EDA++: Estimation of Distribution Algorithms with Feasibility Conserving Mechanisms for Constrained Continuous Optimization

Handling non-linear constraints in continuous optimization is challenging, and finding a feasible solution is usually a difficult task. In the past few decades, various techniques have been developed to deal with linear and non-linear constraints. However, reaching feasible solutions has been a challenging task for most of these methods. In this paper, we adopt the framework of Estimation of Distribution Algorithms (EDAs) and propose a new algorithm (EDA++) equipped with some mechanisms to deal with non-linear constraints. These mechanisms are associated with different stages of the EDA, including seeding, learning and mapping. It is shown that, besides increasing the quality of the solutions in terms of objective values, the feasibility of the final solutions is guaranteed if an initial population of feasible solutions is seeded to the algorithm. The EDA with the proposed mechanisms is applied to two suites of benchmark problems for constrained continuous optimization and its performance is compared with some state-of-the-art algorithms and constraint handling methods. Conducted experiments confirm the speed, robustness and efficiency of the proposed algorithm in tackling various problems with linear and non-linear constraints.La Caixa Foundatio

Evolutionary algorithms to optimize low-thrust trajectory design in spacecraft orbital precession mission

In space environment, perturbations make the spacecraft lose its predefined orbit in space. One of these undesirable changes is the in-plane rotation of space orbit, denominated as orbital precession. To overcome this problem, one option is to correct the orbit direction by employing low-thrust trajectories. However, in addition to the orbital perturbation acting on the spacecraft, a number of parameters related to the spacecraft and its propulsion system must be optimized. This article lays out the trajectory optimization of orbital precession missions using Evolutionary Algorithms (EAs). In this research, the dynamics of spacecraft in the presence of orbital perturbation is modeled. The optimization approach is employed based on the parametrization of the problem according to the space mission. Numerous space mission cases have been studied in low and middle Earth orbits, where various types of orbital perturbations are acted on spacecraft. Consequently, several EAs are employed to solve the optimization problem. Results demonstrate the practicality of different EAs, along with comparing their convergence rates. With a unique trajectory model, EAs prove to be an efficient, reliable and versatile optimization solution, capable of being implemented in conceptual and preliminary design of spacecraft for orbital precession missions.IT-609-13 2013-2018, TIN2016-78365-

An evolutionary discretized Lambert approach for optimal long-range rendezvous considering impulse limit

In this paper, an approach is presented for finding the optimal long-range space rendezvous in terms of fuel and time, considering limited impulse. In this approach , the Lambert problem is expanded towards a discretized multi-impulse transfer. Taking advantage of an analytical form of multi-impulse transfer, a feasible solution that satisfies the impulse limit is calculated. Next, the obtained feasible solution is utilized as a seed for generating individuals for a hybrid self-adaptive evolutionary algorithm to minimize the total time, without violating the impulse limit while keeping the overall fuel mass the same as or less than the one associated with the analytical solution. The algorithm eliminates similar individuals and regenerates them based on a combination of Gaussian and uniform distribution of solutions from the fuel-optimal region during the optimization process. Other enhancements are also applied to the algorithm to make it auto-tuned and robust to the initial and final orbits as well as the impulse limit. Several types of the proposed algorithm are tested considering varieties of rendezvous missions. Results reveal that the approach can successfully reduce the overall transfer time in the multi-impulse transfers while minimizing the fuel mass without violating the impulse limit. Furthermore, the proposed algorithm has superior performance over standard evolutionary algorithms in terms of convergence and optimality.TIN2016-78365

Are the artificially generated instances uniform in terms of difficulty?

In the field of evolutionary computation, it is usual to generate artificial benchmarks of instances that are used as a test-bed to determine the performance of the algorithms at hand. In this context, a recent work on permutation problems analyzed the implications of generating instances uniformly at random (u.a.r.) when building those benchmarks. Particularly, the authors analyzed instances as rankings of the solutions of the search space sorted according to their objective function value. Thus, two instances are considered equivalent when their objective functions induce the same ranking over the search space. Based on the analysis, they suggested that, when some restrictions hold, the probability to create easy rankings is higher than creating difficult ones. In this paper, we continue on that research line by adopting the framework of local search algorithms with the best improvement criterion. Particularly, we empirically analyze, in terms of difficulty, the instances (rankings) created u.a.r. of three popular problems: Linear Ordering Problem, Quadratic Assignment Problem and Flowshop Scheduling Problem. As the neighborhood system is critical for the performance of local search algorithms three different neighborhood systems have been considered: swap, interchange and insert. Conducted experiments reveal that (1) by sampling the parameters uniformly at random we obtain instances with a non-uniform distribution in terms of difficulty, (2) the distribution of the difficulty strongly depends on the pair problem-neighborhood considered, and (3) given a problem, the distribution of the difficulty seems to depend on the smoothness of the landscape induced by the neighborhood and on its size.Research Groups 2013-2018 (IT-609-13) TIN2016-78365-R(Spanish Ministry of Economy, Industry and Competitiveness

