resource management techniques for multicell multi-input single-output (MISO) downlink networks. Specifically, the thesis focuses on developing linear transmit beamforming techniques by optimizing certain quality-of-service (QoS) features, including, spectral efficiency, fairness, and throughput. The problem of weighted sum-rate-maximization (WSRMax) has been identified as a central problem to many network optimization methods, and it is known to be NP-hard. An algorithm based on a branch and bound (BB) technique which globally solves the WSRMax problem with an optimality certificate is proposed. Novel bounding techniques via conic optimization are introduced and their efficiency is illustrated by numerical simulations. The proposed BB based algorithm is not limited to WSRMax only; it can be easily extended to maximize any system performance metric that can be expressed as a Lipschitz continuous and increasing function of the signal-to-interference-plus-noise (SINR) ratio. Beamforming techniques can provide higher spectral efficiency, only when the channel state information (CSI) of users is accurately known. However, in practice the CSI is not perfect. By using an ellipsoidal uncertainty model for CSI errors, both optimal and suboptimal robust beamforming techniques for the worst-case WSRMax problem are proposed. The optimal method is based on a BB technique. The suboptimal algorithm is derived using alternating optimization and sequential convex programming. Through a numerical example it is also shown how the proposed algorithms can be applied to a scenario with statistical channel errors. Next two decentralized algorithms for multicell MISO networks are proposed. The optimization problems considered are: P1) minimization of the total transmission power subject to minimum SINR constraints of each user, and P2) SINR balancing subject to the total transmit power constraint of the base stations. Problem P1 is of great interest for obtaining a transmission strategy with minimal transmission power that can guarantee QoS for users. In a system where the power constraint is a strict system restriction, problem P2 is useful in providing fairness among the users. Decentralized algorithms for both problems are derived by using a consensus based alternating direction method of multipliers. Finally, the problem of spectrum sharing between two wireless operators in a dynamic MISO network environment is investigated. The notion of a two-person bargaining problem is used to model the spectrum sharing problem, and it is cast as a stochastic optimization. For this problem, both centralized and distributed dynamic resource allocation algorithms are proposed. The proposed distributed algorithm is more suitable for sharing the spectrum between the operators, as it requires a lower signaling overhead, compared with centralized one. Numerical results show that the proposed distributed algorithm achieves almost the same performance as the centralized one. / Tiivistelmä Tässä väitöskirjassa tarkastellaan monisoluisten laskevan siirtotien moniantennilähetystä käyttävien verkkojen radioresurssien hallintatekniikoita. Väitöskirjassa keskitytään erityisesti kehittämään lineaarisia siirron keilanmuodostustekniikoita optimoimalla tiettyjä palvelun laadun ominaisuuksia, kuten spektritehokkuutta, tasapuolisuutta ja välityskykyä. Painotetun summadatanopeuden maksimoinnin (WSRMax) ongelma on tunnistettu keskeiseksi monissa verkon optimointitavoissa ja sen tiedetään olevan NP-kova. Tässä työssä esitetään yleinen branch and bound (BB) -tekniikkaan perustuva algoritmi, joka ratkaisee WSRMax-ongelman globaalisti ja tuottaa todistuksen ratkaisun optimaalisuudesta. Samalla esitellään uusia conic-optimointia hyödyntäviä suorituskykyrajojen laskentatekniikoita, joiden tehokkuutta havainnollistetaan numeerisilla simuloinneilla. Ehdotettu BB-perusteinen algoritmi ei rajoitu pelkästään WSRMax-ongelmaan, vaan se voidaan helposti laajentaa maksimoimaan mikä tahansa järjestelmän suorituskykyarvo, joka voidaan ilmaista Lipschitz-jatkuvana ja signaali-(häiriö+kohina) -suhteen (SINR) kasvavana funktiona. Keilanmuodostustekniikat voivat tuottaa suuremman spektritehokkuuden vain, jos käyttäjien kanavien tilatiedot tiedetään tarkasti. Käytännössä kanavan tilatieto ei kuitenkaan ole täydellinen. Tässä väitöskirjassa ehdotetaan WSRMax-ongelman ääritapauksiin sekä optimaalinen että alioptimaalinen keilanmuodostustekniikka soveltaen tilatietovirheisiin ellipsoidista epävarmuusmallia. Optimaalinen tapa perustuu BB-tekniikkaan. Alioptimaalinen algoritmi johdetaan peräkkäistä konveksiohjelmointia käyttäen. Numeerisen esimerkin avulla näytetään, miten ehdotettuja algoritmeja voidaan soveltaa skenaarioon, jossa on tilastollisia kanavavirheitä. Seuraavaksi ehdotetaan kahta hajautettua algoritmia monisoluisiin moniantennilähetyksellä toimiviin verkkoihin. Tarkastelun kohteena olevat optimointiongelmat ovat: P1) lähetyksen kokonaistehon minimointi käyttäjäkohtaisten minimi-SINR-rajoitteiden mukaan ja P2) SINR:n tasapainottaminen tukiasemien kokonaislähetystehorajoitusten mukaisesti. Ongelma P1 on erittäin kiinnostava, kun pyritään kehittämään mahdollisimman pienen lähetystehon vaativa lähetysstrategia, joka pystyy takaamaan käyttäjien palvelun laadun. Ongelma P2 on hyödyllinen tiukasti tehorajoitetussa järjestelmässä, koska se tarjoaa tasapuolisuutta käyttäjien välillä. Molempien ongelmien hajautetut algoritmit johdetaan konsensusperusteisen vuorottelevan kertoimien suuntaustavan avulla. Lopuksi tarkastellaan kahden langattoman operaattorin välisen spektrinjaon ongelmaa dynaamisessa moniantennilähetystä käyttävässä verkkoympäristössä. Spektrinjako-ongelmaa mallinnetaan käyttämällä kahden osapuolen välistä neuvottelua stokastisen optimoinnin näkökulmasta. Tähän ongelmaan ehdotetaan ratkaisuksi sekä keskitettyä että hajautettua resurssien allokoinnin algoritmia. Hajautettu algoritmi sopii paremmin spektrin jakamiseen operaattorien välillä, koska se vaatii vähemmän kontrollisignalointia. Numeeriset tulokset osoittavat, että ehdotetulla hajautetulla algoritmilla saavutetaan lähes sama suorituskyky kuin keskitetyllä algoritmillakin.