The design and applications of the african buffalo algorithm for general optimization problems

Optimization, basically, is the economics of science. It is concerned with the need to maximize profit and minimize cost in terms of time and resources needed to execute a given project in any field of human endeavor. There have been several scientific investigations in the past several decades on discovering effective and efficient algorithms to providing solutions to the optimization needs of mankind leading to the development of deterministic algorithms that provide exact solutions to optimization problems. In the past five decades, however, the attention of scientists has shifted from the deterministic algorithms to the stochastic ones since the latter have proven to be more robust and efficient, even though they do not guarantee exact solutions. Some of the successfully designed stochastic algorithms include Simulated Annealing, Genetic Algorithm, Ant Colony Optimization, Particle Swarm Optimization, Bee Colony Optimization, Artificial Bee Colony Optimization, Firefly Optimization etc. A critical look at these ‘efficient’ stochastic algorithms reveals the need for improvements in the areas of effectiveness, the number of several parameters used, premature convergence, ability to search diverse landscapes and complex implementation strategies. The African Buffalo Optimization (ABO), which is inspired by the herd management, communication and successful grazing cultures of the African buffalos, is designed to attempt solutions to the observed shortcomings of the existing stochastic optimization algorithms. Through several experimental procedures, the ABO was used to successfully solve benchmark optimization problems in mono-modal and multimodal, constrained and unconstrained, separable and non-separable search landscapes with competitive outcomes. Moreover, the ABO algorithm was applied to solve over 100 out of the 118 benchmark symmetric and all the asymmetric travelling salesman’s problems available in TSPLIB95. Based on the successful experimentation with the novel algorithm, it is safe to conclude that the ABO is a worthy contribution to the scientific literature

Optimality conditions for convex stochastic optimization problems in Banach spaces with almost sure state constraint

We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty

Numerical Methods for PDE Constrained Optimization with Uncertain Data

Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization. The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods

Energy resource management in smart buildings considering photovoltaic uncertainty

O aumento do consumo energético em edifícios residenciais tem levado a um maior foco nos métodos de eficiência energética. Deste modo, surge um sistema de gestão de energia residencial que poderá permitir controlar os recursos energéticos em pequena escala dos edifícios, levando a uma diminuição significativa dos custos energéticos através de um escalonamento eficiente. No entanto, a natureza intermitente das fontes de energia renováveis resulta num problema complexo. Para resolver este desafio, esta tese propõe um escalonamento energético baseado na otimização robusta, considerando a incerteza relacionada com a produção fotovoltaica. A otimização robusta é um método emergente e eficaz para lidar com a incerteza e apresenta soluções ótimas considerando o pior cenário da incerteza, ou seja, encontra a melhor solução entre todos os piores cenários possíveis. Um problema de Programação Linear Binária é inicialmente formulado para minimizar os custos do escalonamento energético. De seguida, o objetivo desta tese é transformar o modelo determinístico num problema robusto equivalente para proporcionar-lhe imunidade contra a incerteza associada à produção fotovoltaica. O modelo determinístico é, assim, transformado num modelo do pior cenário possível. Para validar a eficiência e a eficácia do modelo, a metodologia proposta foi implementada em dois cenários sendo cada um deles constituído por três casos de estudo de escalonamento de energia, para um horizonte de escalonamento a curto prazo. Os resultados da simulação demonstram que a abordagem robusta consegue, efetivamente, minimizar os custos totais de eletricidade do edifício, mitigando, simultaneamente, os obstáculos referentes à incerteza relacionada com a produção fotovoltaica. É também demonstrado que a estratégia desenvolvida permite o ajustamento do escalonamento dos recursos energéticos do edifício de acordo com o nível de robustez selecionado.The increase of energy demand in residential buildings has led to a higher focus on energy efficiency methods. This way, the home energy management system arises to control small-scale energy resources on buildings allowing a significant electricity bill decrease throughout efficient scheduling. However, the intermittent and uncertain nature of renewable energy sources results in a complex problem. To solve this challenge, this thesis proposes robust optimization-based scheduling considering the uncertainty in solar generation. Robust Optimization is a very recent and effective technique to deal with uncertainty and provides optimal solutions for the worst-case realization of the uncertain parameter, i.e., it finds the best solution among all the worst scenarios. A Mixed Binary Linear Programming problem is initially formulated to minimize the costs of the energy resource scheduling. Then, this thesis's purpose is to transform the deterministic model into a trackable robust counterpart problem to provide immunity against the photovoltaic output uncertainty. The deterministic model is transformed into the worst-case model. To validate the model’s efficiency and effectiveness, the proposed methodology was implemented in two scenarios with three different energy scheduling case studies for a short-term scheduling horizon. The simulation results demonstrate that the robust approach can effectively minimize the electricity costs of the building while mitigating the drawbacks associated with solar uncertainty. It also proves that the proposed strategy adjusts the energy scheduling according to the selected robustness level

