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The hydrogen production curve of the electrolyzer describes the non-linear and non-convex relationship between its power consumption and hydrogen production. An accurate representation of this curve is essential for the optimal scheduling of the electrolyzer. The current state-of-the-art approach is based on piece-wise linear approximation, which requires binary variables and does not scale well for large-scale problems. To overcome this barrier, we propose two models, both built upon convex relaxations of the hydrogen production curve. The first one is a linear relaxation of the piece-wise linear approximation, while the second one is a conic relaxation of a quadratic approximation. Both relaxations are exact under prevalent operating conditions. We prove this mathematically for the conic relaxation. Using a realistic case study, we show that the conic model, in comparison to the other models, provides a satisfactory trade-off between computational complexity and solution accuracy for large-scale problems.

The rise of flexible payloads on satellites opens a door for controlling satellite resources according to the user demand, user location, and satellite position. In addition to resource management, applying precoding on flexible payloads is essential to obtain high spectral efficiency. However, these cannot be achieved using a conventional resource allocation algorithm that does not consider the user demand. In this paper, we propose a demand-aware algorithm based on multiobjective optimization to jointly design the carrier allocation and precoding for better spectral efficiency and demand matching with proper management of the satellite resources. The optimization problem is non-convex, and we solve it using convex relaxation and successive convex approximation. Then, we evaluate the performance of the proposed algorithm through numerical results. It is shown that the proposed method outperforms the benchmark schemes in terms of resource utilization and demand satisfaction.

This work establishes and validates a Grid Graph Signal Processing (G-GSP) framework for estimating the state vector of a radial distribution feeder. One of the key insights from GSP is the generalization of Shannon's sampling theorem for signals defined over the irregular support of a graph, such as the power grid. Using a GSP interpretation of Ohm's law, we show that the system state can be well approximated with relatively few components that correspond to low-pass Graph Fourier Transform (GFT) frequencies. The target application of this theory is the formulation of a three-phase unbalanced Distribution System State Estimation (DSSE) formulation that recovers the GFT approximation of the system state vector from sparse Advanced Metering Infrastructure (AMI) measurements. To ensure convergence of G-GSP for DSSE, the proposed solution relies on a convex relaxation technique. Furthermore, we propose an optimal sensor placement algorithm for AMI measurements. Numerical results demonstrate the efficacy of the proposed method., QC 20220909

The rise of flexible payloads on satellites opens a door for controlling satellite resources according to the user demand, user location, and satellite position. In addition to resource management, applying precoding on flexible payloads is essential to obtain high spectral efficiency. However, these cannot be achieved using a conventional resource allocation algorithm that does not consider the user demand. In this paper, we propose a demand-aware algorithm based on multiobjective optimization to jointly design the carrier allocation and precoding for better spectral efficiency and demand matching with proper management of the satellite resources. The optimization problem is non-convex, and we solve it using convex relaxation and successive convex approximation. Then, we evaluate the performance of the proposed algorithm through numerical results. It is shown that the proposed method outperforms the benchmark schemes in terms of resource utilization and demand satisfaction.

The rise of flexible payloads on satellites opens a door for controlling satellite resources according to the user demand, user location, and satellite position. In addition to resource management, applying precoding on flexible payloads is essential to obtain high spectral efficiency. However, these cannot be achieved using a conventional resource allocation algorithm that does not consider the user demand. In this paper, we propose a demand-aware algorithm based on multiobjective optimization to jointly design the carrier allocation and precoding for better spectral efficiency and demand matching with proper management of the satellite resources. The optimization problem is non-convex, and we solve it using convex relaxation and successive convex approximation. Then, we evaluate the performance of the proposed algorithm through numerical results. It is shown that the proposed method outperforms the benchmark schemes in terms of resource utilization and demand satisfaction.

We aim to find a solution x is an element of Cn to a system of quadratic equations of the form bi=\aix\2, i=1,2,...,m, e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as IncrePR, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, restart-IncrePR solving three consecutive PhaseLift problems outperforms state-of-the-art nonconvex gradient flow based solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, IncrePR with several consecutive repeats can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex gradient flow based solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.

We aim to find a solution x is an element of Cn to a system of quadratic equations of the form bi=\aix\2, i=1,2,...,m, e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as IncrePR, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, restart-IncrePR solving three consecutive PhaseLift problems outperforms state-of-the-art nonconvex gradient flow based solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, IncrePR with several consecutive repeats can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex gradient flow based solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.

