An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem

This paper presents a new class of multiple-follower bilevel problems and a heuristic approach to solving them. In this new class of problems, the followers may be nonlinear, do not share constraints or variables, and are at most weakly constrained. This allows the leader variables to be partitioned among the followers. We show that current approaches for solving multiple-follower problems are unsuitable for our new class of problems and instead we propose a novel analytics-based heuristic decomposition approach. This approach uses Monte Carlo simulation and k-medoids clustering to reduce the bilevel problem to a single level, which can then be solved using integer programming techniques. The examples presented show that our approach produces better solutions and scales up better than the other approaches in the literature. Furthermore, for large problems, we combine our approach with the use of self-organising maps in place of k-medoids clustering, which significantly reduces the clustering times. Finally, we apply our approach to a real-life cutting stock problem. Here a forest harvesting problem is reformulated as a multiple-follower bilevel problem and solved using our approachThis publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/228

Efficient Algorithms for Computationally Expensive Multifidelity Optimization Problems

Multifidelity optimization problems refer to a class of problems where one is presented with a physical system or mathematical model that can be represented in different levels of fidelity. The term “fidelity” refers to the accuracy of representation, where higher fidelity estimates are more accurate and expensive, while lower fidelity estimates are inaccurate, albeit cheaper. Most common iterative solvers such as those employed in computational fluid dynamics (CFD), finite element analysis (FEA), computational electromagnetics (CEM) etc. can be run with different fine/course meshes or residual error thresholds to yield estimates in various fidelities. In the event an optimization exercise requires their use, it is possible to invoke analysis in various fidelities for different solutions during the course of search. Multifidelity optimization algorithms are the special class of algorithms that are able to deal with analysis in various levels of fidelity. In this thesis, two novel multifidelity optimization algorithms have been developed. The first is to deal with bilevel optimization problems and the second is to deal with robust optimization problems involving iterative solvers. Bilevel optimization problems are particularly challenging as the optimum of an upper level (UL) problem is sought subject to the optimality of a nested lower level (LL) problem. Due to the inherent nested nature, naive implementations consume very significant number of UL and LL evaluations. The proposed multifidelity approach controls the rigour of LL optimization exercise for any given UL solution during the course of search as opposed to undertaking exhaustive LL optimization for every UL solution. Robust optimization problems are yet another class of problems where numerous solutions need to be assessed since the intent is to identify solutions that have both good performance and is also insensitive to unavoidable perturbations in the variable values. Computing the latter metric requires evaluation of numerous solutions in the vicinity of the given solution and not all solutions are worthy of such computation. The proposed multifidelity approach considers pre-converged simulations as lower fidelity estimates and uses them to reduce the computational overhead. While multi-objective optimization problems have long been in existence, there has been limited attempts in the past to deal with problems where the objectives can be independently computed. For example, the weight of a structure and the maximum stress in the structure are two objectives that can be independently computed. For such classes of problems, an efficient algorithm should ideally evaluate either one or both objectives as opposed of always evaluating both objectives. A novel algorithm is introduced that is capable of selectively evaluating the objectives of the infill solutions. The approach exploits principles of non-dominance and sparse subset selection to facilitate decomposition and through maximization of probabilistic dominance (PD) measure, identifies the infill solutions. Thereafter, for each of these infill solutions, one or more objectives are evaluated based on evaluation status of its closest neighbor and the probability of improvement along each objective. Finally, there has been significant research interest in recent years to develop efficient algorithms to deal with multimodal, multi-objective optimization problems (MMOPs). Such problems are particulatly challenging as there is a need to identify well distributed and well converged solutions in the objective space along with diverse solutions in the variable space. Existing algorithms for MMOPs still require prohibitive number of function evaluations (often in several thousands). The algorithms are typically embedded with sophisticated, customized mechanisms that require additional parameters to manage the diversity and convergence in the variable and the objective spaces. A steady-state evolutionary algorithm is introduced in this thesis for solving MMOPs, with a simple design and no additional user-defined parameters that need tuning. All the developments listed above have been studied using well established benchmarks and real-world examples. The results have been compared with existing state-of-the-art approaches to substantiate the benefits

Seeking multiple solutions:an updated survey on niching methods and their applications

Multi-Modal Optimization (MMO) aiming to locate multiple optimal (or near-optimal) solutions in a single simulation run has practical relevance to problem solving across many fields. Population-based meta-heuristics have been shown particularly effective in solving MMO problems, if equipped with specificallydesigned diversity-preserving mechanisms, commonly known as niching methods. This paper provides an updated survey on niching methods. The paper first revisits the fundamental concepts about niching and its most representative schemes, then reviews the most recent development of niching methods, including novel and hybrid methods, performance measures, and benchmarks for their assessment. Furthermore, the paper surveys previous attempts at leveraging the capabilities of niching to facilitate various optimization tasks (e.g., multi-objective and dynamic optimization) and machine learning tasks (e.g., clustering, feature selection, and learning ensembles). A list of successful applications of niching methods to real-world problems is presented to demonstrate the capabilities of niching methods in providing solutions that are difficult for other optimization methods to offer. The significant practical value of niching methods is clearly exemplified through these applications. Finally, the paper poses challenges and research questions on niching that are yet to be appropriately addressed. Providing answers to these questions is crucial before we can bring more fruitful benefits of niching to real-world problem solving

