Low-rank optimization with trace norm penalty

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach that outperforms the naive warm-restart approach on the fixed-rank quotient manifold. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression. © 2013 Society for Industrial and Applied Mathematics.

Manopt, a matlab toolbox for optimization on manifolds

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design effcient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. By dealing internally with most of the differential geometry, the package aims particularly at lowering the entrance barrier. © 2014 Nicolas Boumal.

Design and optimization of highly nonlinear low-dispersion crystal fiber with high birefringence for four-wave mixing

We have proposed a novel type of photonic crystal fiber (PCF) with low dispersion and high nonlinearity for four-wave mixing. This type of fiber is composed of a solid silica core and a cladding with a squeezed hexagonal lattice elliptical airhole along the fiber length. Its dispersion and nonlinearity coefficient are investigated simultaneously by using the full vectorial finite element method. Numerical results show that the proposed highly nonlinear low-dispersion fiber has a total dispersion as low as +/- 2.5 ps nm(-1) km(-1) over an ultrabroad wavelength range from 1.43 to 1.8 mu m, and the corresponding nonlinearity coefficient and birefringence are about 150 W-1 km(-1) and 2.5 x 10(-3) at 1.55 mu m, respectively. The proposed PCF with low ultraflattened dispersion, high nonlinearity, and high birefringence can have important application in four-wave mixing. (C) 2010 Optical Society of America

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation (CNOP), which is a nonlinear generalization of the linear singular vector (LSV), is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming (SQP) algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, and the spectral projected gradients (SPG2) algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

Fast divide-and-conquer algorithms for preemptive scheduling problems with controllable processing times - a polymatroid optimization approach

We consider a variety of preemptive scheduling problems with controllable processing times on a single machine and on identical/uniform parallel machines, where the objective is to minimize the total compression cost. In this paper, we propose fast divide-and-conquer algorithms for these scheduling problems. Our approach is based on the observation that each scheduling problem we discuss can be formulated as a polymatroid optimization problem. We develop a novel divide-and-conquer technique for the polymatroid optimization problem and then apply it to each scheduling problem. We show that each scheduling problem can be solved in $ \O({\rm T}_{\rm feas}(n) \times\log n)$ time by using our divide-and-conquer technique, where n is the number of jobs and Tfeas(n) denotes the time complexity of the corresponding feasible scheduling problem with n jobs. This approach yields faster algorithms for most of the scheduling problems discussed in this paper.

On Infinite Time Performance of Nonlinear Model Predictive Controllers

This paper addresses the problem of infinite time performance of model predictive controllers applied to constrained nonlinear systems. The total performance is compared with a finite horizon optimal cost to reveal performance limits of closed-loop model predictive control systems. Based on the Principle of Optimality, an upper and a lower bound of the ratio between the total performance and the finite horizon optimal cost are obtained explicitly expressed by the optimization horizon. The results also illustrate, from viewpoint of performance, how model predictive controllers approaches to infinite optimal controllers as the optimization horizon increases.

Geometric optimization on visibility problems: metaheuristic and exact solutions

Os problemas de visibilidade têm diversas aplicações a situações reais. Entre os mais conhecidos, e exaustivamente estudados, estão os que envolvem os conceitos de vigilância e ocultação em estruturas geométricas (problemas de vigilância e ocultação). Neste trabalho são estudados problemas de visibilidade em estruturas geométricas conhecidas como polígonos, uma vez que estes podem representar, de forma apropriada, muitos dos objectos reais e são de fácil manipulação computacional. O objectivo dos problemas de vigilância é a determinação do número mínimo de posições para a colocação de dispositivos num dado polígono, de modo a que estes dispositivos consigam “ver” a totalidade do polígono. Por outro lado, o objectivo dos problemas de ocultação é a determinação do número máximo de posições num dado polígono, de modo a que quaisquer duas posições não se consigam “ver”. Infelizmente, a maior parte dos problemas de visibilidade em polígonos são NP-difíceis, o que dá origem a duas linhas de investigação: o desenvolvimento de algoritmos que estabelecem soluções aproximadas e a determinação de soluções exactas para classes especiais de polígonos. Atendendo a estas duas linhas de investigação, o trabalho é dividido em duas partes. Na primeira parte são propostos algoritmos aproximados, baseados essencialmente em metaheurísticas e metaheurísticas híbridas, para resolver alguns problemas de visibilidade, tanto em polígonos arbitrários como ortogonais. Os problemas estudados são os seguintes: “Maximum Hidden Vertex Set problem”, “Minimum Vertex Guard Set problem”, “Minimum Vertex Floodlight Set problem” e “Minimum Vertex k-Modem Set problem”. São também desenvolvidos métodos que permitem determinar a razão de aproximação dos algoritmos propostos. Para cada problema são implementados os algoritmos apresentados e é realizado um estudo estatístico para estabelecer qual o algoritmo que obtém as melhores soluções num tempo razoável. Este estudo permite concluir que as metaheurísticas híbridas são, em geral, as melhores estratégias para resolver os problemas de visibilidade estudados. Na segunda parte desta dissertação são abordados os problemas “Minimum Vertex Guard Set”, “Maximum Hidden Set” e “Maximum Hidden Vertex Set”, onde são identificadas e estudadas algumas classes de polígonos para as quais são determinadas soluções exactas e/ou limites combinatórios.