Massive infrastructure networks such as electric power, natural gas, or water systems play a pivotal role in everyday human lives. Development and operation of these networks is extremely capital-intensive. Moreover, security and reliability of these networks is critical. This work identifies and addresses a diverse class of computationally challenging and time-critical problems pertaining to these networks. This dissertation extends the state of the art on three fronts. First, general proofs of uniqueness for network flow problems are presented, thus addressing open problems. Efficient network flow solvers based on energy function minimizations, convex relaxations, and mixed-integer programming are proposed with performance guarantees. Second, a novel approach is developed for sample-efficient training of deep neural networks (DNN) aimed at solving optimal network dispatch problems. The novel feature here is that the DNNs are trained to match not only the minimizers, but also their sensitivities with respect to the optimization problem parameters. Third, control mechanisms are designed that ensure resilient and stable network operation. These novel solutions are bolstered by mathematical guarantees and extensive simulations on benchmark power, water, and natural gas networks.

Advances in data-driven methods have sparked renewed interest for applications in power systems. Creating datasets for successful application of these methods has proven to be very challenging, especially when considering power system security. This paper proposes a computationally efficient method to create datasets of secure and insecure operating points. We propose an infeasibility certificate based on separating hyperplanes that can a-priori characterize large parts of the input space as insecure, thus significantly reducing both computation time and problem size. Our method can handle an order of magnitude more control variables and creates balanced datasets of secure and insecure operating points, which is essential for data-driven applications. While we focus on N-1 security and uncertainty, our method can extend to dynamic security. For PGLib-OPF networks up to 500 buses and up to 125 control variables, we demonstrate drastic reductions in unclassified input space volumes and computation time, create balanced datasets, and evaluate an illustrative data-driven application.

We aim to find a solution x is an element of Cn to a system of quadratic equations of the form bi=\aix\2, i=1,2,...,m, e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as IncrePR, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, restart-IncrePR solving three consecutive PhaseLift problems outperforms state-of-the-art nonconvex gradient flow based solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, IncrePR with several consecutive repeats can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex gradient flow based solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.

Existing energy networks can foster the integration of uncertain and variable renewable energy sources by providing additional operational flexibility. In this direction, we propose a combined power, heat, and natural gas dispatch model to reveal the maximum potential “network flexibility”, corresponding to the ability of natural gas and district heating pipelines to store energy. To account for both energy transport and linepack in the pipelines in a computational efficient manner, we explore convex quadratic relaxations of the nonconvex flow dynamics of gas and heat. The resulting model is a mixed-integer secondorder cone program. An ex-post analysis ensures feasibility of the heat dispatch, while keeping the relaxation of the gas flow model sufficiently tight. The revealed flexibility is quantified in terms of system cost compared to a dispatch model neglecting the ability of natural gas and district heating networks to store energy.

The combined operation of integrated energy systems is increasingly becoming a crucial topic for renewable energy dominated power systems operation. Flexibility from the district heating system could be used to deal with the uncertainty of renewable energy sources. We formulate a distributionally robust optimization problem for co-optimizing energy and reserve dispatch of the integrated electricity and heating system with a moment-based ambiguity set. The reserve allocation has been modeled through the participation vectors of the controllable generation units. The total reserve capacity has been defined implicitly and is a function of the uncertainty. The proposed model has been transformed into a second-order cone programming (SOCP) optimization problem by applying convex relaxation and linearization of the district heating network equations. Case studies on the integrated six-bus and seven-node system to demonstrate the efficacy of the proposed model.

Power system security assessment methods require large datasets of operating points to train or test their performance. As historical data often contain limited number of abnormal situations, simulation data are necessary to accurately determine the security boundary. Generating such a database is an extremely demanding task, which becomes intractable even for small system sizes. This paper proposes a modular and highly scalable algorithm for computationally efficient database generation. Using convex relaxation techniques and complex network theory, we discard large infeasible regions and drastically reduce the search space. We explore the remaining space by a highly parallelizable algorithm and substantially decrease computation time. Our method accommodates numerous definitions of power system security. Here we focus on the combination of N-k security and small-signal stability. Demonstrating our algorithm on IEEE 14-bus and NESTA 162-bus systems, we show how it outperforms existing approaches requiring less than 10% of the time other methods require.

This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classic

This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classic

In current practice, the day-ahead market-clearing outcomes are not necessarily feasible for distribution networks, i.e, the network constraints might not be satisfied. Hence, the distribution system operator may consider an ex-post re-dispatch mechanism, exploiting potential flexibility of local distributed energy resources (DERs) including demand response (DR) units. Many DR units have an inherent "rebound effect", meaning a decrease in power demand (response) must be followed by an increase (rebound) or vice-versa, due to their underlying physical properties. A naive re-dispatch mechanism relying on DR units with non-negligible rebound effect may fail, since those units may cause another congestion in the rebound period. We propose a mechanism, which models the rebound effect of DR units using asymmetric block offers — this way, those units offer their flexibility using two subsequent blocks (response and rebound), each one representing the load decrease/increase in a time period. We demonstrate that though linear approximations of optimal power flow (OPF) models as potential re-dispatch mechanisms are more computationally efficient, they can result in a different dispatch of the asymmetric blocks than an exact convex relaxation of an AC-OPF model, and therefore, must be used with caution