Development of Methods for Solving Bilevel Optimization Problems

Bilevel optimization, also referred to as bilevel programming, involves solving an upper level problem subject to the optimality of a corresponding lower level problem. The upper and lower level problems are also referred to as the leader and follower problems, respectively. Both levels have their associated objective(s), variable(s) and constraint(s). Such problems model real-life scenarios of cases where the performance of an upper level authority is realizable/sustainable only if the corresponding lower level objective is optimum. A number of practical applications in the field of engineering, logistics, economics and transportation have inherent nested structure that are suited to this type of modelling. The range of applications as well as a rapid increase in the size and complexity of such problems has prompted active interest in the design of efficient algorithms for bilevel optimization. Bilevel optimization problems present a number of unique and interesting challenges to algorithm design. The nested nature of the problem requires optimization of a lower level problem to evaluate each upper level solution, which makes it computationally exorbitant. Theoretically, an upper level solution is considered valid/feasible only if the corresponding lower level variables are the true global optimum of the lower level problem. Global optimality can be reliably asserted in very limited cases, for example convex and linear problems. In deceptive cases, an inaccurate lower level optimum may result in an objective value better than true optimum at the upper level, which poses a severe challenge for ranking/selection strategies used within any optimization technique. In turn, this also makes the performance evaluation very difficult since the performance cannot be judged based on the objective values alone. While the area of bilevel (or more generally, multilevel) programming itself is not very new, most reports in this direction up until about a decade ago considered solving linear or at most quadratic problems at both levels. Correspondingly, the focus on was on development of exact methods to solve such problems. However, such methods typically require assumptions on mathematical properties, which may not always hold in practical applications. With increasing use of computer simulation-based evaluations in a number of disciplines in science and engineering, there is more need than ever to handle problems that are highly nonlinear or even black-box in nature. Metaheuristic algorithms, such as evolutionary algorithms are more suited to this emerging paradigm. The foray of evolutionary algorithms in bilevel programming is relatively recent and there remains scope of substantial development in the field in terms of addressing the aforementioned challenges. The work presented in this thesis is directed towards improving evolutionary techniques to enable them solve generic bilevel problems more accurately using lower number of function evaluations compared to the existing methods. Three key approaches are investigated towards accomplishing this: (a) e active hybridization of global and local search methods during dierent stages of the overall search; (b) use of surrogate models to guide the search using approximations in lieu of true function evaluations, and (c) use of a non-nested re-formulation of the problem. While most of the work is focused on single-objective problems, preliminary studies are also presented on multi-objective bilevel problems. The performance of the proposed approaches is evaluated on a comprehensive suite of mathematical test problems available in the literature, as well as some practical problems. The proposed approaches are observed to achieve a favourable balance between accuracy and computational expense for solving bilevel optimization problems, and thus exhibit suitability for use in real-life applications

Complex System Optimization using Biogeography-Based Optimization

Complex systems are frequently found in modern industry. But with their multisubsystems, multiobjectives, and multiconstraints, the optimization of complex systems is extremely hard. In this paper, a new algorithm adapted from biogeography-based optimization (BBO) is introduced for complex system optimization. BBO/Complex is the combination of BBO with a multiobjective ranking system, an innovative migration approach, and effective diversity control. Based on comparisons with three complex system optimization algorithms (multidisciplinary feasible (MDF), individual discipline feasible (IDF), and collaborative optimization (CO)) on four real-world benchmark problems, BBO/Complex demonstrates competitive performance. BBO/Complex provides the best performance in three of the benchmark problems and the second best in the fourth problem

Reinforcement Learning-assisted Evolutionary Algorithm: A Survey and Research Opportunities

Evolutionary algorithms (EA), a class of stochastic search methods based on the principles of natural evolution, have received widespread acclaim for their exceptional performance in various real-world optimization problems. While researchers worldwide have proposed a wide variety of EAs, certain limitations remain, such as slow convergence speed and poor generalization capabilities. Consequently, numerous scholars actively explore improvements to algorithmic structures, operators, search patterns, etc., to enhance their optimization performance. Reinforcement learning (RL) integrated as a component in the EA framework has demonstrated superior performance in recent years. This paper presents a comprehensive survey on integrating reinforcement learning into the evolutionary algorithm, referred to as reinforcement learning-assisted evolutionary algorithm (RL-EA). We begin with the conceptual outlines of reinforcement learning and the evolutionary algorithm. We then provide a taxonomy of RL-EA. Subsequently, we discuss the RL-EA integration method, the RL-assisted strategy adopted by RL-EA, and its applications according to the existing literature. The RL-assisted procedure is divided according to the implemented functions including solution generation, learnable objective function, algorithm/operator/sub-population selection, parameter adaptation, and other strategies. Finally, we analyze potential directions for future research. This survey serves as a rich resource for researchers interested in RL-EA as it overviews the current state-of-the-art and highlights the associated challenges. By leveraging this survey, readers can swiftly gain insights into RL-EA to develop efficient algorithms, thereby fostering further advancements in this emerging field.Comment: 26 pages, 16 figure