Efficient Optimization Algorithms for Nonlinear Data Analysis

Identification of low-dimensional structures and main sources of variation from multivariate data are fundamental tasks in data analysis. Many methods aimed at these tasks involve solution of an optimization problem. Thus, the objective of this thesis is to develop computationally efficient and theoretically justified methods for solving such problems. Most of the thesis is based on a statistical model, where ridges of the density estimated from the data are considered as relevant features. Finding ridges, that are generalized maxima, necessitates development of advanced optimization methods. An efficient and convergent trust region Newton method for projecting a point onto a ridge of the underlying density is developed for this purpose. The method is utilized in a differential equation-based approach for tracing ridges and computing projection coordinates along them. The density estimation is done nonparametrically by using Gaussian kernels. This allows application of ridge-based methods with only mild assumptions on the underlying structure of the data. The statistical model and the ridge finding methods are adapted to two different applications. The first one is extraction of curvilinear structures from noisy data mixed with background clutter. The second one is a novel nonlinear generalization of principal component analysis (PCA) and its extension to time series data. The methods have a wide range of potential applications, where most of the earlier approaches are inadequate. Examples include identification of faults from seismic data and identification of filaments from cosmological data. Applicability of the nonlinear PCA to climate analysis and reconstruction of periodic patterns from noisy time series data are also demonstrated. Other contributions of the thesis include development of an efficient semidefinite optimization method for embedding graphs into the Euclidean space. The method produces structure-preserving embeddings that maximize interpoint distances. It is primarily developed for dimensionality reduction, but has also potential applications in graph theory and various areas of physics, chemistry and engineering. Asymptotic behaviour of ridges and maxima of Gaussian kernel densities is also investigated when the kernel bandwidth approaches infinity. The results are applied to the nonlinear PCA and to finding significant maxima of such densities, which is a typical problem in visual object tracking.

Column Generation Algorithms for Nonlinear Optimization II: Numerical Investigations

García et al. present a class of column generation (CG) algorithms for nonlinear programs. Its main motivation from a theoretical viewpoint is that under some circumstances, finite convergence can be achieved, in much the same way as for the classic simplicial decomposition method; the main practical motivation is that within the class there are certain nonlinear column generation problems that can accelerate the convergence of a solution approach which generates a sequence of feasible points. This algorithm can, for example, accelerate simplicial decomposition schemes by making the subproblems nonlinear. This paper complements the theoretical study on the asymptotic and finite convergence of these methods given in [1] with an experimental study focused on their computational efficiency. Three types of numerical experiments are conducted. The first group of test problems has been designed to study the parameters involved in these methods. The second group has been designed to investigate the role and the computation of the prolongation of the generated columns to the relative boundary. The last one has been designed to carry out a more complete investigation of the difference in computational efficiency between linear and nonlinear column generation approaches. In order to carry out this investigation, we consider two types of test problems: the first one is the nonlinear, capacitated single-commodity network flow problem of which several large-scale instances with varied degrees of nonlinearity and total capacity are constructed and investigated, and the second one is a combined traffic assignment model

Web-based application programming interface to solve nonlinear optimization problems

Nonlinear Optimization Problems are usual in many engineering fields. Due to its characteristics the objective function of some problems might not be differentiable or its derivatives have complex expressions. There are even cases where an analytical expression of the objective function might not be possible to determine either due to its complexity or its cost (monetary, computational, time, ...). In these cases Nonlinear Optimization methods must be used. An API, including several methods and algorithms to solve constrained and unconstrained optimization problems was implemented. This API can be accessed not only as traditionally, by installing it on the developer and/or user computer, but it can also be accessed remotely using Web Services. As long as there is a network connection to the server where the API is installed, applications always access to the latest API version. Also an Web-based application, using the proposed API, was developed. This application is to be used by users that do not want to integrate methods in applications, and simply want to have a tool to solve Nonlinear Optimization Problems.

A neural system to robust Nonlinear optimization subject to disjoint and constrained sets

The ability of neural networks to realize some complex nonlinear function makes them attractive for system identification. This paper describes a novel method using artificial neural networks to solve robust parameter estimation problems for nonlinear models with unknown-but-bounded errors and uncertainties. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the network convergence to the equilibrium points. A solution for the robust estimation problem with unknown-but-bounded error corresponds to an equilibrium point of the network. Simulation results are presented as an illustration of the proposed approach.