Cette thèse porte sur l'élaboration et l'étude de modèles représentant l'alimentation de porcs en croissance. Elle présente deux objectifs principaux. Un objectif pratique et un objectif théorique. L'objectif pratique consiste à développer de nouvelles méthodes d'alimentation a n de réduire les coûts d'alimentation et de prendre en compte les rejets environnementaux. L'objectif théorique consiste en une étude approfondie du modèle bilinéaire appliqué à l'industrie porcine a n d'obtenir des résultats tant numériques que théoriques. Aujourd'hui, l'alimentation représente plus de 70% du coût de production en engraissement porcin et dans le contexte économique actuel, il est important de parvenir à réduire ce coût. L'alimentation utilisée actuellement dans l'industrie porcine est une alimentation par phase, représentée par un modèle linéaire. L'alimentation par mélanges telle qu'elle a été introduite récemment est représentée par une modèle bilinéaire et permet une réduction du coût de l'alimentation de plus de 4%. Nous introduisons ici une nouvelle méthode de formulation qui est une combinaison de l'alimentation traditionnelle et de l'alimentation par mélanges : la méthode hybride. Nous montrons que la mise en place d'une telle alimentation permet de réaliser une économie de plus de 5%. L'industrie porcine étant fréquemment citée comme activit é polluante, notamment à cause de l'apport important de phosphates et de nitrates lors de l'épandage de lisier, nous devons faire notre possible pour réduire la quantité de phosphore et d'azote contenue dans le lisier. Une façon de réduire ces rejets est d'agir sur l'alimentation. Nous utilisons une approche tricritère, la méthode des ε-contraintes, a n d'obtenir un spectre des solutions possibles permettant de réduire les rejets. Cette approche permet de tracer la surface des courbes de niveaux et nous pouvons ainsi proposer un graphique facilitant la lecture des résultats. Chaque utilisateur pourra alors déterminer le coût d'un, This thesis deals with the development and the study of new models representing the feeding system of growing- nishing pigs. There are two main goals. A practical objective and a theoretical objective, a practical one and a theoretical one. The practical aim consists in developing new feed formulation that reduces feed costs and that consider environmental excretion. The theoretical aim consists in a study of a bilinear diet problem applied in the pig industry to get numerical and theoretical results. Today, feed represents more than 70% of the production cost in growing- nishing pig industry. In the current economic context, it is important to reduce it. The feeding system currently used in the industry is a phase feeding system, modeled in a linear program. The feeding system using feeds introduced recently is modeled by a bilinear program and leads to a feed cost reduction of 4%. We introduce here a new feeding system which is a combination of the traditional feeding system using phases and the feeding system using feeds. We call it the hybrid feeding system. We show that use this feeding system leads to a feed cost reduction of more than 5%. As the pig industry is frequently cited as a major polluting activity, in particular because of the high level of phosphates and nitrates in the manure spreading, we must do our best to reduce the phosphorus and nitrogen excretion. One way to reduce them is to act on the feed. We use a tricriteria approach, the ϵ-constraints method, to get a spectrum of feasible solutions that reduce excretion. Using this approach, we can plot contour lines and thus have a graph facilitating the results reading. Every user can then determine the feed cost that corresponds to the wanted excretion. We applied this method on the feeding system using feeds and the hybrid feeding system. The main part of this manuscript focuses on the global optimization, and then non-convex, that model the feeding system using feeds. The bilinearity ap

The sparse generalized eigenvalue problem (GEP) plays a pivotal role in a large family of high dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis and sufficient dimension reduction. The sparse GEP involves solving a non-convex optimization problem. Most existing methods and theory in the context of specific statistical models that are special cases of the sparse GEP require restrictive structural assumptions on the input matrices. We propose a two-stage computational framework to solve the sparse GEP. At the first stage, we solve a convex relaxation of the sparse GEP. Taking the solution as an initial value, we then exploit a non-convex optimization perspective and propose the truncated Rayleigh flow method (which we call ‘rifle’) to estimate the leading generalized eigenvector. We show that rifle converges linearly to a solution with the optimal statistical rate of convergence. Theoretically, our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices. To achieve this, our analysis involves two key ingredients: a new analysis of the gradient-based method on non-convex objective functions, and a fine-grained characterization of the evolution of sparsity patterns along the solution path. Thorough numerical studies are provided to validate the theoretical results. © 2018 Royal Statistical Society