Optimal control and numerical optimization applied to epidemiological models

A relação entre a epidemiologia, a modelação matemática e as ferramentas computacionais permite construir e testar teorias sobre o desenvolvimento e combate de uma doença. Esta tese tem como motivação o estudo de modelos epidemiológicos aplicados a doenças infeciosas numa perspetiva de Controlo Ótimo, dando particular relevância ao Dengue. Sendo uma doença tropical e subtropical transmitida por mosquitos, afecta cerca de 100 milhões de pessoas por ano, e é considerada pela Organização Mundial de Saúde como uma grande preocupação para a saúde pública. Os modelos matemáticos desenvolvidos e testados neste trabalho, baseiam-se em equações diferenciais ordinárias que descrevem a dinâmica subjacente à doença nomeadamente a interação entre humanos e mosquitos. É feito um estudo analítico dos mesmos relativamente aos pontos de equilíbrio, sua estabilidade e número básico de reprodução. A propagação do Dengue pode ser atenuada através de medidas de controlo do vetor transmissor, tais como o uso de inseticidas específicos e campanhas educacionais. Como o desenvolvimento de uma potencial vacina tem sido uma aposta mundial recente, são propostos modelos baseados na simulação de um hipotético processo de vacinação numa população. Tendo por base a teoria de Controlo Ótimo, são analisadas as estratégias ótimas para o uso destes controlos e respetivas repercussões na redução/erradicação da doença aquando de um surto na população, considerando uma abordagem bioeconómica. Os problemas formulados são resolvidos numericamente usando métodos diretos e indiretos. Os primeiros discretizam o problema reformulando-o num problema de optimização não linear. Os métodos indiretos usam o Princípio do Máximo de Pontryagin como condição necessária para encontrar a curva ótima para o respetivo controlo. Nestas duas estratégias utilizam-se vários pacotes de software numérico. Ao longo deste trabalho, houve sempre um compromisso entre o realismo dos modelos epidemiológicos e a sua tratabilidade em termos matemáticos.

Self-adaptive combination of global tabu search and local search for nonlinear equations

Solving systems of nonlinear equations is a very important task since the problems emerge mostly through the mathematical modelling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a self-adaptive combination of a metaheuristic with a classical local search method is able to converge to some difficult problems that are not solved by Newton-type methods.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

RBF equalizer design using directional evolutionary multi-objective optimization

Whilst radial basis function (RBF) equalizers have been employed to combat the linear and nonlinear distortions in modern communication systems, most of them do not take into account the equalizer's generalization capability. In this paper, it is firstly proposed that the. model's generalization capability can be improved by treating the modelling problem as a multi-objective optimization (MOO) problem, with each objective based on one of several training sets. Then, as a modelling application, a new RBF equalizer learning scheme is introduced based on the directional evolutionary MOO (EMOO). Directional EMOO improves the computational efficiency of conventional EMOO, which has been widely applied in solving MOO problems, by explicitly making use of the directional information. Computer simulation demonstrates that the new scheme can be used to derive RBF equalizers with good performance not only on explaining the training samples but on predicting the unseen samples.

Solving optimal control problems with state constraints using nonlinear programming and simulation tools

This paper illustrates how nonlinear programming and simulation tools, which are available in packages such as MATLAB and SIMULINK, can easily be used to solve optimal control problems with state- and/or input-dependent inequality constraints. The method presented is illustrated with a model of a single-link manipulator. The method is suitable to be taught to advanced undergraduate and Master's level students in control engineering.

On combination of SMOTE and particle swarm optimization based radial basis function classifier for imbalanced problems

The combination of the synthetic minority oversampling technique (SMOTE) and the radial basis function (RBF) classifier is proposed to deal with classification for imbalanced two-class data. In order to enhance the significance of the small and specific region belonging to the positive class in the decision region, the SMOTE is applied to generate synthetic instances for the positive class to balance the training data set. Based on the over-sampled training data, the RBF classifier is constructed by applying the orthogonal forward selection procedure, in which the classifier structure and the parameters of RBF kernels are determined using a particle swarm optimization algorithm based on the criterion of minimizing the leave-one-out misclassification rate. The experimental results on both simulated and real imbalanced data sets are presented to demonstrate the effectiveness of our proposed algorithm.