Spacecraft Trajectory Optimization: A review of Models, Objectives, Approaches and Solutions

This article is a survey paper on solving spacecraft trajectory optimization problems. The solving process is decomposed into four key steps of mathematical modeling of the problem, defining the objective functions, development of an approach and obtaining the solution of the problem. Several subcategories for each step have been identified and described. Subsequently, important classifications and their characteristics have been discussed for solving the problems. Finally, a discussion on how to choose an element of each step for a given problem is provided.La Caixa, TIN2016-78365-

Distance-based exponential probability models on constrained combinatorial optimization problems

Estimation of distribution algorithms have already demonstrated their utility when solving a broad range of combinatorial problems. However, there is still room for methodological improvements when approaching constrained type problems. The great majority of works in the literature implement external repairing or penalty schemes, or use ad-hoc sampling methods in order to avoid unfeasible solutions. In this work, we present a new way to develop EDAs for this type of problems by implementing distance-based exponential probability models defined exclusively on the set of feasible solutions. In order to illustrate this procedure, we take the 2-partition balanced Graph Partitioning Problem as a case of study, and design efficient learning and sampling methods in order to use these distance-based probability models in EDAs

Optimal multi-impulse space rendezvous considering limited impulse using a discretized Lambert problem combined with evolutionary algorithms

In this paper, a direct approach is presented to tackle the multi-impulse rendezvous problem considering the impulse limit. Particularly, the standard Lambert problem is extended toward several consequential orbit transfers for the rendezvous problem. A number of different evolutionary algorithms are taken into consideration. It is shown that the proposed approach can lead to the optimal multi-impulse transfer maneuver that has the minimum amount of fuel similar to the traditional two-impulse transfer without violating the impulse limitation. Results also indicate that the approach is efficient even when the number of stages increases due to lower impulse limitations

Trajectory optimization of space vehicle in rendezvous proximity operation with evolutionary feasibility conserving techniques

In this paper, a direct approach is developed for discovering optimal transfer trajectories of close-range rendezvous of satellites considering disturbances in elliptical orbits. The control vector representing the inputs is parameterized via different interpolation methods, and an Estimation of Distribution Algorithm (EDA) that implements mixtures of probability models is presented. To satisfy the terminal conditions, which are represented as non-linear inequality constraints, several feasibility conserving mechanisms associated with learning and sampling methods of the EDAs are proposed, which guarantee the feasibility of the explored solutions. They include a particular implementation of a clustering algorithm, outlier detection, and several heuristic mapping methods. The combination of the proposed operators guides the optimization process in achieving the optimal solution by surfing the regions of the search domain associated with feasible solutions. Numerical simulations confirm that space transfer trajectories with minimum-fuel consumption for the chaser spacecraft can be obtained with terminal condition satisfaction in rendezvous proximity operation.KK-2021/00065 KK-2022/00106; PID2019-104933GB-10/AEI/10.13039/501100011033 PID2019-106453GAI00/AEI/10.13039/501100011033 IT1504-2

A note on the Boltzmann distribution and the linear ordering problem

The Boltzmann distribution plays a key role in the field of optimization as it directly connects this field with that of probability. Basically, given a function to optimize, the Boltzmann distribution associated to this function assigns higher probability to the candidate solutions with better quality. Therefore, an efficient sampling of the Boltzmann distribution would turn optimization into an easy task. However, inference tasks on this distribution imply performing operations over an exponential number of terms, which hinders its applicability. As a result, the scientific community has investigated how the structure of objective functions is translated to probabilistic properties in order to simplify the corresponding Boltzmann distribution. In this paper, we elaborate on the properties induced in the Boltzmann distribution associated to permutation-based combinatorial optimization problems. Particularly, we prove that certain characteristics of the linear ordering problem are translated as conditional independence relations to the Boltzmann distribution in the form of L − decomposability

Implementing the Cumulative Difference Plot in the IOHanalyzer

The IOHanalyzer is a web-based framework that enables an easy visualization and comparison of the quality of stochastic optimization algorithms. IOHanalyzer offers several graphical and statistical tools analyze the results of such algorithms. In this work, we implement the cumulative difference plot in the IOHanalyzer. The cumulative difference plot is a graphical approach that compares two samples through the first-order stochastic dominance. It improves upon other graphical approaches with the ability to distinguish between a small magnitude of difference and high uncertainty.The data science and artificial intelligence chair for digitalized industry and services, Telecom Paris, Institut Polytechnique de Paris, Research Groups 20192021 (IT1244-19), Spanish Ministry of Economy and Competitiveness, through the research project PID2019-106453GAI00/AEI/10.13039/501100011033