Under the envisioned smart city paradigm, there is an increasing demand for the coordinated operation of our infrastructure networks. In this context, this thesis puts forth a comprehensive toolbox for the optimization of electric power and water distribution networks. On the analytical front, the toolbox consists of novel mixed-integer (non)-linear program (MINLP) formulations; convex relaxations with optimality guarantees; and the powerful technique of McCormick linearization. On the application side, the developed tools support the operation of each of the infrastructure networks independently, but also towards their joint operation. Starting with water distribution networks, the main difficulty in solving any (optimal-) water flow problem stems from a piecewise quadratic pressure drop law. To efficiently handle these constraints, we have first formulated a novel MINLP, and then proposed a relaxation of the pressure drop constraints to yield a mixed-integer second-order cone program. Further, a novel penalty term is appended to the cost that guarantees optimality and exactness under pre-defined network conditions. This contribution can be used to solve the WF problem; the OWF task of minimizing the pumping cost satisfying operational constraints; and the task of scheduling the operation of tanks to maximize the water service time in an area experiencing electric power outage. Regarding electric power systems, a novel MILP formulation for distribution restoration using binary indicator vectors on graph properties alongside exact McCormick linearization is proposed. This can be used to minimize the restoration time of an electric system under critical operational constraints, and to enable a coordinated response with the water utilities during outages.

The sparse generalized eigenvalue problem (GEP) plays a pivotal role in a large family of high dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis and sufficient dimension reduction. The sparse GEP involves solving a non-convex optimization problem. Most existing methods and theory in the context of specific statistical models that are special cases of the sparse GEP require restrictive structural assumptions on the input matrices. We propose a two-stage computational framework to solve the sparse GEP. At the first stage, we solve a convex relaxation of the sparse GEP. Taking the solution as an initial value, we then exploit a non-convex optimization perspective and propose the truncated Rayleigh flow method (which we call ‘rifle’) to estimate the leading generalized eigenvector. We show that rifle converges linearly to a solution with the optimal statistical rate of convergence. Theoretically, our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices. To achieve this, our analysis involves two key ingredients: a new analysis of the gradient-based method on non-convex objective functions, and a fine-grained characterization of the evolution of sparsity patterns along the solution path. Thorough numerical studies are provided to validate the theoretical results. © 2018 Royal Statistical Society

In this paper, we propose a data-driven preventive security-constrained AC optimal power flow (SC-OPF), which ensures small-signal stability and N-1 security. Our approach can be used by both system and market operators for optimizing redispatch or AC based market-clearing auctions. We derive decision trees from large datasets of operating points, which capture all security requirements and allow to define tractable decision rules that are implemented in the SC-OPF using mixed-integer nonlinear programming (MINLP). We propose a second-order cone relaxation for the non-convex MINLP, which allows us to translate the non-convex and possibly disjoint feasible space of secure system operation to a convex mixed-integer OPF formulation. Our case study shows that the proposed approach increases the feasible space represented in the SC-OPF compared to conventional methods, can identify the global optimum as opposed to tested MINLP solvers and significantly reduces computation time due to a decreased problem size.

Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results.

Contex: Spectrum polling is universally known as the main Cognitive Radio (CR) enabler, since it provides the CR device with the ability to know the radio environment.Objetive: This article presents an algorithm design to perform the broadband spectrum probing in Cognitive Radio systems. Method: Spectrum polling is based on Compressive Sensing (CS), by which cognitive users minimize the amount of samples to be processed, without the need for a priori knowledge of signal characteristics in the radio environment. In this way, it is possible to proportionally reduce detection times, power consumption and processing capacity required in cognitive radio devices (CRD).Results: The performance of the proposed algorithm is evaluated by obtaining the probability of detection, the probability of non-detection, the probability of false alarm, and the Receiver Operating Characteristics (ROC), and comparing it with other algorithms proposed in the state of the art.Conclusion: The simulation results demonstrate that the proposed method allows the efficient sampling of the spectrum. This improves the probe performance based on the probability of detection and the Receiver Operating Characteristic ROC, and it is better than the other proposed algorithms based on sub-Nyquist sampling., Contexto: El sondeo de espectro es universalmente conocido como el principal habilitador de radio cognitivo (CR), dado que dota al dispositivo de CR de la capacidad de conocimiento del entorno de radio.Objetivo: En este artículo se presenta un algoritmo diseñado para realizar el sondeo de espectro de banda ancha en sistemas de radio cognitivo (CR).Método: El sondeo de espectro está basado en sondeo compresivo (CS), mediante el cual los usuarios cognitivos, sin necesidad de conocimiento a priori de características de la señal en el entorno de radio, minimizan la cantidad de muestras a procesar y, con ello, reducen de forma proporcional los tiempos de detección, consumos de energía y capacidades de procesamiento requeridas en los dispositivos de radio cognitivo (CRD).Resultados: El desempeño del algoritmo propuesto se evalúa mediante la obtención de las curvas de probabilidad de detección, probabilidad de omisión de detección, probabilidad de falsa alarma y las características operativas del receptor (ROC) en comparación con otros algoritmos propuestos en el estado del arte.Conclusión: Los resultados de simulación demuestran que el método propuesto permite realizar el sondeo de espectro de manera eficiente, mejorando el desempeño del sondeo en función de la probabilidad de detección y de las características operacionales del receptor con respecto a otros algoritmos propuestos basados en muestreo sub-Nyquist.

