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Dissertations / Theses on the topic 'Nonlinear optimization'
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Skrobanski, Jerzy Jan. "Optimization subject to nonlinear constraints." Thesis, Imperial College London, 1986. http://hdl.handle.net/10044/1/7331.
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Strandberg, Mattias. "Portfolio Optimization with NonLinear Instruments." Thesis, Umeå universitet, Institutionen för fysik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-137233.
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Denton, Trip Shokoufandeh Ali. "Subset selection using nonlinear optimization /." Philadelphia, Pa. : Drexel University, 2007. http://hdl.handle.net/1860/1763.
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Robinson, Daniel P. "Primal-dual methods for nonlinear optimization." Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2007. http://wwwlib.umi.com/cr/ucsd/fullcit?p3274512.
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Thesis (Ph. D.)--University of California, San Diego, 2007.<br>Title from first page of PDF file (viewed October 4, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 173-175).
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Raj, Ashish. "Evolutionary Optimization Algorithms for Nonlinear Systems." DigitalCommons@USU, 2013. http://digitalcommons.usu.edu/etd/1520.
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Many real world problems in science and engineering can be treated as optimization problems with multiple objectives or criteria. The demand for fast and robust stochastic algorithms to cater to the optimization needs is very high. When the cost function for the problem is nonlinear and non-differentiable, direct search approaches are the methods of choice. Many such approaches use the greedy criterion, which is based on accepting the new parameter vector only if it reduces the value of the cost function. This could result in fast convergence, but also in misconvergence where it could lead the vectors to get trapped in local minima. Inherently, parallel search techniques have more exploratory power. These techniques discourage premature convergence and consequently, there are some candidate solution vectors which do not converge to the global minimum solution at any point of time. Rather, they constantly explore the whole search space for other possible solutions. In this thesis, we concentrate on benchmarking three popular algorithms: Real-valued Genetic Algorithm (RGA), Particle Swarm Optimization (PSO), and Differential Evolution (DE). The DE algorithm is found to out-perform the other algorithms in fast convergence and in attaining low-cost function values. The DE algorithm is selected and used to build a model for forecasting auroral oval boundaries during a solar storm event. This is compared against an established model by Feldstein and Starkov. As an extended study, the ability of the DE is further put into test in another example of a nonlinear system study, by using it to study and design phase-locked loop circuits. In particular, the algorithm is used to obtain circuit parameters when frequency steps are applied at the input at particular instances.
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Chryssochoos, Ioannis. "Optimization based control of nonlinear systems." Thesis, Imperial College London, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.399165.
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Wilson, Simon Paul. "Aircraft routing using nonlinear global optimization." Thesis, University of Hertfordshire, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275117.
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Soto, Jonathan. "Nonlinear contraction tools for constrained optimization." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/62538.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 77-78).<br>This thesis derives new results linking nonlinear contraction analysis, a recent stability theory for nonlinear systems, and constrained optimization theory. Although dynamic systems and optimization are both areas that have been extensively studied [21], few results have been achieved in this direction because strong enough tools for dynamic systems were not available. Contraction analysis provides the necessary mathematical background. Based on an operator that projects the speed of the system on the tangent space of the constraints, we derive generalizations of Lagrange parameters. After presenting some initial examples that show the relations between contraction and optimization, we derive a contraction theorem for nonlinear systems with equality constraints. The method is applied to examples in differential geometry and biological systems. A new physical interpretation of Lagrange parameters is provided. In the autonomous case, we derive a new algorithm to solve minimization problems. Next, we state a contraction theorem for nonlinear systems with inequality constraints. In the autonomous case, the algorithm solves minimization problems very fast compared to standard algorithms. Finally, we state another contraction theorem for nonlinear systems with time-varying equality constraints. A new generalization of time varying Lagrange parameters is given. In the autonomous case, we provide a solution for a new class of optimization problems, minimization with time-varying constraints.<br>by Jonathan Soto.<br>S.M.
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Prokopyev, Oleg A. "Nonlinear integer optimization and applications in biomedicine." [Gainesville, Fla.] : University of Florida, 2006. http://purl.fcla.edu/fcla/etd/UFE0015226.
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Zhang, Hongchao. "Gradient methods for large-scale nonlinear optimization." [Gainesville, Fla.] : University of Florida, 2006. http://purl.fcla.edu/fcla/etd/UFE0013703.
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Grothey, Andreas. "Decomposition methods for nonlinear nonconvex optimization problems." Thesis, University of Edinburgh, 2001. http://hdl.handle.net/1842/12065.
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The subject of this thesis is the development of ways to solve structured nonlinear nonconvex programming problems by a decomposition procedure. This thesis extends the existing decomposition methods for linear or convex problems to the nonconvex nonlinear case. The algorithms presented are in principle applicable to a general nonlinear problem, although in order to be efficient compared with a nondecomposed method a certain structure is highly advantageous. Two main ideas are explored. In the first augmented Lagrangians are employed to relax some key constraints of the subproblems, thus guaranteeing that they are feasible for all choices of complicating variables. The resulting formulation is then decomposed by a generalized Benders decomposition scheme, resulting in a three-level problem. As an alternative a more direct generalization of Benders decomposition is considered. The problem of infeasible subproblems is overcome here by using feasibility cuts that build up a local approximation of the (nonconvex) feasible region in the master problem. Apart from the issue of infeasible subproblems, there are various differences from the linear/convex case, which are addressed. The subproblem value functions are shown to be piecewise differentiable nonconvex functions, whose subgradients can in general be obtained as certain Lagrange multipliers at the solution of the subproblems. Efficient ways of obtaining first and second derivatives of the value function from the subproblems are derived. A bundle method is used to solve the master problems at the top and middle level of the decomposition. The bundle concept is extended to cope with nonconvex functions and to incorporate second order information of the value function as well as its subgradient. The resulting method is demonstrated to converge superlinearly. The proposed bundle method can also be used outside the decomposition framework to minimize a nonconvex nonsmooth function subject to smooth constraints.
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Marsden, Christopher J. "Nonlinear dynamics of pattern recognition and optimization." Thesis, Loughborough University, 2012. https://dspace.lboro.ac.uk/2134/10694.
