Optimization of Constrained Function Using Genetic Algorithm

Optimization is the process of finding the minimum or maximum value that a particular function attains which also means finding the value for the independent variables of a function for which function is minimum or maximum. As real world problems are different and can be represented by different types of functions, so are the optimization algorithms. From last few decades, lot of research has been invested in developing different techniques of optimization suitable for different types of functions. These methods are broadly classified in to calculus based and search based methods.  After brief description on optimization and classification of different optimization problems, this study focuses on constrained optimization problem and the use of Genetic Algorithm to optimize  such problems. Keywords: Optimization, Genetic Algorithm, Penalty functio

Solving MDPs with thresholded lexicographic ordering using reinforcement learning

Includes bibliographical references.2022 Fall.Multiobjective problems with a strict importance order over the objectives occur in many real-life scenarios. While Reinforcement Learning (RL) is a promising approach with a great potential to solve many real-life problems, the RL literature focuses primarily on single-objective tasks, and approaches that can directly address multiobjective with importance order have been scarce. The few proposed approach were noted to be heuristics without theoretical guarantees. However, we found that their practical applicability is very limited as they fail to find a good solution even in very common scenarios. In this work, we first investigate these shortcomings of the existing approaches and propose some solutions that could improve their practical performance. Finally, we propose a completely different approach based on policy optimization using our Lexicographic Projection Optimization (LPO) algorithm and show its performance on some benchmark problems

Exact Pareto Optimal Search for Multi-Task Learning and Multi-Criteria Decision-Making

Given multiple non-convex objective functions and objective-specific weights, Chebyshev scalarization (CS) is a well-known approach to obtain an Exact Pareto Optimal (EPO), i.e., a solution on the Pareto front (PF) that intersects the ray defined by the inverse of the weights. First-order optimizers that use the CS formulation to find EPO solutions encounter practical problems of oscillations and stagnation that affect convergence. Moreover, when initialized with a PO solution, they do not guarantee a controlled trajectory that lies completely on the PF. These shortcomings lead to modeling limitations and computational inefficiency in multi-task learning (MTL) and multi-criteria decision-making (MCDM) methods that utilize CS for their underlying non-convex multi-objective optimization (MOO). To address these shortcomings, we design a new MOO method, EPO Search. We prove that EPO Search converges to an EPO solution and empirically illustrate its computational efficiency and robustness to initialization. When initialized on the PF, EPO Search can trace the PF and converge to the required EPO solution at a linear rate of convergence. Using EPO Search we develop new algorithms: PESA-EPO for approximating the PF in a posteriori MCDM, and GP-EPO for preference elicitation in interactive MCDM; experiments on benchmark datasets confirm their advantages over competing alternatives. EPO Search scales linearly with the number of decision variables which enables its use for training deep networks. Empirical results on real data from personalized medicine, e-commerce and hydrometeorology demonstrate the efficacy of EPO Search for deep MTL

Multiobjective evolutionary algorithm based on vector angle neighborhood

Selection is a major driving force behind evolution and is a key feature of multiobjective evolutionary algorithms. Selection aims at promoting the survival and reproduction of individuals that are most fitted to a given environment. In the presence of multiple objectives, major challenges faced by this operator come from the need to address both the population convergence and diversity, which are conflicting to a certain extent. This paper proposes a new selection scheme for evolutionary multiobjective optimization. Its distinctive feature is a similarity measure for estimating the population diversity, which is based on the angle between the objective vectors. The smaller the angle, the more similar individuals. The concept of similarity is exploited during the mating by defining the neighborhood and the replacement by determining the most crowded region where the worst individual is identified. The latter is performed on the basis of a convergence measure that plays a major role in guiding the population towards the Pareto optimal front. The proposed algorithm is intended to exploit strengths of decomposition-based approaches in promoting diversity among the population while reducing the user's burden of specifying weight vectors before the search. The proposed approach is validated by computational experiments with state-of-the-art algorithms on problems with different characteristics. The obtained results indicate a highly competitive performance of the proposed approach. Significant advantages are revealed when dealing with problems posing substantial difficulties in keeping diversity, including many-objective problems. The relevance of the suggested similarity and convergence measures are shown. The validity of the approach is also demonstrated on engineering problems.This work was supported by the Portuguese Fundacao para a Ciencia e Tecnologia under grant PEst-C/CTM/LA0025/2013 (Projecto Estrategico - LA 25 - 2013-2014 - Strategic Project - LA 25 - 2013-2014).info:eu-repo/semantics/publishedVersio

Weighted stress function method for multiobjective evolutionary algorithm based on decomposition

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) is a well established state-of-the-art framework. Major concerns that must be addressed when applying MOEA/D are the choice of an appropriate scalarizing function and setting the values of main control parameters. This study suggests a weighted stress function method (WSFM) for fitness assignment in MOEA/D. WSFM establishes analogy between the stress-strain behavior of thermoplastic vulcanizates and scalarization of a multiobjective optimization problem. The experimental results suggest that the proposed approach is able to provide a faster convergence and a better performance of final approximation sets with respect to quality indicators when compared with traditional methods. The validity of the proposed approach is also demonstrated on engineering problems.This work has been supported by FCT - Fundação para a Ciência e Tecnologia in the scope of the project: PEst-OE/EEI/UI0319/2014.info:eu-repo/semantics/publishedVersio