Contex: Spectrum polling is universally known as the main Cognitive Radio (CR) enabler, since it provides the CR device with the ability to know the radio environment.Objetive: This article presents an algorithm design to perform the broadband spectrum probing in Cognitive Radio systems. Method: Spectrum polling is based on Compressive Sensing (CS), by which cognitive users minimize the amount of samples to be processed, without the need for a priori knowledge of signal characteristics in the radio environment. In this way, it is possible to proportionally reduce detection times, power consumption and processing capacity required in cognitive radio devices (CRD).Results: The performance of the proposed algorithm is evaluated by obtaining the probability of detection, the probability of non-detection, the probability of false alarm, and the Receiver Operating Characteristics (ROC), and comparing it with other algorithms proposed in the state of the art.Conclusion: The simulation results demonstrate that the proposed method allows the efficient sampling of the spectrum. This improves the probe performance based on the probability of detection and the Receiver Operating Characteristic ROC, and it is better than the other proposed algorithms based on sub-Nyquist sampling., Contexto: El sondeo de espectro es universalmente conocido como el principal habilitador de radio cognitivo (CR), dado que dota al dispositivo de CR de la capacidad de conocimiento del entorno de radio.Objetivo: En este artículo se presenta un algoritmo diseñado para realizar el sondeo de espectro de banda ancha en sistemas de radio cognitivo (CR).Método: El sondeo de espectro está basado en sondeo compresivo (CS), mediante el cual los usuarios cognitivos, sin necesidad de conocimiento a priori de características de la señal en el entorno de radio, minimizan la cantidad de muestras a procesar y, con ello, reducen de forma proporcional los tiempos de detección, consumos de energía y capacidades de procesamiento requeridas en los dispositivos de radio cognitivo (CRD).Resultados: El desempeño del algoritmo propuesto se evalúa mediante la obtención de las curvas de probabilidad de detección, probabilidad de omisión de detección, probabilidad de falsa alarma y las características operativas del receptor (ROC) en comparación con otros algoritmos propuestos en el estado del arte.Conclusión: Los resultados de simulación demuestran que el método propuesto permite realizar el sondeo de espectro de manera eficiente, mejorando el desempeño del sondeo en función de la probabilidad de detección y de las características operacionales del receptor con respecto a otros algoritmos propuestos basados en muestreo sub-Nyquist.

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the present paper is to provide a Berry-Esseen bound for the normal approximation of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincar\'e inequality, (3) concentration estimates, and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy and Tropp (2014) about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the present paper is to provide a Berry-Esseen bound for the normal approximation of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincar\'e inequality, (3) concentration estimates, and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy and Tropp (2014) about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses given noisy measurements of a subset of their pairwise relative transforms. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up t

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses given noisy measurements of a subset of their pairwise relative transforms. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up t

Renewable energies are increasingly integrated in electric distribution networks and will cause severe overvoltage issues. Smart grid technologies make it possible to use coordinated control to mitigate the overvoltage issues and the optimal power flow (OPF) method is proven to be efficient in the applications such as curtailment management and reactive power control. Nonconvex nature of the OPF makes it difficult to solve and convex relaxation is a promising method to solve the OPF very efficiently. This paper investigates the geometry of the power flows and the convex-relaxed power flows when high penetration level of renewables is present in the distribution networks. The geometry study helps understand the fundamental nature of the OPF and its convex-relaxed problem, such as the second-order cone programming (SOCP) problem. A case study based on a three-node system is used to illustrate the geometry profile of the feasible sub-injection (injection of nodes excluding the root/substation node) region

It is commonly observed that over the lifetime of most model predictive controllers, the achieved performance degrades over time. This effect can often be attributed to the fact that the dynamics of the controlled plant change as the plant ages, due to wear and tear, refurbishment and design changes of the plant, to name a few factors. These changes mean that re-identification is necessary to restore the desired performance of the controller. An extension of existing predictive controllers, capable of producing signals suitable for closed loop re-identification, is presented in this article. The main contribution is an extensive experimental evaluation of the proposed controller for closed loop re-identification on an industrial depropanizer distillation column in simulations and in real experiments. The plant experiments are conducted on the depropanizer during normal plant operations. In the simulations, as well as in the experiments, the updated models from closed loop re-identification result in improvement of the performance. The algorithm used combines regular model predictive control with ideas from applications oriented input design and linear matrix inequality based convex relaxation techniques. Even though the experiments show promising result, some implementation problems arise and are discussed., QC 20150602

In the application of the Expectation Maximization (EM) algorithm to identification of dynamical systems, latent variables are typically taken as system states, for simplicity. In this work, we propose a different choice of latent variables, namely, system disturbances. Such a formulation is shown, under certain circumstances, to improve the fidelity of bounds on the likelihood, and circumvent difficulties related to intractable model transition densities. To access these benefits, we propose a Lagrangian relaxation of the challenging optimization problem that arises when formulating over latent disturbances, and fully develop the method for linear models.