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We associate learning in living systems with the shaping of the velocity vector field of a dynamical system in response to external, generally random, stimuli. We consider various approaches to implement a system that is able to adapt the whole vector field, rather than just parts of it - a drawback of the most common current learning systems: artificial neural networks. This leads us to propose the mathematical concept of self-shaping dynamical systems. To begin, there is an empty phase space with no attractors, and thus a zero velocity vector field. Upon receiving the random stimulus, the vector field deforms and eventually becomes smooth and deterministic, despite the random nature of the applied force, while the phase space develops various geometrical objects. We consider the simplest of these - gradient self-shaping systems, whose vector field is the gradient of some energy function, which under certain conditions develops into the multi-dimensional probability density distribution of the input. We explain how self-shaping systems are relevant to artificial neural networks. Firstly, we show that they can potentially perform pattern recognition tasks typically implemented by Hopfield neural networks, but without any supervision and on-line, and without developing spurious minima in the phase space. Secondly, they can reconstruct the probability density distribution of input signals, like probabilistic neural networks, but without the need for new training patterns to have to enter the network as new hardware units. We therefore regard self-shaping systems as a generalisation of the neural network concept, achieved by abandoning the "rigid units - flexible couplings'' paradigm and making the vector field fully flexible and amenable to external force. It is not clear how such systems could be implemented in hardware, and so this new concept presents an engineering challenge. It could also become an alternative paradigm for the modelling of both living and learning systems. Mathematically it is interesting to find how a self shaping system could develop non-trivial objects in the phase space such as periodic orbits or chaotic attractors. We investigate how a delayed vector field could form such objects. We show that this method produces chaos in a class systems which have very simple dynamics in the non-delayed case. We also demonstrate the coexistence of bounded and unbounded solutions dependent on the initial conditions and the value of the delay. Finally, we speculate about how such a method could be used in global optimization.
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Boroson, Ethan Rain. "Optimization Under Uncertainty of Nonlinear Energy Sinks." Thesis, The University of Arizona, 2015. http://hdl.handle.net/10150/595972.
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Nonlinear Energy Sinks (NESs) are a promising technique for passively reducing the amplitude of vibrations. Through nonlinear stiffness properties, a NES is able to passively absorb energy. Unlike a traditional Tuned Mass Damper (TMD), NESs do not require a specific tuning and absorb energy from a wide range of frequencies. However, each NES is only efficient over a limited range of excitations. In addition, NES efficiency is extremely sensitive to perturbations in design parameters or loading, demonstrating a nearly discontinuous efficiency. Therefore, in order to optimally design a NES, uncertainties must be accounted for. This thesis focuses on optimally selecting parameters to design an effective NES system through optimization under uncertainty. For this purpose, a specific algorithm is introduced that makes use of clustering techniques to segregate efficient and inefficient NES behavior. SVM and Kriging approximations as well as new adaptive sampling techniques are used for the optimization under uncertainty. The variables of the problems are either random design variables or aleatory variables. For example, the excitation applied to the main vibrating system is treated as aleatory. In an effort to increase the range of excitations for which NESs are effective, a combination of NESs configured in parallel is considered. Optimization under uncertainty is performed on several examples with varying design parameters as well as different numbers of NESs (from 1 to 10). Results show that combining NESs in parallel is an effective method to increase the excitation range over which a NES is effective.
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Shapoval, Andriy. "Topics in linear and nonlinear discrete optimization." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/53460.
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This work contributes to modeling, theoretical, and practical aspects of structured Mathematical Programming problems. Many real-world applications have nonlinear characteristics and can be modeled as Mixed Integer Nonlinear Programming problems (MINLP). Modern global solvers have significant difficulty handling large-scale instances of them. Several convexification and underestimation techniques were proposed in the last decade as a part of the solution process, and we join this trend. The thesis has three major parts.
The first part considers MINLP problems containing convex (in the sense of continuous relaxations) and posynomial terms (also called monomials), i.e. products of variables with some powers. Recently, a linear Mixed Integer Programming (MIP) approach was introduced for minimization the number of variables and transformations for convexification and underestimation of these structured problems. We provide polyhedral analysis together with separation for solving our variant of this minimization subproblem, containing binary and bounded continuous variables. Our novel mixed hyperedge method allows to outperform modern commercial MIP software, providing new families of facet-defining inequalities. As a byproduct, we introduce a new research area called mixed conflict hypergraphs. It merges mixed conflict graphs and 0-1 conflict hypergraphs.
The second part applies our mixed hyperedge method to a linear subproblem of the same purpose for another class of structured MINLP problems. They contain signomial terms, i.e. posynomial terms of both positive and negative signs. We obtain new facet-defining inequalities in addition to those families from the first part.
The final part is dedicated to managing guest flow in Georgia Aquarium after the Dolphin Tales opening with applying a large-scale MINLP. We consider arrival and departure processes related to scheduled shows and develop three stochastic models for them. If demand for the shows is high, all processes become interconnected and require a generalized model. We provide and solve a Signomial Programming problem with mixed variables for minimization resources to prevent and control congestions.
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Boccia, Andrea. "Optimization based control of nonlinear constrained systems." Thesis, Imperial College London, 2014. http://hdl.handle.net/10044/1/24700.
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This thesis is in the field of Optimal Control. It addresses research questions concerning both the properties of optimal controls and also schemes for control system stabilization based on the solution of optimal control problems. The first part is concerned with the derivation of necessary conditions of optimality for two classes of optimal control problems not covered by earlier theory. The first is the class of optimal control problems with a combination of mixed control-state constraints and pure state constraints in which the dynamics are described by a differential inclusion under weaker hypotheses than have previously been considered. The second is the class of optimal control problems in which the dynamics take the form of a non-smooth differential equation with delays, and where the end-time is included in the decision variables. We shall demonstrate that these new optimality conditions lead to algorithms for solution of certain optimal control problems not amenable to earlier theory. Model Predictive Control (MPC) is an approach to control system design based on solving, at each control update time, an optimal control problem. This is the subject matter of the second part of the thesis. We derive new MPC algorithms for constrained linear and nonlinear systems which, in certain significant respect, are simpler to implement than standard schemes, and which achieve performance specification under more general conditions than has previously been demonstrated. These include stability and feasibility.
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Chow, Raymond W. L. Carleton University Dissertation Information and Systems Science. "Gate level transistor sizing by nonlinear optimization." Ottawa, 1992.
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Callen, Bryan. "NONLINEAR OPTIMIZATION AS IT APPLIES TO CURVEFITTING." OpenSIUC, 2014. https://opensiuc.lib.siu.edu/theses/1494.
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The purpose of this thesis is to examine different methods of curve-fitting through the process of nonlinear least squares. Specifically, Newton's method, Gauss Newton's method, Levenberg-Marquardt, Quasi Newton methods, and Nonlinear Conjugate Gradient method. The differing convergence rates, as well as necessary initial conditions are explored for a variety of methods. The concepts of the line search and trust region are also examined.