Numerical methods in stochastic and two-scale shape optimization

In this thesis, three different models of elastic shape optimization are described. All models use phase fields to describe the elastic shapes and regularize the interface length on some level or scale to control fine-scale structures. First, the paradigm of stochastic dominance is transferred from finite dimensional stochastic programming to elastic shape optimization under stochastic loads. The shapes are optimized under the constraint, that they dominate a given benchmark shape in a certain stochastic order. This allows for a flexible risk aversion comparison. Risk aversion is handled in the constraint rather than the objective functional, which results in an optimization over a subset of admissible shapes only. First and second order stochastic dominance constraints are examined and compared. An (adaptive) Q1 finite element scheme is used, that was implemented for two of the models described in this thesis and is introduced here. Several stochastic loads setups and benchmark variables are discretized and optimized. Starting with the observation that unregularized elastic shape optimization methods create arbitrarily fine micro-structures in many scenarios, domains composited of a number of geometrical subdomains with prescribed boundary conditions are considered in the second model. A reference subdomain is mapped to each type of geometrical subdomains to optimize computational complexity. These are suitable to model fine-scale elastic structures, that are widespread in nature. Examples are fine-scale structures in bones or plants, resulting from the need for a stiff and low-weight structure. The subdomains are coupled to simulate fine-scale structures as they appear e. g. in bones (branching periodic structures). The elastic shape is optimized only for those reference subdomains, simulating periodically repeating structures in one or more coordinate directions. The stress is supposed to be continuous over the domain. A stress-based finite volume discretization and an alternating optimization algorithm are used to find optimal elastic structures for compression and shear loads. Finally, a model considering a fine-scale material in which the elastic shape is modeled by a phase field on the microscale is introduced. This approach further investigates the fine-scale structures mentioned above and allows for a comparison with laminated materials and previous work on homogenization. A short introduction into homogenization is given and the two-scale energies required for the optimization are derived and discretized. An estimation of the scale between macro- and microscale is derived and a finite element discretization using the Heterogeneous Multiscale Method is introduced. Numerical results for compression and shear loads are presented

Variational Methods in Shape Space

This dissertation deals with the application of variational methods in spaces of geometric shapes. In particular, the treated topics include shape averaging, principal component analysis in shape space, computation of geodesic paths in shape space, as well as shape optimisation. Chapter 1 provides a brief overview over the employed models of shape space. Geometric shapes are identified with two- or three-dimensional, deformable objects. Deformations will be described via physical models; in particular, the objects will be interpreted as consisting of either a hyperelastic solid or a viscous liquid material. Furthermore, the description of shapes via phase fields or level sets is briefly introduced. Chapter 2 reviews different and related approaches to shape space modelling. References to related topics in image segmentation and registration are also provided. Finally, the relevant shape optimisation literature is introduced. Chapter 3 recapitulates the employed concepts from continuum mechanics and phase field modelling and states basic theoretical results needed for the later analysis. Chapter 4 addresses the computation of shape averages, based on a hyperelastic notion of shape dissimilarity: The dissimilarity between two shapes is measured as the minimum deformation energy required to deform the first into the second shape. A corresponding phase-field model is introduced, analysed, and finally implemented numerically via finite elements. A principal component analysis of shapes, which is consistent with the previously introduced average, is considered in Chapter 5. Elastic boundary stresses on the average shape are used as representatives of the input shapes in a linear vector space. On these linear representatives, a standard principal component analysis can be performed, where the employed covariance metric should be properly chosen to depend on the input shapes. Chapter 6 interprets shapes as belonging to objects made of a viscous liquid and correspondingly defines geodesic paths between shapes. The energy of a path is given as the total physical dissipation during the deformation of an object along the path. A rigid body motion invariant time discretisation is achieved by approximating the dissipation along a path segment by the deformation energy of a small solid deformation. The numerical implementation is based on level sets. Chapter 7 is concerned with the optimisation of the geometry and topology of solid structures that are subject to a mechanical load. Given the load configuration, the structure rigidity, its volume, and its surface area shall be optimally balanced. A phase field model is devised and analysed for this purpose. In this context, the use of nonlinear elasticity allows to detect buckling phenomena which would be ignored in linearised elasticity