We propose a graph cut based global minimization method for image segmentation by representing the segmentation label function with a series of nested binary super-level set functions. This representation enables us to use K - 1 binary functions to partition any images into K phases. Both continuous and discretized formulations will be treated. For the discrete model, we propose a new graph cut algorithm which is faster than the existing graph cut methods to obtain the exact global solution. In the continuous case, we further improve the segmentation accuracy using a number of techniques that are unique to the continuous segmentation models. With the convex relaxation and the dual method, the related continuous dual model is convex and we can mathematically show that the global minimization can be achieved. The corresponding continuous max-flow algorithms are easy and stable. Experimental results show that our model is very competitive to some existing methods. (C) 2014 Elsevier Inc. All rights reserved.

We propose a graph cut based global minimization method for image segmentation by representing the segmentation label function with a series of nested binary super-level set functions. This representation enables us to use K - 1 binary functions to partition any images into K phases. Both continuous and discretized formulations will be treated. For the discrete model, we propose a new graph cut algorithm which is faster than the existing graph cut methods to obtain the exact global solution. In the continuous case, we further improve the segmentation accuracy using a number of techniques that are unique to the continuous segmentation models. With the convex relaxation and the dual method, the related continuous dual model is convex and we can mathematically show that the global minimization can be achieved. The corresponding continuous max-flow algorithms are easy and stable. Experimental results show that our model is very competitive to some existing methods. (C) 2014 Elsevier Inc. All rights reserved.

This paper proposes a novel adversarial optimization approach to efficient outlier removal in computer vision. We characterize the outlier removal problem as a game that involves two players of conflicting interests, namely, model optimizer and outliers. Such an adversarial view not only brings new insights into some existing methods, but also gives rise to a general optimization framework that provably unifies them. Under the proposed framework, we develop a new outlier removal approach that is able to offer a much needed control over the trade-off between reliability and speed, which is usually not available in previous methods. Underlying the proposed approach is a mixed-integer minmax (convex-concave) problem formulation. Although a minmax problem is generally not amenable to efficient optimization, we show that for some commonly used vision objective functions, an equivalent Linear Program reformulation exists. This significantly simplifies the optimization. We demonstrate our method on two representative multiview geometry problems. Experiments on real image data illustrate superior practical performance of our method over recent techniques.

We propose a graph cut based global minimization method for image segmentation by representing the segmentation label function with a series of nested binary super-level set functions. This representation enables us to use K - 1 binary functions to partition any images into K phases. Both continuous and discretized formulations will be treated. For the discrete model, we propose a new graph cut algorithm which is faster than the existing graph cut methods to obtain the exact global solution. In the continuous case, we further improve the segmentation accuracy using a number of techniques that are unique to the continuous segmentation models. With the convex relaxation and the dual method, the related continuous dual model is convex and we can mathematically show that the global minimization can be achieved. The corresponding continuous max-flow algorithms are easy and stable. Experimental results show that our model is very competitive to some existing methods. (C) 2014 Elsevier Inc. All rights reserved.

Many problems in machine learning can be naturally formulated as non-convex optimization problems. However, such direct nonconvex formulations have been largely replaced by convex relaxation methods (such as support vector machines for classification or L1 regularization for sparse learning) to avoid the usual local optima issues. Many significant theoretical results have been developed in recent years to show that the convex relaxation approach solves the original problem asymptotically. However in practice, the standard simple convex relaxation schemes can be sub-optimal. In the proposed work, we consider a more general framework of multi-stage convex relaxation methods, which remedies the above gap between theory and practice. The method is derived from concave duality, and involves solving a sequence of convex relaxation problems, leading to better and better approximations to the original nonconvex formulation. We will develop theoretical properties of this method and algorithmic consequences. Related convex and nonconvex machine learning methods will also be investigated.