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Sitharaman, Sai Ganesh. "Nonlinear continuous feedback controllers." Texas A&M University, 2004. http://hdl.handle.net/1969.1/363.
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Packet-switched communication networks such as today's Internet are built with several interconnected core and distribution packet forwarding routers and several sender and sink transport agents. In order to maintain stability and avoid congestion collapse in the network, the sources control their rate behavior and voluntarily adjust their sending rates to accommodate other sources in the network. In this thesis, we study one class of sender rate control that is modeled using continuous first-order differential equation of the sending rates. In order to adjust the rates appropriately, the network sends continuous packet-loss feedback to the sources. We study a form of closed-loop feedback congestion controllers whose rate adjustments exhibit a nonlinear form.
There are three dimensions to our work in this thesis. First, we study the network optimization problem in which sources choose utilities to maximize their underlying throughput. Each sender maximizes its utility proportional to the throughput achieved. In our model, sources choose a utility function to define their level of satisfaction of the underlying resource usages. The objective of this direction is to establish the properties of source utility functions using inequality constrained bounded sets and study the functional forms of utilities against a chosen rate differential equation.
Second, stability of the network and tolerance to perturbation are two essential factors that keep communication networks operational around the equilibrium point. Our objective in this part of the thesis is to analytically understand the existence of local asymptotic stability of delayed-feedback systems under homogeneous network delays.
Third, we propose a novel tangential controller for a generic maximization function and study its properties using nonlinear optimization techniques. We develop the necessary theoretical background and the properties of our controller to prove that it is a better rate adaptation algorithm for logarithmic utilities compared to the well-studied proportional controllers. We establish the asymptotic local stability of our controller with upper bounds on the increase / decrease gain parameters.
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Mehlman, Stephanie A. "Modeling mixtures in chemistry, some nonlinear optimization problems." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ36370.pdf.
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Claewplodtook, Pana. "Optimization of nonlinear dynamic systems without Lagrange multipliers." Ohio : Ohio University, 1996. http://www.ohiolink.edu/etd/view.cgi?ohiou1178654973.
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NediÌc, Jelena. "A dynamical systems view of nonlinear optimization problems." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408684.
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Bremberg, Sebastian. "Calibration of Multilateration Positioning Systems via Nonlinear Optimization." Thesis, KTH, Optimeringslära och systemteori, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-173224.
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This master thesis presents an evaluation of a method for improving performance of sensor positioning in a network of emitters. The positioning method used for the analysis is Time Difference of Arrival, TDOA, a multilateration technique based on measurements of differences in signal travel time between a pair of synchronous and spatially separated pairs of emitters and a sensor. The method in question aims at reducing positioning errors caused by errors in initially reported emitter positions as well as network synchronization errors by using already known sensor positions to re-calibrate the network of emitters. This is done by minimizing the difference between signal based TDOA measurements from the system and estimated TDOA measurements made by calculations based on given sensor positions by means of nonlinear least squares optimization. Alterations of the method with different settings and error contributions and with varying amount of sensors and emitters are tested throughout several simulations. The proposed method shows apparent results of improving the system parameters and also copes well with contributing errors provided that the amount of measurements is sufficiently large.<br>I denna masteruppsats utvärderas en metod syftande till att förbättra noggrannheten i den funktion som positionerar sensorer i ett trådlöst transmissionsnätverk. Den positioneringsmetod som har legat till grund för analysen är TDOA (Time Difference of Arrival), en multilaterations-teknik som baseras på mätning av tidsskillnaden av en radiosignal från två rumsligt separerade och synkrona transmittorer till en mottagande sensor. Metoden syftar till att reducera positioneringsfel som orsakats av att de ursprungliga positionsangivelserna varit felaktiga samt synkroniseringsfel i nätet. För rekalibrering av transmissionsnätet används redan kända sensorpositioner. Detta uppnås genom minimering av skillnaden mellan signalbaserade TDOA-mätningar från systemet och uppskattade TDOA-mått vilka erhållits genom beräkningar av en given sensorposition baserat på optimering via en ickelinjär minstakvadratanpassning. Genom ett antal simuleringar testas sedan den föreslagna metoden med olika grundinställningar och olika grad av mätbrus samt ett varierande antal sensorer och transmittorer. Denna metod ger tydlig förbättring för estimering av systemparametrar och klarar även av att hantera multipla felkällor förutsatt att antalet mätningar ¨ar tillräckligt stort.
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Li, Zhongwei. "Reliability-Based Design Optimization of Nonlinear Beam-Columns." Diss., Virginia Tech, 2018. http://hdl.handle.net/10919/82958.
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This dissertation addresses the ultimate strength analysis of nonlinear beam-columns under axial compression, the sensitivity of the ultimate strength, structural optimization and reliability analysis using ultimate strength analysis, and Reliability-Based Design Optimization (RBDO) of the nonlinear beam-columns. The ultimate strength analysis is based on nonlinear beam theory with material and geometric nonlinearities. Nonlinear constitutive law is developed for elastic-perfectly-plastic beam cross-section consisting of base plate and T-bar stiffener. The analysis method is validated using commercial nonlinear finite element analysis. A new direct solving method is developed, which combines the original governing equations with their derivatives with respect to deformation matric and solves for the ultimate strength directly. Structural optimization and reliability analysis use a gradient-based algorithm and need accurate sensitivities of the ultimate strength to design variables. Semi-analytic sensitivity of the ultimate strength is calculated from a linear set of analytical sensitivity equations which use the Jacobian matrix of the direct solving method. The derivatives of the structural residual equations in the sensitivity equation set are calculated using complex step method. The semi-analytic sensitivity is more robust and efficient as compared to finite difference sensitivity. The design variables are the cross-sectional geometric parameters. Random variables include material properties, geometric parameters, initial deflection and nondeterministic load. Failure probabilities calculated by ultimate strength reliability analysis are validated by Monte Carlo Simulation. Double-loop RBDO minimizes structural weight with reliability index constraint. The sensitivity of reliability index with respect to design variables is calculated from the gradient of limit state function at the solution of reliability analysis. By using the ultimate strength direct solving method, semi-analytic sensitivity and gradient-based optimization algorithm, the RBDO method is found to be robust and efficient for nonlinear beam-columns. The ultimate strength direct solving method, semi-analytic sensitivity, structural optimization, reliability analysis, and RBDO method can be applied to more complicated engineering structures including stiffened panels and aerospace/ocean structures.<br>Ph. D.