Numerical Methods for Optimal Transport and Elastic Shape Optimization

In this thesis, we consider a novel unbalanced optimal transport model incorporating singular sources, we develop a numerical computation scheme for an optimal transport distance on graphs, we propose a simultaneous elastic shape optimization problem for bone tissue engineering, and we investigate optimal material distributions on thin elastic objects. The by now classical theory of optimal transport admits a metric between measures of the same total mass. Various generalizations of this so-called Wasserstein distance have been recently studied in the literature. In particular, these have been motivated by imaging applications, where the mass-preserving condition is too restrictive. Based on the Benamou Brenier formulation we present a novel unbalanced optimal transport model by introducing a source term in the continuity equation, which is incorporated in the path energy by a squared L2-norm in time of a functional with linear growth in space. As a key advantage of our model, this source term functional allows singular sources in space. We demonstrate the existence of constant speed geodesics in the space of Radon measures. Furthermore, for a numerical computation scheme, we apply a proximal splitting algorithm for a finite element discretization. On discrete spaces, Maas introduced a Benamou Brenier formulation, where a kinetic energy is defined via an appropriate (e.g., logarithmic) averaging of mass on nodes and momentum on edges. Concerning a numerical optimization scheme, this, unfortunately, couples all these variables on the graph. We propose a conforming finite element discretization in time and prove convergence of corresponding path energy minimizing curves. To apply a proximal splitting algorithm, we introduce suitable auxiliary variables. Besides similar projections as for the classical optimal transport distance and additional simple operations, this allows us to separate the nonlinearity given by the averaging operator to projections onto three-dimensional convex sets, the associated (e.g., logarithmic) cones. In elastic shape optimization, we are usually concerned with finding a subdomain maximizing the mechanical stability w.r.t. given forces acting onto a larger domain of interest. Motivated by a biomechanical application in bone tissue engineering, where recently biologically degradable polymers have been explored as bone substitutes, we propose a simultaneous elastic shape optimization problem to guarantee stiffness of the polymer implant and of the complementary set where new bone tissue will grow first. Under the assumption that the microstructure of the scaffold is periodic, we optimize a single microcell. We define a novel cost functional depending on specific entries of the homogenized elasticity tensors of polymer and regrown bone. Additionally, the perimeter is penalized for regularizing the interface of the scaffold. For a numerical optimization scheme, we choose a phase-field model, which allows a diffuse approximation of the elastic objects and the perimeter by the Modica Mortola functional. We also incorporate further biomechanically relevant constraints like the diffusivity of the regrown bone. Finally, we investigate shape optimization problems for thin elastic objects. For a numerical discretization, we take into account the discrete Kirchhoff triangle (DKT) element for parametric surfaces and approximate the material distribution by a phase-field. To describe equilibrium deformations for a given force, we study different corresponding state equations. In particular, we consider nonlinear elasticity combining membrane and bending models. Furthermore, a special focus is on pure bending isometries, which can be efficiently approximated by the DKT element. We also analyze a one-dimensional model of nonlinear elastic planar beams, where our numerical simulations confirm and extend a theoretical classification result of the optimal design