In this paper, we study the problem of learning from weakly labeled data, where labels of the training examples are incomplete. This includes, for example, (i) semi-supervised learning where labels are partially known; (ii) multi-instance learning where labels are implicitly known; and (iii) clustering where labels are completely unknown. Unlike supervised learning, learning with weak labels involves a difficult Mixed-Integer Programming (MIP) problem. Therefore, it can suffer from poor scalability and may also get stuck in local minimum. In this paper, we focus on SVMs and propose the WELLSVM via a novel label generation strategy. This leads to a convex relaxation of the original MIP, which is at least as tight as existing convex Semi-Definite Programming (SDP) relaxations. Moreover, the WELLSVM can be solved via a sequence of SVM subproblems that are much more scalable than previous convex SDP relaxations. Experiments on three weakly labeled learning tasks, namely, (i) semi-supervised learning; (ii) multi-instance learning for locating regions of interest in content-based information retrieval; and (iii) clustering, clearly demonstrate improved performance, and WELLSVM is also readily applicable on large data sets.

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):2540-2561, 2011), Leung and Zhao (J. Comput. Phys. 228:7706-7728, 2009a, J. Comput. Phys. 228:2993-3024, 2009b, Commun. Comput. Phys. 8:758-796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.

In this paper, we study the problem of learning from weakly labeled data, where labels of the training examples are incomplete. This includes, for example, (i) semi-supervised learning where labels are partially known; (ii) multi-instance learning where labels are implicitly known; and (iii) clustering where labels are completely unknown. Unlike supervised learning, learning with weak labels involves a difficult Mixed-Integer Programming (MIP) problem. Therefore, it can suffer from poor scalability and may also get stuck in local minimum. In this paper, we focus on SVMs and propose the WELLSVM via a novel label generation strategy. This leads to a convex relaxation of the original MIP, which is at least as tight as existing convex Semi-Definite Programming (SDP) relaxations. Moreover, the WELLSVM can be solved via a sequence of SVM subproblems that are much more scalable than previous convex SDP relaxations. Experiments on three weakly labeled learning tasks, namely, (i) semi-supervised learning; (ii) multi-instance learning for locating regions of interest in content-based information retrieval; and (iii) clustering, clearly demonstrate improved performance, and WELLSVM is also readily applicable on large data sets.

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):2540-2561, 2011), Leung and Zhao (J. Comput. Phys. 228:7706-7728, 2009a, J. Comput. Phys. 228:2993-3024, 2009b, Commun. Comput. Phys. 8:758-796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.

We investigate a class of variational problems that incorporate in some sense curvature information of the level lines. The functionals we consider incorporate metrics defined on the orientations of pairs of line segments that meet in the vertices of the level lines. We discuss two particular instances: One instance that minimizes the total number of vertices of the level lines and another instance that minimizes the total sum of the absolute exterior angles between the line segments. In case of smooth level lines, the latter corresponds to the total absolute curvature. We show that these problems can be solved approximately by means of a tractable convex relaxation in higher dimensions. In our numerical experiments we present preliminary results for image segmentation, image denoising and image inpainting.

We investigate a class of variational problems that incorporate in some sense curvature information of the level lines. The functionals we consider incorporate metrics defined on the orientations of pairs of line segments that meet in the vertices of the level lines. We discuss two particular instances: One instance that minimizes the total number of vertices of the level lines and another instance that minimizes the total sum of the absolute exterior angles between the line segments. In case of smooth level lines, the latter corresponds to the total absolute curvature. We show that these problems can be solved approximately by means of a tractable convex relaxation in higher dimensions. In our numerical experiments we present preliminary results for image segmentation, image denoising and image inpainting.

In this paper, we study the problem of learning from weakly labeled data, where labels of the training examples are incomplete. This includes, for example, (i) semi-supervised learning where labels are partially known; (ii) multi-instance learning where labels are implicitly known; and (iii) clustering where labels are completely unknown. Unlike supervised learning, learning with weak labels involves a difficult Mixed-Integer Programming (MIP) problem. Therefore, it can suffer from poor scalability and may also get stuck in local minimum. In this paper, we focus on SVMs and propose the WELLSVM via a novel label generation strategy. This leads to a convex relaxation of the original MIP, which is at least as tight as existing convex Semi-Definite Programming (SDP) relaxations. Moreover, the WELLSVM can be solved via a sequence of SVM subproblems that are much more scalable than previous convex SDP relaxations. Experiments on three weakly labeled learning tasks, namely, (i) semi-supervised learning; (ii) multi-instance learning for locating regions of interest in content-based information retrieval; and (iii) clustering, clearly demonstrate improved performance, and WELLSVM is also readily applicable on large data sets.

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):2540-2561, 2011), Leung and Zhao (J. Comput. Phys. 228:7706-7728, 2009a, J. Comput. Phys. 228:2993-3024, 2009b, Commun. Comput. Phys. 8:758-796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.