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LIN, JEEN. "SHAPE OPTIMIZATION OF NONLINEAR STRUCTURES UNDER FATIGUE LOADING." University of Cincinnati / OhioLINK, 2001. http://rave.ohiolink.edu/etdc/view?acc_num=ucin982764930.
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Wang, Chen. "Variants of Deterministic and Stochastic Nonlinear Optimization Problems." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112294/document.
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Les problèmes d’optimisation combinatoire sont généralement réputés NP-difficiles, donc il n’y a pas d’algorithmes efficaces pour les résoudre. Afin de trouver des solutions optimales locales ou réalisables, on utilise souvent des heuristiques ou des algorithmes approchés. Les dernières décennies ont vu naitre des méthodes approchées connues sous le nom de métaheuristiques, et qui permettent de trouver une solution approchées. Cette thèse propose de résoudre des problèmes d’optimisation déterministe et stochastique à l’aide de métaheuristiques. Nous avons particulièrement étudié la méthode de voisinage variable connue sous le nom de VNS. Nous avons choisi cet algorithme pour résoudre nos problèmes d’optimisation dans la mesure où VNS permet de trouver des solutions de bonne qualité dans un temps CPU raisonnable. Le premier problème que nous avons étudié dans le cadre de cette thèse est le problème déterministe de largeur de bande de matrices creuses. Il s’agit d’un problème combinatoire difficile, notre VNS a permis de trouver des solutions comparables à celles de la littérature en termes de qualité des résultats mais avec temps de calcul plus compétitif. Nous nous sommes intéressés dans un deuxième temps aux problèmes de réseaux mobiles appelés OFDMA-TDMA. Nous avons étudié le problème d’affectation de ressources dans ce type de réseaux, nous avons proposé deux modèles : Le premier modèle est un modèle déterministe qui permet de maximiser la bande passante du canal pour un réseau OFDMA à débit monodirectionnel appelé Uplink sous contraintes d’énergie utilisée par les utilisateurs et des contraintes d’affectation de porteuses. Pour ce problème, VNS donne de très bons résultats et des bornes de bonne qualité. Le deuxième modèle est un problème stochastique de réseaux OFDMA d’affectation de ressources multi-cellules. Pour résoudre ce problème, on utilise le problème déterministe équivalent auquel on applique la méthode VNS qui dans ce cas permet de trouver des solutions avec un saut de dualité très faible. Les problèmes d’allocation de ressources aussi bien dans les réseaux OFDMA ou dans d’autres domaines peuvent aussi être modélisés sous forme de problèmes d’optimisation bi-niveaux appelés aussi problèmes d’optimisation hiérarchique. Le dernier problème étudié dans le cadre de cette thèse porte sur les problèmes bi-niveaux stochastiques. Pour résoudre le problème lié à l’incertitude dans ce problème, nous avons utilisé l’optimisation robuste plus précisément l’approche appelée « distributionnellement robuste ». Cette approche donne de très bons résultats légèrement conservateurs notamment lorsque le nombre de variables du leader est très supérieur à celui du suiveur. Nos expérimentations ont confirmé l’efficacité de nos méthodes pour l’ensemble des problèmes étudiés<br>Combinatorial optimization problems are generally NP-hard problems, so they can only rely on heuristic or approximation algorithms to find a local optimum or a feasible solution. During the last decades, more general solving techniques have been proposed, namely metaheuristics which can be applied to many types of combinatorial optimization problems. This PhD thesis proposed to solve the deterministic and stochastic optimization problems with metaheuristics. We studied especially Variable Neighborhood Search (VNS) and choose this algorithm to solve our optimization problems since it is able to find satisfying approximated optimal solutions within a reasonable computation time. Our thesis starts with a relatively simple deterministic combinatorial optimization problem: Bandwidth Minimization Problem. The proposed VNS procedure offers an advantage in terms of CPU time compared to the literature. Then, we focus on resource allocation problems in OFDMA systems, and present two models. The first model aims at maximizing the total bandwidth channel capacity of an uplink OFDMA-TDMA network subject to user power and subcarrier assignment constraints while simultaneously scheduling users in time. For this problem, VNS gives tight bounds. The second model is stochastic resource allocation model for uplink wireless multi-cell OFDMA Networks. After transforming the original model into a deterministic one, the proposed VNS is applied on the deterministic model, and find near optimal solutions. Subsequently, several problems either in OFDMA systems or in many other topics in resource allocation can be modeled as hierarchy problems, e.g., bi-level optimization problems. Thus, we also study stochastic bi-level optimization problems, and use robust optimization framework to deal with uncertainty. The distributionally robust approach can obtain slight conservative solutions when the number of binary variables in the upper level is larger than the number of variables in the lower level. Our numerical results for all the problems studied in this thesis show the performance of our approaches
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Tran, Ngoc Nguyen. "Infeasibility detection and regularization strategies in nonlinear optimization." Thesis, Limoges, 2018. http://www.theses.fr/2018LIMO0059/document.
