Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last 7 years.

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CasADi: a software framework for nonlinear optimization and optimal control

Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems

Many criteria for statistical parameter estimation, such as maximum likelihood, are formulated as a nonlinear optimization problem. Automatic Differentiation Model Builder (ADMB) is a programming framework based on automatic differentiation, aimed at highly nonlinear models with a large number of parameters. The benefits of using AD are computational efficiency and high numerical accuracy, both crucial in many practical problems. We describe the basic components and the underlying philosophy of ADMB, with an emphasis on functionality found in no other statistical software. One example of such a feature is the generic implementation of Laplace approximation of high-dimensional integrals for use in latent variable models. We also review the literature in which ADMB has been used, and discuss future development of ADMB as an open source project. Overall, the main advantages of ADMB are flexibility, speed, precision, stability and built-in methods to quantify uncertainty.

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AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models

In this paper the software environment and algorithm collection ACADO Toolkit is presented, which implements tools for automatic control and dynamic optimization. It provides a general framework for using a great variety of algorithms for direct optimal control, including model predictive control as well as state and parameter estimation. The ACADO Toolkit is implemented as a self-contained C++ code, while the object-oriented design allows for convenient coupling of existing optimization packages and for extending it with user-written optimization routines. We discuss details of the software design of the ACADO Toolkit 1.0 and describe its main software modules. Along with that we highlight a couple of algorithmic features, in particular its functionality to handle symbolic expressions. The user-friendly syntax of the ACADO Toolkit to set up optimization problems is illustrated with two tutorial examples: an optimal control and a parameter estimation problem. Copyright © 2010 John Wiley & Sons, Ltd.

ACADO toolkit—An open-source framework for automatic control and dynamic optimization

Earlier work has shown that the efficient subspace method can be employed to reduce the effective size of the input data stream for high-dimensional models when the effective rank of the first-order sensitivity matrix is orders of magnitude smaller than the size of the input data. Here, the method is extended to handle nonlinear models, where the evaluation of higher-order derivatives is important but also challenging because the number of derivatives increases exponentially with the size of the input data streams. A recently developed hybrid approach is employed to combine reverse-mode automatic differentiation to calculate first-order derivatives and perform the required reduction in the input data stream, followed by forward-mode automatic differentiation to calculate higher-order derivatives with respect only to the reduced input variables. Three test cases illustrate the viability of the approach.

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Combining Automatic Differentiation Methods for High-Dimensional Nonlinear Models

We propose two frameworks to deal with problem settings in which both structured and unstructured data are available. Structured data problems are best solved by traditional machine learning models such as boosting and tree-based algorithms, whereas deep learning has been widely applied to problems dealing with images, text, audio, and other unstructured data sources. However, for the setting in which both structured and unstructured data are accessible, it is not obvious what the best modeling approach is to enhance performance on both data sources simultaneously. Our proposed frameworks allow joint learning on both kinds of data by integrating the paradigms of boosting models and deep neural networks. The first framework, the boosted-feature-vector deep learning network, learns features from the structured data using gradient boosting and combines them with embeddings from unstructured data via a two-branch deep neural network. Secondly, the two-weak-learner boosting framework extends the boosting paradigm to the setting with two input data sources. We present and compare first- and second-order methods of this framework. Our experimental results on both public and real-world datasets show performance gains achieved by the frameworks over selected baselines by magnitudes of 0.1% - 4.7%.

Gradient-Boosted Based Structured and Unstructured Learning

There are classification tasks that take as inputs groups of images rather than single images. In order to address such situations, we introduce a nested multi-view deep network. The approach is generic in that it is applicable to general data instances, not just images. The network has several convolutional neural networks grouped together at different stages. This primarily differs from other previous works in that we organize instances into relevant groups that are treated differently. We also introduce a method to replace instances that are missing which successfully creates neutral input instances and consistently outperforms standard fill-in methods in real world use cases. In addition, we propose a method for manual dropout when a whole group of instances is missing that allows us to use sparser training data and obtain higher accuracy at the end of training. With specific pretraining, we find that the model works to great effect on our real world and public datasets compared to baseline methods, with our improvements ranging from 1% to 5%.

Nested Multi-view Image Classification

Convolutional neural networks (CNNs) and transformers, which are composed of multiple processing layers and blocks to learn the representations of data with multiple abstract levels, are the most successful machine learning models in recent years. However, millions of parameters and many blocks make them difficult to be trained, and sometimes several days or weeks are required to find an ideal architecture or tune the parameters. Within this paper, we propose a new algorithm for boosting Deep Convolutional Neural Networks (BoostCNN) to combine the merits of dynamic feature selection and BoostCNN, and another new family of algorithms combining boosting and transformers. To learn these new models, we introduce subgrid selection and importance sampling strategies and propose a set of algorithms to incorporate boosting weights into a deep learning architecture based on a least squares objective function. These algorithms not only reduce the required manual effort for finding an appropriate network architecture but also result in superior performance and lower running time. Experiments show that the proposed methods outperform benchmarks on several finegrained classification tasks. 1 ar X iv :2 20 3. 00 76 1v 1 [ cs .L G ] 1 M ar 2 02 2

Tricks and Plugins to GBM on Images and Sequences

We propose a graph-oriented attention-based explainability method for tabular data. Tasks involving tabular data have been solved mostly using traditional tree-based machine learning models which have the challenges of feature selection and engineering. With that in mind, we consider a transformer architecture for tabular data, which is amenable to explainability, and present a novel way to leverage self-attention mechanism to provide explanations by taking into account the attention matrices of all layers as a whole. The matrices are mapped to a graph structure where groups of features correspond to nodes and attention values to arcs. By finding the maximum probability paths in the graph, we identify groups of features providing larger contributions to explain the model's predictions. To assess the quality of multi-layer attention-based explanations, we compare them with popular attention-, gradient-, and perturbation-based explanability methods.

Jean Utke

Papers

Combining Automatic Differentiation Methods for High-Dimensional Nonlinear Models

Gradient-Boosted Based Structured and Unstructured Learning

Nested Multi-view Image Classification

Tricks and Plugins to GBM on Images and Sequences

Multi-Layer Attention-Based Explainability via Transformers for Tabular Data