Locating a radiating source based on range or range measurements obtained from a network of passive sensors has been a subject of research over the past two decades due to the problem’s importance in applications in wireless communications, surveillance, navigation, geosciences, and several other fields. In this thesis, we develop new solution methods for the problem of localizing a single radiating source based on range and range-difference measurements. Iterative re-weighting algorithms are developed for both range-based and range-difference-based least squares localization. Then we propose a penalty convex-concave procedure for finding an approximate solution to nonlinear least squares problems that are related to the range measurements. Finally, the sequential convex relaxation procedures are proposed to obtain the nonlinear least squares estimate of source coordinates. Localization in wireless sensor network, where the RF signals are used to derive the ranging measurements, is the primary application area of this work. However, the solution methods proposed are general and could be applied to range and range-difference measurements derived from other types of signals. / Graduate / 0544 / ismailds@uvic.ca

Les travaux présentés dans ce manuscrit portent sur la conception des lignes de désassemblageen contexte incertain. Une ligne de désassemblage consiste en unesuccession de postes de travail où les tâches sont exécutées séquentiellement au niveau de chaque poste. La conception d'un tel système, de revalorisationdes produits en fin de vie, peut être ramenée à un problème d'optimisation combinatoire.Ce dernier cherche à obtenir une configuration permettant d'optimiser certains objectifs enrespectant des contraintes techniques, économiques et écologiques.Dans un premier temps, nous décrivons les activités principales de la revalorisation des produitsen fin de vie, en particulier le désassemblage. Puis, après présentation des travaux de la littératureportant sur la prise en compte des incertitudes des durées opératoires lors de la conception des lignesde production, nous nous focalisons sur l'étude des incertitudes des durées opératoires des tâches de désassemblage.Ainsi, nous présentons trois modélisations principales avec leurs approches de résolution.La première s'intéresse à la minimisation des arrêts de la ligne causés par les incertitudes des durées des opérationsde désassemblage. La deuxième cherche à garantir un niveau opérationnel de la ligne lié à sa cadence de fonctionnement.Le but de la troisième modélisation est l'intégration des problématiques de conception des ligneset de séquencement des tâches de désassemblage. Enfin, les performances des méthodes de résolutionproposées sont présentées en analysant les résultats d'optimisation sur un ensemble d'instances de taille industrielle. / This thesis is dedicated to the problem of disassembly line design in uncertain context. A disassembly linecan be represented as a succession of workstations where tasks are performed sequentially at each workstation.The design of such a product recovery system can be reduced to a combinatorial optimization problem which seeksa line configuration that optimizes certain objectives under technical, economical and environmental constraints.We begin by describing the principal product recovery activities especially disassembly. Then, after a literaturereview on the design of production lines under uncertainty of task processing times, we focus our study on the consideration of the disassembly task time uncertainties. Hence, we present three main models as well as the associatedsolution approaches. The first one is interested in minimizing the line stoppages caused by the task processing timeuncertainties. The second one seeks to guarantee an operational level closely related with the line speed. The goal of thethird model is to integrate the line design and sequencing problems. At last, the performances of the proposed solutionapproaches are presented by analyzing the optimization results on a set of instances of industrial size.

Neste trabalho, realizamos uma extensão da chamada condição Aproximadamente Karush-Kuhn-Tucker (AKKT), inicialmente introduzida em programação não linear [AHM11], para os problemas de otimização sob cones simétricos não linear. Uma condição nova, a qual chamamos Trace AKKT (TAKKT), também foi apresentada para o problema de programação semidefinida não linear. TAKKT se mostrou mais prática que AKKT para programação semidefinida não linear. Provamos que, tanto a condição AKKT como a condição TAKKT são condições de otimalidade. Resultados de convergência global para o método de Lagrangiano aumentado foram obtidos. Condições de qualificação estritas foram introduzidas para medir a força dos resultados de convergência global apresentados. Através destas condições de qualificação estritas, foi pos- sível verificar que nossos resultados de convergência global se mostraram melhores do que os conhecidos na literatura. Também apresentamos uma prova para um caso particular da conjectura feita em [AMS07]. Palavras-chave: condições sequenciais de otimalidade, programação semidefinida não linear, programação sob cones simétricos não linear, condições de qualificação estritas. / In this work, we perform an extension of the so-called Approximate Karush-Kuhn-Tucker (AKKT) condition, initially introduced in nonlinear programming [AHM11], for nonlinear symmetric cone pro- gramming. A new condition, which we call Trace AKKT (TAKKT), was also presented for the nonlinear semidefinite programming problem. TAKKT proved to be more practical than AKKT for nonlinear semi- definite programming. We prove that both the AKKT condition and the TAKKT condition are optimality conditions. Results of global convergence for the augmented Lagrangian method were obtained. Strict qua- lification conditions were introduced to measure the strength of the overall convergence results presented. Through these strict qualification conditions, it was possible to verify that our results of global convergence proved to be better than those known in the literature. We also present a proof for a particular case of the conjecture made in [AMS07].