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Dans cette thèse, nous nous étudions des algorithmes d’optimisation non linéaire. D’une part nous proposons des techniques de détection rapide de la non-réalisabilité d’un problème à résoudre. D’autre part, nous analysons le comportement local des algorithmes pour la résolution de problèmes singuliers. Dans la première partie, nous présentons une modification d’un algorithme de lagrangien augmenté pour l’optimisation avec contraintes d’égalité. La convergence quadratique du nouvel algorithme dans le cas non-réalisable est démontrée théoriquement et numériquement. La seconde partie est dédiée à l’extension du résultat précédent aux problèmes d’optimisation non linéaire généraux avec contraintes d’égalité et d’inégalité. Nous proposons une modification d’un algorithme de pénalisation mixte basé sur un lagrangien augmenté et une barrière logarithmique. Les résultats théoriques de l’analyse de convergence et quelques tests numériques montrent l’avantage du nouvel algorithme dans la détection de la non-réalisabilité. La troisième partie est consacrée à étudier le comportement local d’un algorithme primal-dual de points intérieurs pour l’optimisation sous contraintes de borne. L’analyse locale est effectuée sans l’hypothèse classique des conditions suffisantes d’optimalité de second ordre. Celle-ci est remplacée par une hypothèse plus faible basée sur la notion de borne d’erreur locale. Nous proposons une technique de régularisation de la jacobienne du système d’optimalité à résoudre. Nous démontrons ensuite des propriétés de bornitude de l’inverse de ces matrices régularisées, ce qui nous permet de montrer la convergence superlinéaire de l’algorithme. La dernière partie est consacrée à l’analyse de convergence locale de l’algorithme primal-dual qui est utilisé dans les deux premières parties de la thèse. En pratique, il a été observé que cet algorithme converge rapidement même dans le cas où les contraintes ne vérifient l’hypothèse de qualification de Mangasarian-Fromovitz. Nous démontrons la convergence superlinéaire et quadratique de cet algorithme, sans hypothèse de qualification des contraintes<br>This thesis is devoted to the study of numerical algorithms for nonlinear optimization. On the one hand, we propose new strategies for the rapid infeasibility detection. On the other hand, we analyze the local behavior of primal-dual algorithms for the solution of singular problems. In the first part, we present a modification of an augmented Lagrangian algorithm for equality constrained optimization. The quadratic convergence of the new algorithm in the infeasible case is theoretically and numerically demonstrated. The second part is dedicated to extending the previous result to the solution of general nonlinear optimization problems with equality and inequality constraints. We propose a modification of a mixed logarithmic barrier-augmented Lagrangian algorithm. The theoretical convergence results and the numerical experiments show the advantage of the new algorithm for the infeasibility detection. In the third part, we study the local behavior of a primal-dual interior point algorithm for bound constrained optimization. The local analysis is done without the standard assumption of the second-order sufficient optimality conditions. These conditions are replaced by a weaker assumption based on a local error bound condition. We propose a regularization technique of the Jacobian matrix of the optimality system. We then demonstrate some boundedness properties of the inverse of these regularized matrices, which allow us to prove the superlinear convergence of our algorithm. The last part is devoted to the local convergence analysis of the primal-dual algorithm used in the first two parts of this thesis. In practice, it has been observed that this algorithm converges rapidly even in the case where the constraints do not satisfy the Mangasarian-Fromovitz constraint qualification. We demonstrate the superlinear and quadratic convergence of this algorithm without any assumption of constraint qualification
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Wu, Dawen. "Solving Some Nonlinear Optimization Problems with Deep Learning." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASG083.
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Cette thèse considère quatre types de problèmes d'optimisation non linéaire, à savoir les jeux de bimatrice, les équations de projection non linéaire (NPEs), les problèmes d'optimisation convexe non lisse (NCOPs) et les jeux à contraintes stochastiques (CCGs). Ces quatre classes de problèmes d'optimisation non linéaire trouvent de nombreuses applications dans divers domaines tels que l'ingénierie, l'informatique, l'économie et la finance. Notre objectif est d'introduire des algorithmes basés sur l'apprentissage profond pour calculer efficacement les solutions optimales de ces problèmes d'optimisation non linéaire.Pour les jeux de bimatrice, nous utilisons des réseaux neuronaux convolutionnels (CNNs) pour calculer les équilibres de Nash. Plus précisément, nous concevons une architecture de CNN où l'entrée est un jeu de bimatrice et la sortie est l'équilibre de Nash prédit pour le jeu. Nous générons un ensemble de jeux de bimatrice suivant une distribution de probabilité donnée et utilisons l'algorithme de Lemke-Howson pour trouver leurs véritables équilibres de Nash, constituant ainsi un ensemble d'entraînement. Le CNN proposé est formé sur cet ensemble de données pour améliorer sa précision. Une fois l'apprentissage terminée, le CNN est capable de prédire les équilibres de Nash pour des jeux de bimatrice inédits. Les résultats expérimentaux démontrent l'efficacité computationnelle exceptionnelle de notre approche basée sur CNN, au détriment de la précision.Pour les NPEs, NCOPs et CCGs, qui sont des problèmes d'optimisation plus complexes, ils ne peuvent pas être directement introduits dans les réseaux neuronaux. Par conséquent, nous avons recours à des outils avancés, à savoir l'optimisation neurodynamique et les réseaux neuronaux informés par la physique (PINNs), pour résoudre ces problèmes. Plus précisément, nous utilisons d'abord une approche neurodynamique pour modéliser un problème d'optimisation non linéaire sous forme de système d'équations différentielles ordinaires (ODEs). Ensuite, nous utilisons un modèle basé sur PINN pour résoudre le système d'ODE résultant, où l'état final du modèle représente la solution prédite au problème d'optimisation initial. Le réseau neuronal est formé pour résoudre le système d'ODE, résolvant ainsi le problème d'optimisation initial. Une contribution clé de notre méthode proposée réside dans la transformation d'un problème d'optimisation non linéaire en un problème d'entraînement de réseau neuronal. En conséquence, nous pouvons maintenant résoudre des problèmes d'optimisation non linéaire en utilisant uniquement PyTorch, sans compter sur des solveurs d'optimisation convexe classiques tels que CVXPY, CPLEX ou Gurobi<br>This thesis considers four types of nonlinear optimization problems, namely bimatrix games, nonlinear projection equations (NPEs), nonsmooth convex optimization problems (NCOPs), and chance-constrained games (CCGs).These four classes of nonlinear optimization problems find extensive applications in various domains such as engineering, computer science, economics, and finance.We aim to introduce deep learning-based algorithms to efficiently compute the optimal solutions for these nonlinear optimization problems.For bimatrix games, we use Convolutional Neural Networks (CNNs) to compute Nash equilibria.Specifically, we design a CNN architecture where the input is a bimatrix game and the output is the predicted Nash equilibrium for the game.We generate a set of bimatrix games by a given probability distribution and use the Lemke-Howson algorithm to find their true Nash equilibria, thereby constructing a training dataset.The proposed CNN is trained on this dataset to improve its accuracy. Upon completion of training, the CNN is capable of predicting Nash equilibria for unseen bimatrix games.Experimental results demonstrate the exceptional computational efficiency of our CNN-based approach, at the cost of sacrificing some accuracy.For NPEs, NCOPs, and CCGs, which are more complex optimization problems, they cannot be directly fed into neural networks.Therefore, we resort to advanced tools, namely neurodynamic optimization and Physics-Informed Neural Networks (PINNs), for solving these problems.Specifically, we first use a neurodynamic approach to model a nonlinear optimization problem as a system of Ordinary Differential Equations (ODEs).Then, we utilize a PINN-based model to solve the resulting ODE system, where the end state of the model represents the predicted solution to the original optimization problem.The neural network is trained toward solving the ODE system, thereby solving the original optimization problem.A key contribution of our proposed method lies in transforming a nonlinear optimization problem into a neural network training problem.As a result, we can now solve nonlinear optimization problems using only PyTorch, without relying on classical convex optimization solvers such as CVXPY, CPLEX, or Gurobi
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Lin, Chin-Yee. "Interior point methods for convex optimization." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/15044.
