A Collection of Challenging Optimization Problems in Science, Engineering and Economics

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE Congress on Evolutionary Computation (at Sendai International Center, Sendai, Japan, from 25th to 28th May, 2015.

Improving Tatonnement Methods for Solving Heterogeneous Agent Models

This paper modifies standard block Gauss-Seidel iterations used by tatonnement methods for solving large scale deterministic heterogeneous agent models. The composite method between first- and second-order tatonnement methods is shown to considerably improve convergence both in terms of speed as well as robustness relative to conventional first-order tatonnement methods. In addition, the relative advantage of the modified algorithm increases in the size and complexity of the economic model. Therefore, the algorithm allows significant reductions in computational time when solving large models. The algorithm is particularly attractive since it is easy to implement - it only augments conventional and intuitive tatonnement iterations with standard numerical methods.

Improving Tatonnement Methods of Solving Heterogeneous Agent Models

This paper modifies standard block Gauss-Seidel iterations used by tatonnement methods for solving large scale deterministic heterogeneous agent models. The composite method between first- and second-order tatonnement methods is shown to considerably improve convergence both in terms of speed as well as robustness relative to conventional first-order tatonnement methods. In addition, the relative advantage of the modified algorithm increases in the size and complexity of the economic model. Therefore, the algorithm allows significant reductions in computational time when solving large models. The algorithm is particularly attractive since it is easy to implement – it only augments conventional and intuitive tatonnement iterations with standard numerical methods.

A hybrid GA–PS–SQP method to solve power system valve-point economic dispatch problems

This study presents a new approach based on a hybrid algorithm consisting of Genetic Algorithm (GA), Pattern Search (PS) and Sequential Quadratic Programming (SQP) techniques to solve the well-known power system Economic dispatch problem (ED). GA is the main optimizer of the algorithm, whereas PS and SQP are used to fine tune the results of GA to increase confidence in the solution. For illustrative purposes, the algorithm has been applied to various test systems to assess its effectiveness. Furthermore, convergence characteristics and robustness of the proposed method have been explored through comparison with results reported in literature. The outcome is very encouraging and suggests that the hybrid GA–PS–SQP algorithm is very efficient in solving power system economic dispatch problem

Scalable numerical approach for the steady-state ab initio laser theory

We present an efficient and flexible method for solving the non-linear lasing equations of the steady-state ab initio laser theory. Our strategy is to solve the underlying system of partial differential equations directly, without the need of setting up a parametrized basis of constant flux states. We validate this approach in one-dimensional as well as in cylindrical systems, and demonstrate its scalability to full-vector three-dimensional calculations in photonic-crystal slabs. Our method paves the way for efficient and accurate simulations of lasing structures which were previously inaccessible.Comment: 17 pages, 8 figure

Methods for suspensions of passive and active filaments

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid--structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation

Swarm Intelligence and Metaphorless Algorithms for Solving Nonlinear Equation Systems

The simplicity, flexibility, and ease of implementation have motivated the use of population-based metaheuristic optimization algorithms. By focusing on two classes of such algorithms, particle swarm optimization (PSO) and the metaphorless Jaya algorithm, this thesis proposes to explore the capacity of these algorithms and their respective variants to solve difficult optimization problems, in particular systems of nonlinear equations converted into nonlinear optimization problems. For a numerical comparison to be made, the algorithms and their respective variants were implemented and tested several times in order to achieve a large sample that could be used to compare these approaches as well as find common methods that increase the effectiveness and efficiency of the algorithms. One of the approaches that was explored was dividing the solution search space into several subspaces, iteratively running an optimization algorithm on each subspace, and comparing those results to a greatly increased initial population. The insights from these previous experiments were then used to create a new hybrid approach to enhance the capabilities of the previous algorithms, which was then compared to preexisting alternatives.A simplicidade, flexibilidade e facilidade de implementa¸c˜ao motivou o uso de algoritmos metaheur´ısticos de optimiza¸c˜ao baseados em popula¸c˜oes. Focando-se em dois destes algoritmos, optimiza¸c˜ao por exame de part´ıculas (PSO) e no algoritmo Jaya, esta tese prop˜oe explorar a capacidade destes algoritmos e respectivas variantes para resolver problemas de optimiza¸c˜ao de dif´ıcil resolu¸c˜ao, em particular sistemas de equa¸c˜oes n˜ao lineares convertidos em problemas de optimiza¸c˜ao n˜ao linear. Para que fosse poss´ıvel fazer uma compara¸c˜ao num´erica, os algoritmos e respectivas variantes foram implementados e testados v´arias vezes, de modo a que fosse obtida uma amostra suficientemente grande de resultados que pudesse ser usada para comparar as diferentes abordagens, assim como encontrar m´etodos que melhorem a efic´acia e a eficiˆencia dos algoritmos. Uma das abordagens exploradas foi a divis˜ao do espa¸co de procura em v´arios subespa¸cos, iterativamente correndo um algoritmo de optimiza¸c˜ao em cada subespa¸co, e comparar esses resultados a um grande aumento da popula¸c˜ao inicial, o que melhora a qualidade da solu¸c˜ao, por´em com um custo computacional acrescido. O conhecimento resultante dessas experiˆencias foi utilizado na cria¸c˜ao de uma nova abordagem hibrida para melhorar as capacidades dos algoritmos anteriores, a qual foi comparada a alternativas pr´e-existentes

A sharp-front moving boundary model for malignant invasion

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction-diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c \ge c^* > 0$, the moving boundary model leads to travelling wave solutions with $|c| < c^{**}$. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.Comment: 48 pages, 21 figure

Numerical computation of rare events via large deviation theory

An overview of rare events algorithms based on large deviation theory (LDT) is presented. It covers a range of numerical schemes to compute the large deviation minimizer in various setups, and discusses best practices, common pitfalls, and implementation trade-offs. Generalizations, extensions, and improvements of the minimum action methods are proposed. These algorithms are tested on example problems which illustrate several common difficulties which arise e.g. when the forcing is degenerate or multiplicative, or the systems are infinite-dimensional. Generalizations to processes driven by non-Gaussian noises or random initial data and parameters are also discussed, along with the connection between the LDT-based approach reviewed here and other methods, such as stochastic field theory and optimal control. Finally, the integration of this approach in importance sampling methods using e.g. genealogical algorithms is explored

Improving Tatonnement Methods for Solving Heterogeneous Agent Models

This paper modifies standard block Gauss-Seidel iterations used by tatonnement methods for solving large scale deterministic heterogeneous agent models. The composite method between first- and second-order tatonnement methods is shown to considerably improve convergence both in terms of speed as well as robustness relative to conventional first-order tatonnement methods. In addition, the relative advantage of the modified algorithm increases in the size and complexity of the economic model. Therefore, the algorithm allows significant reductions in computational time when solving large models. The algorithm is particularly attractive since it is easy to implement - it only augments conventional and intuitive tatonnement iterations with standard numerical methods