New Optimization Algorithms for Structural Reliability Analysis

Solution of structural reliability problems by the First Order method require optimization algorithms to find the smallest distance between a limit state function and the origin of standard Gaussian space. The Hassofer-Lind-Rackwitz-Fiessler (HLRF) algorithm, developed specifically for this purpose, has been shown to be efficient but not robust, as it fails to converge for a significant number of problems. On the other hand, recent developments in general (augmented Lagrangian) optimization techniques have not been tested in aplication to structural reliability problems. In the present article, three new optimization algorithms for structural reliability analysis are presented. One algorithm is based on the HLRF, but uses a new differentiable merit function with Wolfe conditions to select step length in linear search. It is shown in the article that, under certain assumptions, the proposed algorithm generates a sequence that converges to the local minimizer of the problem. Two new augmented Lagrangian methods are also presented, which use quadratic penalties to solve nonlinear problems with equality constraints. Performance and robustness of the new algorithms is compared to the classic augmented Lagrangian method, to HLRF and to the improved HLRF (iHLRF) algorithms, in the solution of 25 benchmark problems from the literature. The new proposed HLRF algorithm is shown to be more robust than HLRF or iHLRF, and as efficient as the iHLRF algorithm. The two augmented Lagrangian methods proposed herein are shown to be more robust and more efficient than the classical augmented Lagrangian method.

Nonlinear optimization method for short-term hydro scheduling considering head-dependency

This paper is on the problem of short-term hydro scheduling, particularly concerning head-dependent reservoirs under competitive environment. We propose a new nonlinear optimization method to consider hydroelectric power generation as a function of water discharge and also of the head. Head-dependency is considered on short-term hydro scheduling in order to obtain more realistic and feasible results. The proposed method has been applied successfully to solve a case study based on one of the main Portuguese cascaded hydro systems, providing a higher profit at a negligible additional computation time in comparison with a linear optimization method that ignores head-dependency.

Derivative-free nonlinear optimization filter simplex

The filter method is a technique for solving nonlinear programming problems. The filter algorithm has two phases in each iteration. The first one reduces a measure of infeasibility, while in the second the objective function value is reduced. In real optimization problems, usually the objective function is not differentiable or its derivatives are unknown. In these cases it becomes essential to use optimization methods where the calculation of the derivatives or the verification of their existence is not necessary: direct search methods or derivative-free methods are examples of such techniques. In this work we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of simplex and filter methods. This method neither computes nor approximates derivatives, penalty constants or Lagrange multipliers.

Towards Human-like Bimanual Movements in Anthropomorphic Robots: A Nonlinear Optimization Approach

Previously we have presented a model for generating human-like arm and hand movements on an unimanual anthropomorphic robot involved in human-robot collaboration tasks. The present paper aims to extend our model in order to address the generation of human-like bimanual movement sequences which are challenged by scenarios cluttered with obstacles. Movement planning involves large scale nonlinear constrained optimization problems which are solved using the IPOPT solver. Simulation studies show that the model generates feasible and realistic hand trajectories for action sequences involving the two hands. The computational costs involved in the planning allow for real-time human robot-interaction. A qualitative analysis reveals that the movements of the robot exhibit basic characteristics of human movements.

Global vs. local nonlinear optimization techniques for human-like movement of an anthropomorphic robot

In this paper a comparison between using global and local optimization techniques for solving the problem of generating human-like arm and hand movements for an anthropomorphic dual arm robot is made. Although the objective function involved in each optimization problem is convex, there is no evidence that the admissible regions of these problems are convex sets. For the sequence of movements for which the numerical tests were done there were no significant differences between the optimal solutions obtained using the global and the local techniques. This suggests that the optimal solution obtained using the local solver is indeed a global solution.

Nonlinear optimization for human-like synchronous movements of a dual arm-hand robotic system

In previous work we have presented a model capable of generating human-like movements for a dual arm-hand robot involved in human-robot cooperative tasks. However, the focus was on the generation of reach-to-grasp and reach-to-regrasp bimanual movements and no synchrony in timing was taken into account. In this paper we extend the previous model in order to accomplish bimanual manipulation tasks by synchronously moving both arms and hands of an anthropomorphic robotic system. Specifically, the new extended model has been designed for two different tasks with different degrees of difficulty. Numerical results were obtained by the implementation of the IPOPT solver embedded in our MATLAB simulator.