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Handley-Schachler, Sybille H. "Applications of parallel processing to optimization." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240512.
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Lee, Haewon. "Nonlinear evolution equations and optimization problems in Banach spaces." Ohio : Ohio University, 2005. http://www.ohiolink.edu/etd/view.cgi?ohiou1127498683.
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Yamakawa, Yuya. "Studies on Optimization Methods for Nonlinear Semidefinite Programming Problems." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/199446.
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Kwok, Terence 1973. "Neural networks with nonlinear system dynamics for combinatorial optimization." Monash University, School of Business Systems, 2001. http://arrow.monash.edu.au/hdl/1959.1/8928.
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Schlueter, Martin. "Nonlinear mixed integer based optimization technique for space applications." Thesis, University of Birmingham, 2012. http://etheses.bham.ac.uk//id/eprint/3101/.
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In this thesis a new algorithm for mixed integer nonlinear programming (MINLP) is developed and applied to several real world applications with special focus on space applications. The algorithm is based on two main components, which are an extension of the Ant Colony Optimization metaheuristic and the Oracle Penalty Method for constraint handling. A sophisticated implementation (named MIDACO) of the algorithm is used to numerically demonstrate the usefulness and performance capabilities of the here developed novel approach on MINLP. An extensive amount of numerical results on both, comprehensive sets of benchmark problems (with up to 100 test instances) and several real world applications, are presented and compared to results obtained by concurrent methods. It can be shown, that the here developed approach is not only fully competitive with established MINLP algorithms, but is even able to outperform those regarding global optimization capabilities and cpu runtime performance. Furthermore, the algorithm is able to solve challenging space applications, that are considered here as mixed integer problems for the very first time.
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Dunatunga, Manimelwadu Samson 1958. "SUCCESSIVE TWO SEGMENT SEPARABLE PROGRAMMING FOR NONLINEAR MINIMAX OPTIMIZATION." Thesis, The University of Arizona, 1986. http://hdl.handle.net/10150/275509.
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Tsao, Lu-Ping 1959. "INTERACTIVE NONLINEAR PROGRAMMING (OPTIMIZATION, NLP, DARE/INTERACTIVE, DEVELOPMENT SYSTEM)." Thesis, The University of Arizona, 1986. http://hdl.handle.net/10150/291293.
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Foley, Dawn Christine. "Short horizon optimal control of nonlinear systems via discrete state space realization." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/16803.
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Satir, Sarp. "Modeling and optimization of capacitive micromachined ultrasonic transducers." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/54303.
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The objective of this research is to develop large signal modeling and optimization methods for Capacitive Micromachined Ultrasonic Transducers (CMUTs), especially when they are used in an array configuration. General modeling and optimization methods that cover a large domain of CMUT designs are crucial, as many membrane and array geometry combinations are possible using existing microfabrication technologies. Currently, large signal modeling methods for CMUTs are not well established and nonlinear imaging techniques utilizing linear piezoelectric transducers are not applicable to CMUTs because of their strong nonlinearity. In this work, the nonlinear CMUT behavior is studied, and a feedback linearization method is proposed to reduce the CMUT nonlinearity. This method is shown to improve the CMUT performance for continuous wave applications, such as high-intensity focused ultrasound or harmonic imaging, where transducer linearity is crucial. In the second part of this dissertation, a large signal model is developed that is capable of transient modeling of CMUT arrays with arbitrary electrical terminations. The developed model is suitable for iterative design optimization of CMUTs and CMUT based imaging systems with arbitrary membrane and array geometries for a variety of applications. Finally, a novel multi-pulse method for nonlinear tissue and contrast agent imaging with CMUTs is presented. It is shown that the nonlinear content can be successfully extracted from echo signals in a CMUT based imaging system using a multiple pulse scheme. The proposed method is independent of the CMUT geometry and valid for large signal operation. Experimental results verifying the developed large signal CMUT array model, proposed gap feedback and multi-pulse techniques are also presented.
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Akteke-ozturk, Basak. "New Approaches To Desirability Functions By Nonsmooth And Nonlinear Optimization." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612649/index.pdf.
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Desirability Functions continue to attract attention of scientists and researchers working in
the area of multi-response optimization. There are many versions of such functions, differing
mainly in formulations of individual and overall desirability functions. Derringer and
Suich&rsquo<br>s desirability functions being used throughout this thesis are still the most preferred
ones in practice and many other versions are derived from these. On the other hand, they have
a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current
approaches to their optimization, which are based on derivative-free search techniques and
modification of the functions by higher-degree polynomials, need to be diversified considering
opportunities offered by modern nonlinear (global) optimization techniques and related
softwares. A first motivation of this work is to develop a new efficient solution strategy for the
maximization of overall desirability functions which comes out to be a nonsmooth composite
constrained optimization problem by nonsmooth optimization methods.
We observe that individual desirability functions used in practical computations are of mintype,
a subclass of continuous selection functions. To reveal the mechanism that gives rise to
a variation in the piecewise structure of desirability functions used in practice, we concentrate
on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of
desirability functions via piecewise max-type functions.
In this thesis, we introduce adjusted desirability functions based on a reformulation of the
individual desirability functions by a binary integer variable in order to deal with their piecewise
definition. We define a constraint on the binary variable to obtain a continuous optimization
problem of a nonlinear objective function including nondifferentiable points with
the constraints of bounds for factors and responses. After describing the adjusted desirability
functions on two well-known problems from the literature, we implement modified subgradient
algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for
solving the corresponding optimization problems. Moreover, BARON solver of GAMS is
used to solve these optimization problems including adjusted desirability functions. Numerical
applications with BARON show that this is a more efficient alternative solution strategy
than the current desirability maximization approaches.
We apply negative logarithm to the desirability functions and consider the properties of the
resulting functions when they include more than one nondifferentiable point. With this approach
we reveal the structure of the functions and employ the piecewise max-type functions
as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure
of the individual functions over their compact and connected interval that yield our
so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have
efficient structural and topological properties. We present the structural stability, optimality
and constraint qualification properties of GDFs using that of max-type functions.
As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach
for multi-objective optimization problems, based on a separation of the parameters:
in y-space (optimization) and in x-space (representation). This approach is about calculating
the factor variables corresponding to the ideal solutions of each individual functions in y, and
then finding a set of compromised solutions in x by considering the convex hull of the ideal
factors. This is an early attempt of a new multi-objective optimization method. Our first results
show that global optimum of the overall problem may not be an element of the set of
compromised solution.