<div>Optimization of atmospheric flight control has long been performed on the ground, prior to mission flight due to large computational requirements used to solve non-linear programming problems. Onboard trajectory optimization enables the creation of new reference trajectories and updates to guidance coefficients in real time. This thesis summarizes the methods involved in solving optimal control problems in real time using convexification and Sequential Convex Programming (SCP). The following investigation provided insight in assessing the use of state of the art SCP optimization architectures and convexification of the hypersonic equations of motion[ 1 ]–[ 3 ] with different control schemes for the purposes of enabling on-board trajectory optimization capabilities.</div><div>An architecture was constructed to solve convexified optimal control problems using direct population of sparse matrices in triplet form and an embedded conic solver to enable rapid turn around of optimized trajectories. The results of this show that convexified optimal control problems can be solved quickly and efficiently which holds promise in autonomous trajectory design to better overcome unexpected environments and mission parameter changes. It was observed that angle of attack control problems can be successfully convexified and solved using SCP methods. However, the use of multiple coupled controls is not guaranteed to be successful with this method when they act in the same plane as one another. The results of this thesis demonstrate that state of the art SCP methods have the capacity to enable onboard trajectory optimization with both angle of attack control and bank angle control schemes.</div><div><br></div>

Dans cette thèse, nous étudions trois types de problèmes stochastiques : les problèmes avec contraintes probabilistes, les problèmes distributionnellement robustes et les problèmes avec recours. Les difficultés des problèmes stochastiques sont essentiellement liées aux problèmes de convexité du domaine des solutions, et du calcul de l’espérance mathématique ou des probabilités qui nécessitent le calcul complexe d’intégrales multiples. A cause de ces difficultés majeures, nous avons résolu les problèmes étudiées à l’aide d’approximations efficaces.Nous avons étudié deux types de problèmes stochastiques avec des contraintes en probabilités, i.e., les problèmes linéaires avec contraintes en probabilité jointes (LLPC) et les problèmes de maximisation de probabilités (MPP). Dans les deux cas, nous avons supposé que les variables aléatoires sont normalement distribués et les vecteurs lignes des matrices aléatoires sont indépendants. Nous avons résolu LLPC, qui est un problème généralement non convexe, à l’aide de deux approximations basée sur les problèmes coniques de second ordre (SOCP). Sous certaines hypothèses faibles, les solutions optimales des deux SOCP sont respectivement les bornes inférieures et supérieures du problème du départ. En ce qui concerne MPP, nous avons étudié une variante du problème du plus court chemin stochastique contraint (SRCSP) qui consiste à maximiser la probabilité de la contrainte de ressources. Pour résoudre ce problème, nous avons proposé un algorithme de Branch and Bound pour calculer la solution optimale. Comme la relaxation linéaire n’est pas convexe, nous avons proposé une approximation convexe efficace. Nous avons par la suite testé nos algorithmes pour tous les problèmes étudiés sur des instances aléatoires. Pour LLPC, notre approche est plus performante que celles de Bonferroni et de Jaganathan. Pour MPP, nos résultats numériques montrent que notre approche est là encore plus performante que l’approximation des contraintes probabilistes individuellement.La deuxième famille de problèmes étudiés est celle relative aux problèmes distributionnellement robustes où une partie seulement de l’information sur les variables aléatoires est connue à savoir les deux premiers moments. Nous avons montré que le problème de sac à dos stochastique (SKP) est un problème semi-défini positif (SDP) après relaxation SDP des contraintes binaires. Bien que ce résultat ne puisse être étendu au cas du problème multi-sac-à-dos (MKP), nous avons proposé deux approximations qui permettent d’obtenir des bornes de bonne qualité pour la plupart des instances testées. Nos résultats numériques montrent que nos approximations sont là encore plus performantes que celles basées sur les inégalités de Bonferroni et celles plus récentes de Zymler. Ces résultats ont aussi montré la robustesse des solutions obtenues face aux fluctuations des distributions de probabilités. Nous avons aussi étudié une variante du problème du plus court chemin stochastique. Nous avons prouvé que ce problème peut se ramener au problème de plus court chemin déterministe sous certaine hypothèses. Pour résoudre ce problème, nous avons proposé une méthode de B&B où les bornes inférieures sont calculées à l’aide de la méthode du gradient projeté stochastique. Des résultats numériques ont montré l’efficacité de notre approche. Enfin, l’ensemble des méthodes que nous avons proposées dans cette thèse peuvent s’appliquer à une large famille de problèmes d’optimisation stochastique avec variables entières. / In this thesis, we studied three types of stochastic problems: chance constrained problems, distributionally robust problems as well as the simple recourse problems. For the stochastic programming problems, there are two main difficulties. One is that feasible sets of stochastic problems is not convex in general. The other main challenge arises from the need to calculate conditional expectation or probability both of which are involving multi-dimensional integrations. Due to the two major difficulties, for all three studied problems, we solved them with approximation approaches.We first study two types of chance constrained problems: linear program with joint chance constraints problem (LPPC) as well as maximum probability problem (MPP). For both problems, we assume that the random matrix is normally distributed and its vector rows are independent. We first dealt with LPPC which is generally not convex. We approximate it with two second-order cone programming (SOCP) problems. Furthermore under mild conditions, the optimal values of the two SOCP problems are a lower and upper bounds of the original problem respectively. For the second problem, we studied a variant of stochastic resource constrained shortest path problem (called SRCSP for short), which is to maximize probability of resource constraints. To solve the problem, we proposed to use a branch-and-bound framework to come up with the optimal solution. As its corresponding linear relaxation is generally not convex, we give a convex approximation. Finally, numerical tests on the random instances were conducted for both problems. With respect to LPPC, the numerical results showed that the approach we proposed outperforms Bonferroni and Jagannathan approximations. While for the MPP, the numerical results on generated instances substantiated that the convex approximation outperforms the individual approximation method.Then we study a distributionally robust stochastic quadratic knapsack problems, where we only know part of information about the random variables, such as its first and second moments. We proved that the single knapsack problem (SKP) is a semedefinite problem (SDP) after applying the SDP relaxation scheme to the binary constraints. Despite the fact that it is not the case for the multidimensional knapsack problem (MKP), two good approximations of the relaxed version of the problem are provided which obtain upper and lower bounds that appear numerically close to each other for a range of problem instances. Our numerical experiments also indicated that our proposed lower bounding approximation outperforms the approximations that are based on Bonferroni's inequality and the work by Zymler et al.. Besides, an extensive set of experiments were conducted to illustrate how the conservativeness of the robust solutions does pay off in terms of ensuring the chance constraint is satisfied (or nearly satisfied) under a wide range of distribution fluctuations. Moreover, our approach can be applied to a large number of stochastic optimization problems with binary variables.Finally, a stochastic version of the shortest path problem is studied. We proved that in some cases the stochastic shortest path problem can be greatly simplified by reformulating it as the classic shortest path problem, which can be solved in polynomial time. To solve the general problem, we proposed to use a branch-and-bound framework to search the set of feasible paths. Lower bounds are obtained by solving the corresponding linear relaxation which in turn is done using a Stochastic Projected Gradient algorithm involving an active set method. Meanwhile, numerical examples were conducted to illustrate the effectiveness of the obtained algorithm. Concerning the resolution of the continuous relaxation, our Stochastic Projected Gradient algorithm clearly outperforms Matlab optimization toolbox on large graphs.