Multi-Objective Archiving

Most multi-objective optimisation algorithms maintain an archive explicitly or implicitly during their search. Such an archive can be solely used to store high-quality solutions presented to the decision maker, but in many cases may participate in the search process (e.g., as the population in evolutionary computation). Over the last two decades, archiving, the process of comparing new solutions with previous ones and deciding how to update the archive/population, stands as an important issue in evolutionary multi-objective optimisation (EMO). This is evidenced by constant efforts from the community on developing various effective archiving methods, ranging from conventional Pareto-based methods to more recent indicator-based and decomposition-based ones. However, the focus of these efforts is on empirical performance comparison in terms of specific quality indicators; there is lack of systematic study of archiving methods from a general theoretical perspective. In this paper, we attempt to conduct a systematic overview of multi-objective archiving, in the hope of paving the way to understand archiving algorithms from a holistic perspective of theory and practice, and more importantly providing a guidance on how to design theoretically desirable and practically useful archiving algorithms. In doing so, we also present that archiving algorithms based on weakly Pareto compliant indicators (e.g., epsilon-indicator), as long as designed properly, can achieve the same theoretical desirables as archivers based on Pareto compliant indicators (e.g., hypervolume indicator). Such desirables include the property limit-optimal, the limit form of the possible optimal property that a bounded archiving algorithm can have with respect to the most general form of superiority between solution sets.Comment: 21 pages, 4 figures, journa

Regularization-free multicriteria optimization of polymer viscoelasticity model

This paper introduces a multiobjective optimization (MOP) method for nonlinear regression analysis which is capable of simultaneously minimizing the model order and estimating parameter values without the need of exogenous regularization constraints. The method is introduced through a case study in polymer rheology modeling. Prevailing approaches in this field tackle conflicting optimization goals as a monobjective problem by aggregating individual regression errors on each dependent variable into a single weighted scalarization function. In addition, their supporting deterministic numerical methods often rely on assumptions which are extrinsic to the problem, such as regularization constants and restrictions on parameter distribution, thereby introducing methodology inherent biases into the model. Our proposed non-deterministic MOP strategy, on the other hand, aims at finding the Pareto-front of all optimal solutions with respect not only to individual regression errors, but also to the number of parameters needed to fit the data, automatically reducing the model order. The evolutionary computation approach does not require arbitrary constraints on objective weights, regularization parameters or other exogenous assumptions to handle the ill-posed inverse problem. The article discusses the method rationales, implementation, simulation experiments, and comparison with other methods, with experimental evidences that it can outperform state-of-art techniques. While the discussion focuses on the study case, the introduced method is general and immediately applicable to other problem domains.This work is funded by National Funds through FCT - Portuguese Foundation for Science and Technology, References UIDB/05256/2020 and UIDP/05256/2020 and the European project MSCA-RISE-2015, NEWEX, Reference 734205

Multiparametric Continuous and Mixed-Integer Nonlinear Optimization with Parameters in General Locations

Convex programming has been a research topic for a long time, both theoretically and algorithmically. Frequently, these programs lack complete data or contain rapidly shifting data. In response, we consider solving parametric programs, which allow for fast evaluation of the optimal solutions once the data is known. It has been established that, when the objective and constraint functions are convex in both variables and parameters, the optimal solutions can be estimated via linear interpolation. Many applications of parametric optimization violate the necessary convexity assumption. However, the linear interpolation is still useful; as such, we extend this interpolation to more general parametric programs in which the objective and constraint functions are biconvex. The resulting algorithm can be applied to scalarized multiobjective problems, which are inherently parametric, or be used in a gradient dual ascent method. We also provide two termination conditions and perform a numerical study on synthetic parametric biconvex optimization problems to compare their effectiveness

Parallel Multi-Objective Evolutionary Algorithms: A Comprehensive Survey

Multi-Objective Evolutionary Algorithms (MOEAs) are powerful search techniques that have been extensively used to solve difficult problems in a wide variety of disciplines. However, they can be very demanding in terms of computational resources. Parallel implementations of MOEAs (pMOEAs) provide considerable gains regarding performance and scalability and, therefore, their relevance in tackling computationally expensive applications. This paper presents a survey of pMOEAs, describing a refined taxonomy, an up-to-date review of methods and the key contributions to the field. Furthermore, some of the open questions that require further research are also briefly discussed

Ensemble Multi-Objective Biogeography-Based Optimization with Application to Automated Warehouse Scheduling

This paper proposes an ensemble multi-objective biogeography-based optimization (EMBBO) algorithm, which is inspired by ensemble learning, to solve the automated warehouse scheduling problem. First, a real-world automated warehouse scheduling problem is formulated as a constrained multi-objective optimization problem. Then EMBBO is formulated as a combination of several multi-objective biogeography-based optimization (MBBO) algorithms, including vector evaluated biogeography-based optimization (VEBBO), non-dominated sorting biogeography-based optimization (NSBBO), and niched Pareto biogeography-based optimization (NPBBO). Performance is tested on a set of 10 unconstrained multi-objective benchmark functions and 10 constrained multi-objective benchmark functions from the 2009 Congress on Evolutionary Computation (CEC), and compared with single constituent MBBO and CEC competition algorithms. Results show that EMBBO is better than its constituent algorithms, and among the best CEC competition algorithms, for the benchmark functions studied in this paper. Finally, EMBBO is successfully applied to the automated warehouse scheduling problem, and the results show that EMBBO is a competitive algorithm for automated warehouse scheduling