The overall problem in both x and y is extended to a new refined (disjunctive) generalized
semi-infinite problem, herewith analyzing the stability and robustness properties of the objective
function. In this course, we introduce the so-called robust optimization of desirability
functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability
functions.
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Thekale, Alexander [Verfasser]. "Trust-Region Methods for Simulation Based Nonlinear Optimization / Alexander Thekale." Aachen : Shaker, 2011. http://d-nb.info/1071528785/34.
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Mitradjieva-Daneva, Maria. "Feasible Direction Methods for Constrained Nonlinear Optimization : Suggestions for Improvements." Doctoral thesis, Linköping : Department of Mathematics, Linköping University, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-8811.
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Pajot, Joseph M. "Topology optimization of geometrically nonlinear structures including thermo-mechanical coupling." Diss., Connect to online resource, 2006. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3219000.
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Petersson, Daniel. "A Nonlinear Optimization Approach to H2-Optimal Modeling and Control." Doctoral thesis, Linköpings universitet, Reglerteknik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-93324.
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Mathematical models of physical systems are pervasive in engineering. These models can be used to analyze properties of the system, to simulate the system, or synthesize controllers. However, many of these models are too complex or too large for standard analysis and synthesis methods to be applicable. Hence, there is a need to reduce the complexity of models. In this thesis, techniques for reducing complexity of large linear time-invariant (lti) state-space models and linear parameter-varying (lpv) models are presented. Additionally, a method for synthesizing controllers is also presented. The methods in this thesis all revolve around a system theoretical measure called the H2-norm, and the minimization of this norm using nonlinear optimization. Since the optimization problems rapidly grow large, significant effort is spent on understanding and exploiting the inherent structures available in the problems to reduce the computational complexity when performing the optimization. The first part of the thesis addresses the classical model-reduction problem of lti state-space models. Various H2 problems are formulated and solved using the proposed structure-exploiting nonlinear optimization technique. The standard problem formulation is extended to incorporate also frequency-weighted problems and norms defined on finite frequency intervals, both for continuous and discrete-time models. Additionally, a regularization-based method to account for uncertainty in data is explored. Several examples reveal that the method is highly competitive with alternative approaches. Techniques for finding lpv models from data, and reducing the complexity of lpv models are presented. The basic ideas introduced in the first part of the thesis are extended to the lpv case, once again covering a range of different setups. lpv models are commonly used for analysis and synthesis of controllers, but the efficiency of these methods depends highly on a particular algebraic structure in the lpv models. A method to account for and derive models suitable for controller synthesis is proposed. Many of the methods are thoroughly tested on a realistic modeling problem arising in the design and flight clearance of an Airbus aircraft model. Finally, output-feedback H2 controller synthesis for lpv models is addressed by generalizing the ideas and methods used for modeling. One of the ideas here is to skip the lpv modeling phase before creating the controller, and instead synthesize the controller directly from the data, which classically would have been used to generate a model to be used in the controller synthesis problem. The method specializes to standard output-feedback H2 controller synthesis in the lti case, and favorable comparisons with alternative state-of-the-art implementations are presented.
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Joham, Michael [Verfasser]. "Optimization of Linear and Nonlinear Transmit Signal Processing / Michael Joham." Aachen : Shaker, 2004. http://d-nb.info/1170537405/34.
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Vasudevan, Deepak. "Water Distribution Networks: Leakage Management using Nonlinear Optimization of Pressure." Thesis, KTH, Skolan för elektroteknik och datavetenskap (EECS), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-246095.
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The eective management of water distribution systems is gaining tremendousfocus within the scientic community thanks to the Internet of Things.This report is an exploration attempt into the mathematical optimization ofa water distribution network under the inuence of leakages caused by deteriorationwithin the distribution network. The optimization process is carriedout in a two phase manner involving the optimal placement of pressure valves,followed by the optimal control of the valves in the presence of known leakswithin the network. The optimization framework addresses the minimisationof the average network pressure in an extended time setting by imposingthe hydraulic equations as nonlinear constraints. The hydraulic components,namely the pressure reduction valves, are modelled as integer variables leadingto a non-convex and non-linear optimization problem that falls under theclass of optimization problems known as mixed-integer nonlinear programming(MINLP). The exercise implements two reformulation methods that solves theMINLP problem as a sequence of regular nonlinear programs (NLPs), and alsopresents the hydraulic simulation results of the implementation. While thereis sucient research on water network optimization using various mathematicalmethods, this study endeavours to combine a leakage model within theoptimization framework and presents the ndings of the analysis. In addition,the report also includes the outcomes of the simulation on a real distributionnetwork simulated under varying demand conditions.<br>Effektiv förvaltning av vattendistributionssystem har skaffat stor inriktning inom det vetenskapliga samhället på grund av Sakernas internet (engelska IoT). En matematisk optimering av ett vattendistributionsnät som påverkades av läckage på grund av försämring undersöktes i den här rapporten. Optimeringsprocessen utfördes i tvåstegs med den optimala placeringen av tryckventiler i första steget följt av optimal styrning av ventilerna i närvaro av kända läckor inom nätverket. Optimeringsramen behandlar minimering av det genomsnittliga nätverkstrycket i en förlängd tidsinställning genom att införa hydrauliska ekvationer som icke-linjära begränsningar. De hydrauliska komponenterna, nämligen tryckreduceringsventilerna, modellerades som integervariabler somleder till ett icke-konvex och icke-linjärt optimeringsproblem som kallas mixedinteger icke-linjära programmering (engelska MINLP). Ö vningen genomförtvå reformuleringsmetoder som löser MINLP-problemet som regelbundna ickelinjära program (NLP) och presenterar också hydrauliska simuleringsresultatav det. Ä ven om det finns tillräcklig forskning om optimering av vattennätverkmed hjälp av olika matematiska metoder strävar det här arbetet efter att kombinera en hydraulisk läckagemodell inom nätverkoptimeringsramen och presenterar analysens resultat. Dessutom innehåller rapporten även simuleringsresultaten på ett riktigt distributionsnät simulerad under varierande vat-tenefterfrågan.