Towards human-like bimanual movements in anthropomorphic robots: a nonlinear optimization approach

Previously we have presented a model for generating human-like arm and hand movements on an unimanual anthropomorphic robot involved in human-robot collaboration tasks. The present paper aims to extend our model in order to address the generation of human-like bimanual movement sequences which are challenged by scenarios cluttered with obstacles. Movement planning involves large scale nonlinear constrained optimization problems which are solved using the IPOPT solver. Simulation studies show that the model generates feasible and realistic hand trajectories for action sequences involving the two hands. The computational costs involved in the planning allow for real-time human robot-interaction. A qualitative analysis reveals that the movements of the robot exhibit basic characteristics of human movements.

Revisiting optimization algorithms for maximum likelihood estimation

Parmi les méthodes d’estimation de paramètres de loi de probabilité en statistique, le maximum de vraisemblance est une des techniques les plus populaires, comme, sous des conditions l´egères, les estimateurs ainsi produits sont consistants et asymptotiquement efficaces. Les problèmes de maximum de vraisemblance peuvent être traités comme des problèmes de programmation non linéaires, éventuellement non convexe, pour lesquels deux grandes classes de méthodes de résolution sont les techniques de région de confiance et les méthodes de recherche linéaire. En outre, il est possible d’exploiter la structure de ces problèmes pour tenter d’accélerer la convergence de ces méthodes, sous certaines hypothèses. Dans ce travail, nous revisitons certaines approches classiques ou récemment d´eveloppées en optimisation non linéaire, dans le contexte particulier de l’estimation de maximum de vraisemblance. Nous développons également de nouveaux algorithmes pour résoudre ce problème, reconsidérant différentes techniques d’approximation de hessiens, et proposons de nouvelles méthodes de calcul de pas, en particulier dans le cadre des algorithmes de recherche linéaire. Il s’agit notamment d’algorithmes nous permettant de changer d’approximation de hessien et d’adapter la longueur du pas dans une direction de recherche fixée. Finalement, nous évaluons l’efficacité numérique des méthodes proposées dans le cadre de l’estimation de modèles de choix discrets, en particulier les modèles logit mélangés.

Global Optimization Algorithms and their Application to Distributed Systems

In this report, we discuss the application of global optimization and Evolutionary Computation to distributed systems. We therefore selected and classified many publications, giving an insight into the wide variety of optimization problems which arise in distributed systems. Some interesting approaches from different areas will be discussed in greater detail with the use of illustrative examples.

Optimization algorithms in FMRF model-based segmentation for LIDAR data and co-registered bands

In this paper, a fuzzy Markov random field (FMRF) model is used to segment land-objects into free, grass, building, and road regions by fusing remotely, sensed LIDAR data and co-registered color bands, i.e. scanned aerial color (RGB) photo and near infra-red (NIR) photo. An FMRF model is defined as a Markov random field (MRF) model in a fuzzy domain. Three optimization algorithms in the FMRF model, i.e. Lagrange multiplier (LM), iterated conditional mode (ICM), and simulated annealing (SA), are compared with respect to the computational cost and segmentation accuracy. The results have shown that the FMRF model-based ICM algorithm balances the computational cost and segmentation accuracy in land-cover segmentation from LIDAR data and co-registered bands.

Distributed parallelization of greedy Mobile Network Optimization algorithms

The Mobile Network Optimization (MNO) technologies have advanced at a tremendous pace in recent years. And the Dynamic Network Optimization (DNO) concept emerged years ago, aimed to continuously optimize the network in response to variations in network traffic and conditions. Yet, DNO development is still at its infancy, mainly hindered by a significant bottleneck of the lengthy optimization runtime. This paper identifies parallelism in greedy MNO algorithms and presents an advanced distributed parallel solution. The solution is designed, implemented and applied to real-life projects whose results yield a significant, highly scalable and nearly linear speedup up to 6.9 and 14.5 on distributed 8-core and 16-core systems respectively. Meanwhile, optimization outputs exhibit self-consistency and high precision compared to their sequential counterpart. This is a milestone in realizing the DNO. Further, the techniques may be applied to similar greedy optimization algorithm based applications.