<p>In many practical multi-channel active sound control (ASC) applications, such as active noise control (ANC), various constraints need to be satisfied, such as the robust stability constraint, noise amplification constraint, controller output power constraints, etc. One way to enforce these constraints is to add a regularization term to the Wiener filter formulation, which, by tuning only a single parameter, can over-satisfy many constraints and degrade the ANC performance. Another approach for non-adaptive ANC filter design that can produce better ANC performance is to directly solve the constrained optimization problem formulated based on the <em>H</em>₂/<em>H</em>_inf control framework. However, such a formulation does not result in a convex optimization problem and its practicality can be limited by the significant computation time required in the solving process. In this dissertation, the traditional <em>H</em>₂/<em>H</em>_inf formulation is convexified and a global minimum is guaranteed. It is then further reformulated into a cone programming formulation and simplified by exploiting the problem structure in its dual form to obtain a more numerically efficient and stable formulation. A warmstarting strategy is also proposed to further reduce the required iterations. Results show that, compared with the traditional methods, the proposed method is more reliable and the computation time can be reduced from the order of days to seconds. When the acoustic feedback path is not strong enough to cause instability, then only constraints that prevent noise amplification outside the desired noise control band are needed. A singular vector filtering method is proposed to maintain satisfactory noise control performance in the desired noise reduction bands while mitigating noise amplification.</p> <p><br></p> <p>The proposed convex conic formulation can be used for a wide range of ASC applications. For example, the improvement in numerical efficiency and stability makes it possible to apply the proposed method to adaptive ANC filter design. Results also show that compared with the conventional constrained adaptive ANC method (leaky FxLMS), the proposed method can achieve a faster convergence rate and better steady-state noise control performance. The proposed conic method can also be used to design the room equalization filter for sound field reproduction and the hear-through filter design for earphones.</p> <p><br></p> <p>Besides efficient filter design methods, efficient filter implementation methods are also developed to reduce real-time computations in implementing designed control filters. A polyphase-structure-based filter design and implementation method is developed for ANC systems that can reduce the computation load for high sampling rate real-time filter implementation but does not introduce an additional time delay. Results show that, compared with various traditional low sampling rate implementations, the proposed method can significantly improve the noise control performance. Compared with the non-polyphase high-sampling rate method, the real-time computations that increase with the sampling rate are improved from quadratically to linearly. Another efficient filter implementation method is to use the infinite impulse response (IIR) filter structure instead of the finite impulse response (FIR) filter structure. A stable IIR filter design approach that does not need the computation and relocation of poles is improved to be applicable in the ANC applications. The result demonstrated that the proposed method can achieve better fitting accuracy and noise control performance in high-order applications.</p>