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Coffee, Thomas Merritt. "Validated global multiobjective optimization of trajectories in nonlinear dynamical systems." Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/98584.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2015.<br>This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 315-349).<br>We introduce a new approach for global multiobjective optimization of trajectories in continuous nonlinear dynamical systems that can provide rigorous, arbitrarily tight bounds on the objective values and state paths realized by (Pareto-)optimal trajectories. By controlling all sources of error, our resulting method is the first global trajectory optimization method that can reliably handle nonconvex nonlinear dynamical systems with substantial instabilities, such as the notoriously ill-behaved multi-body gravitational systems governing interplanetary space trajectories. Rigorous finite-dimensional global optimization methods based on space partitioning (branch and bound) do not directly extend to infinite-dimensional problems of trajectory optimization, lacking a way to exhaustively partition an infinite-dimensional space. Thus existing generic methods for deterministic global trajectory optimization rely on direct discretization of the control variables, if not also the state variables. While the resulting errors may prove inconsequential for relatively stable (conservative/dissipative) systems, they severely influence results in unstable systems that arise in many aerospace applications, and whose chaotic sensitivities offer great potential for inexpensive trajectory control. In order to achieve higher accuracy, current programs for interplanetary trajectory optimization typically use problem-specific control parameterizations with local optimization methods (commonly, multiple shooting with sequential quadratic programming), combined with stochastic or expert-guided sampling to seek global optimality. This approach substantially relies on pre-existing intuition about the character of optimal solutions, and provides no guarantees on the global optimality of solutions obtained. The requirements for expert guidance and judgment of uncertainties tend to drive up costs and restrict innovation for the trajectory solutions that play a crucial role in early conceptual design for deep space missions. The thesis takes a new approach to avoid unaccountable discretization errors. Using a specially designed exhaustive partition of the (finite-dimensional) state space into subregions, we construct a finite transition graph between these subregions, such that each trajectory of interest maps to a finite path (transition sequence) in the graph, where each transition trajectory lies in a local state space neighborhood of its corresponding subregions. For any such path, the cost of any corresponding trajectory can be bounded below by the sum of lower bounds on the cost of each stepwise transition. Provided that the transition bounds converge to exact bounds with increasing refinement of the state space partition, an adaptive refinement can produce asymptotically convergent bounds on optimal trajectories. We compute a lower bound on each stepwise transition between state subregions by a novel "interval linearization" technique that simultaneously considers all possible trajectories between two subregions that lie within a local neighborhood. This technique first linearizes the dynamics on the local neighborhood, and replaces the remaining nonlinear terms by interval enclosures of their values over the neighborhood. We then derive a nonlinear system-of-equations solution to a corresponding pointwise generalized linear optimal control problem with time-varying coefficients. Finally, using interval methods, we compute enclosures to the solutions of these equations as the coefficients for the nonlinear terms range over the previously computed enclosures on the neighborhood. This technique effectively confines the difficulties of the infinite-dimensional trajectory space to a local neighborhood, where they can be contained by rigorous approximation. While our approach can in principle be applied to compute a complete optimal control policy over the entire state space for a given target, practical efficiency in most cases demands adaptive restriction of the state space to trajectories between particular start and goal subregions. We introduce a bidirectional "bounded path" algorithm, generalizing efficient graph shortest path algorithms, which permits simultaneously identifying the shortest path(s) in the transition graph-to direct adaptive refinement-and identifying state space subregions whose intersecting path bounds exceed a threshold-to prune subregions that cannot intersect optimal trajectories. By expanding a generally nonconvex dynamical flow to a finite graph admitting this Dijkstra-like search procedure, the transition graph may be seen as "unfolding" the state space to leverage some of the same efficiencies as level-set methods for convex dynamical systems. The structure of our method yields additional practical advantages. It is the first global trajectory optimization method indifferent to the forms of the optimal controls, requiring no prior knowledge and dealing naturally with unbounded controls, singular arcs, and certain types of control constraints. By augmenting the state space to represent additional objective functions, it can provide adaptive sampling enclosures of a bounded Pareto front, directly according to the refinement of the state space and independent of further user input. Finally, its persistent data structures built on state space decomposition can provide reusable "maps" indicating regions of interest, that can jump-start refinement for related trajectory optimization problems with small variations in their defining parameters, as may readily arise in engineering design. We demonstrate the behavior of our method first on two simple trajectory optimization problems (single- and multiobjective) for illustrative purposes, and then on two more complex problems (single- and multiobjective) related to current problems of interest in astrodynamics and robotics (respectively). In each case, our results prove consistent with known or strongly conjectured solutions for these problems obtained from highly problem-specific analysis, and overcome the apparent limitations of a benchmark direct multiple shooting method. We also discuss the potential for our method to address important open problems in spaceflight trajectory optimization, given future work to improve the scalability of our implementation.<br>by Thomas Merritt Coffee.<br>Ph. D.
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Lee, Hyesuk Kwon. "Optimization Based Domain Decomposition Methods for Linear and Nonlinear Problems." Diss., Virginia Tech, 1997. http://hdl.handle.net/10919/30696.
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Optimization based domain decomposition methods for the solution of partial differential equations are considered. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations.
First, we consider a linear constraint. The existence of optimal solutions for the optimization problem is shown as is its convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as is an eminently parallelizable gradient method for solving the optimality system. The linear constraint minimization problem is also recast as a linear least squares problem and is solved by a conjugate gradient method.
The domain decomposition method can be extended to nonlinear problems such as the Navier-Stokes equations. This results from the fact that the objective functional for the minimization problem involves the jump in dependent variables across the interfaces between subdomains. Thus, the method does not require that the partial differential equations themselves be derivable through an extremal problem.
An optimality system is derived by applying a Lagrange multiplier rule to a constrained optimization problem. Error estimates for finite element approximations are presented as is a gradient method to solve the optimality system. We also use a Gauss-Newton method to solve the minimization problem with the nonlinear constraint.<br>Ph. D.
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Haskara, Ibrahim. "Sliding mode estimation and optimization methods in nonlinear control problems." The Ohio State University, 1999. http://rave.ohiolink.edu/etdc/view?acc_num=osu1250272986.
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Haskara, ?brahim. "Sliding mode estimation and optimization methods in nonlinear control problems /." The Ohio State University, 1999. http://rave.ohiolink.edu/etdc/view?acc_num=osu1488192960166775.
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Herbold, Eric B. "Optimization of the dynamic behavior of strongly nonlinear heterogeneous materials." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2008. http://wwwlib.umi.com/cr/ucsd/fullcit?p3320788.
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Thesis (Ph. D.)--University of California, San Diego, 2008.<br>Title from first page of PDF file (viewed November 13, 2008). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 284-298).
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Haskara, İbrahim. "Sliding mode estimation and optimization methods in nonlinear control problems /." Connect to resource, 1999. http://rave.ohiolink.edu/etdc/view.cgi?acc%5Fnum=osu1250272986.
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