In this paper, we address the dependent joint bit allocation problem in H.264/AVC statistical multiplexing. In most existing methods, to improve the overall visual quality, the bit allocation is based upon the instantaneous relative frame complexity of different video programs. However, due to the temporal prediction employed in H. 264, the influence of a frame on the rate-distortion (characteristics of the future frames should be taken into account as well. In this paper, we use our previously proposed simple, but accurate, inter-frame dependency model (IFDM) to quantitatively measure the coding dependency between the current frame and its reference frame. Based on the IFDM, we formulate the dependent joint bit allocation problem, considering both the inter-program relative frame complexity and intra-program coding dependency. We prove that the dependent joint bit allocation (DeJoBA) problem can actually be formulated and relaxed into a convex optimization problem, which can be optimally and efficiently solved. Experimental results suggest that the proposed DeJoBA method can achieve 36.81% and 13.11% bitrate reduction, on average, compared with the equal bit allocation and optimal independent joint bit allocation methods, respectively.

Projecte realitzat en el marc d’un programa de mobilitat amb la Fakultät für Informätik de la Technischen Universität München, In this thesis a superresolution method consisting in a deblurring and an upsampling is used to improve the quality of a 3D reconstruction using data sampled with a Kinect. Results show that there is a significant quality improve.

In many practical situations, it is highly desirable to estimate an accurate mathematical model of a real system using as few parameters as possible. This can be motivated either from appealing to a parsimony principle (Occam's razor) or from the view point of the utilization complexity in terms of control synthesis, prediction, etc. At the same time, the need for an accurate description of the system behavior without knowing its complete dynamical structure often leads to model parameterizations describing a rich set of possible hypotheses; an unavoidable choice, which suggests sparsity of the desired parameter estimate. An elegant way to impose this expectation of sparsity is to estimate the parameters by penalizing the criterion with the ℓ 0 norm of the parameters, which is often implemented as solving an optimization program based on a convex relaxation (e.g. ℓ 1/ LASSO, nuclear norm, ⋯). However, in order to apply these methods, the (unpenalized) cost function must be convex. This imposes a severe constraint on the types of model structures or estimation methods on which these relaxations can be applied. In this paper, we extend the use of convex relaxation techniques for sparsity to general rational plant model structures estimated by using prediction error minimization. This is done by combining the LASSO and the Steiglitz-McBride approaches. To demonstrate the advantages of the proposed solution an extensive simulation study is provided., QC 20121122

The active contours without edges model of Chan and Vese (IEEE Transactions on Image Processing 10(2):266-277, 2001) is a popular method for computing the segmentation of an image into two phases, based on the piecewise constant Mumford-Shah model. The minimization problem is non-convex even when the optimal region constants are known a priori. In (SIAM Journal of Applied Mathematics 66(5):1632-1648, 2006), Chan, Esedolu, and Nikolova provided a method to compute global minimizers by showing that solutions could be obtained from a convex relaxation. In this paper, we propose a convex relaxation approach to solve the case in which both the segmentation and the optimal constants are unknown for two phases and multiple phases. In other words, we propose a convex relaxation of the popular K-means algorithm. Our approach is based on the vector-valued relaxation technique developed by Goldstein et al. (UCLA CAM Report 09-77, 2009) and Brown et al. (UCLA CAM Report 10-43, 2010). The idea is to consider the optimal constants as functions subject to a constraint on their gradient. Although the proposed relaxation technique is not guaranteed to find exact global minimizers of the original problem, our experiments show that our method computes tight approximations of the optimal solutions. Particularly, we provide numerical examples in which our method finds better solutions than the method proposed by Chan et al. (SIAM Journal of Applied Mathematics 66(5):1632-1648, 2006), whose quality of solutions depends on the choice of the initial condition. © 2011 Springer Science+Business Media, LLC.

The active contours without edges model of Chan and Vese (IEEE Transactions on Image Processing 10(2):266-277, 2001) is a popular method for computing the segmentation of an image into two phases, based on the piecewise constant Mumford-Shah model. The minimization problem is non-convex even when the optimal region constants are known a priori. In (SIAM Journal of Applied Mathematics 66(5):1632-1648, 2006), Chan, Esedolu, and Nikolova provided a method to compute global minimizers by showing that solutions could be obtained from a convex relaxation. In this paper, we propose a convex relaxation approach to solve the case in which both the segmentation and the optimal constants are unknown for two phases and multiple phases. In other words, we propose a convex relaxation of the popular K-means algorithm. Our approach is based on the vector-valued relaxation technique developed by Goldstein et al. (UCLA CAM Report 09-77, 2009) and Brown et al. (UCLA CAM Report 10-43, 2010). The idea is to consider the optimal constants as functions subject to a constraint on their gradient. Although the proposed relaxation technique is not guaranteed to find exact global minimizers of the original problem, our experiments show that our method computes tight approximations of the optimal solutions. Particularly, we provide numerical examples in which our method finds better solutions than the method proposed by Chan et al. (SIAM Journal of Applied Mathematics 66(5):1632-1648, 2006), whose quality of solutions depends on the choice of the initial condition. © 2011 Springer Science+Business Media, LLC.