Infeasible Solutions Research Articles

Localization estimation of two leaks in pipelines through Monte Carlo simulations and hydraulic-spatial constraints

The problem of localizing two leaks using only flow rate and pressure measurements at the boundaries of a pipeline, and under steady-state flow conditions, is ill-posed due to the undetermined nature of the inverse problem, which involves two coupled equations with four unknowns associated with the presence of the leaks: two emitter coefficients and two locations. Therefore, attempting to solve this problem using any method without imposing additional constraints leads to an unbounded solution space, which contains solutions that may be physically meaningless. In this article, we propose a four-algorithm method that incorporates both spatial and hydraulic constraints to filter out physically infeasible solutions. The proposed method is based on Monte Carlo simulations, which use input data from hydraulic instruments installed at the boundaries of the pipeline, as well as random values with predefined probabilities that are bounded by the hydraulic-spatial constraints. The outputs of the method are probability distributions for the four unknowns. To demonstrate the feasibility of the method, results obtained through simulations and experimental testing on a test bed are presented.

Improved SQP and SLSQP algorithms for feasible path-based process optimisation

Feasible path algorithms have been widely used for process optimisation due to its good convergence. The sequential quadratic programming (SQP) algorithm is usually used to drive the feasible path algorithms towards optimality. However, existing SQP algorithms may suffer from inconsistent quadratic programming (QP) subproblems and numerical noise, especially for ill-conditioned process optimisation problems, leading to a suboptimal or infeasible solution. In this work, we propose an improved SQP algorithm (I-SQP) and an improved sequential least squares programming algorithm (I-SLSQP) that solves a least squares (LSQ) subproblem at each major iteration. A hybrid method through the combination of two existing relaxations is proposed to solve the inconsistent subproblems for better convergence and higher efficiency. We find that a certain part of the dual LSQ algorithm suffers from serious cancellation errors, resulting in an inaccurate search direction or no viable search direction generated. Therefore, the QP solver is used to solve LSQ subproblems in such a situation. The computational results indicates that I-SLSQP is more robust than fmincon in MATLAB, IPOPT, Py-SLSQP and I-SQP. It is also shown that I-SLSQP and Py-SLSQP is superior to I-SQP for ill-conditioned process optimisation problems, whilst I-SQP is more computationally efficient than I-SLSQP and Py-SLSQP for well-conditioned problems.

An effective multi-level memetic search with neighborhood reduction for the clustered team orienteering problem

The Clustered Team Orienteering Problem (CluTOP) extends the classic Clustered Orienteering Problem by considering the use of multiple vehicles. The problem is known to be NP-hard and can be used to formulate many real-life applications. This work presents a highly effective multi-level memetic search for CluTOP that combines a backbone-based edge assembly crossover to generate promising offspring solutions with an effective bilevel synergistic local search procedure at both cluster and customer levels to improve offspring solutions. Other novel features of the proposed approach include a joint use of three specific hash functions to identify the tabu status of candidate solutions at the cluster level, a multi-neighborhood search with inter-route and intra-route optimization at the customer level, a pre-processing neighborhood reduction strategy to avoid examining non-promising candidate solutions, and a strategy for controlled exploration of infeasible solutions. Extensive experimental results on 1848 benchmark instances convincingly demonstrate high competitiveness of the approach in terms of both solution quality and computational time, compared to the state-of-the-art heuristics from the literature. In particular, the proposed algorithm improves upon the existing best-known solutions for 294 instances, while matching the previous best-known results for all but 3 of the remaining instances. To gain further insights into the algorithm’s performance, additional experiments are conducted to analyze its main components.

Modified effective butterfly optimizer for solving optimal power flow problem

The optimal power flow (OPF) problem remains a popular and challenging work in optimizing power systems. Although researchers have suggested many optimization algorithms to solve this problem in the literature, their comparison studies lack fairness and transparency. As these studies increase in number, they deviate from a standard test system, considering a common security and technical constraints., there is a growing trend away from a standard test system. Different studies used different search ranges for the same decision and constraint parameters, different than the standard ranges suggested by IEEE systems. This caused many unfair comparisons in literature. Furthermore, these studies are generally not transparent enough so that their results cannot be verified. This has resulted in numerous infeasible solutions in the literature, violating the limits of constraint parameters. The recent incorporating of renewable energy sources in OPF studies has made this situation more complicated. Sorting through the literature and identifying those OPF applications having exactly the same test conditions is a challenging process. The main contribution is this paper adapts the modified effective butterfly algorithm (MEBO) to solve OPF problem under the common parameter constraints and sufficient transparency. The focus is on a transparent comparison with works in the literature with the same constraint values. This paper compares the performance of the proposed algorithm with other state-of-the-art algorithms in the literature, focusing on the wind energy and without wind energy IEEE 30-bus and IEEE 57-bus systems and the most commonly used constraints. The results demonstrate the efficiency and superiority of the proposed algorithm. For instance, in the 30-bus test system, compared to the initial case, fuel cost has been reduced by 11.42 %, emission by 14.33 %, L-index by 45.10 %, active power losses by 51.60 %, and voltage deviation by 92.70 %.

A novel penalty function-based interval constrained multi-objective optimization algorithm for uncertain problems

To directly address problems involving uncertain objectives and constraints, a novel penalty function-based interval constrained multi-objective optimization algorithm (PF-ICMOA) was developed. It comprises three main components: an uncertainty propagation-based interval constraint solver, an interval constraint violation function, and a two-stage penalty function. The uncertainty violation of interval constraints can be accurately evaluated by the proposed advanced multi-objective optimization algorithm using a segmentation function without requiring multiple interval analyses. Additionally, by applying penalties to infeasible solutions in the two-stage process, partially feasible and superior infeasible solutions are retained in the early stages of evolution, while superior feasible solutions are favored in the later stages. A novel benchmark was designed and the effect of the imprecision factor on the Pareto front was examined in details. Comparisons with seven deterministic constraint handling techniques demonstrated the effectiveness of the proposed method for solving interval constrained multi-objective optimization problems, demonstrating that PF-ICMOA is a highly competitive method for balancing the feasibility, convergence, and diversity performance characteristics.

A cost-effective seed selection model for multi-constraint influence maximization in social networks

The Influence Maximization (IM) problem aims to maximize the diffusion of information or adoption of products among users within a social network by identifying and activating a set of initial users. It is not unrealistic to have a higher activation cost for more influential users in real-life applications. However, existing works on IM focus solely on finding the most influential users as the seed set, without considering either the activation costs of individual nodes and the total budget or the size of the seed set. This oversight may lead to infeasible solutions, particularly from financial and managerial perspectives, respectively. To address these issues, we propose a more realistic and generalized formulation called Multi-Constraint Influence Maximization (MCIM) to achieve a cost-effective solution under both budgetary and cardinality constraints. The MCIM model allows for variable-length solutions, necessitating the exclusion of seed nodes from the influence spread estimation. Consequently, unlike existing IM formulations, the spread function under the MCIM model is no longer monotonic but submodular. As it has also been proven to be an NP-hard problem, we propose a Simple Additive Weighting (SAW)-assisted Differential Evolution (DE) algorithm for solving large-size real-world IM problems. Experimental results on four datasets demonstrate the effectiveness of the proposed formulation and algorithm in finding optimal and cost-effective solutions.

An accelerated benders decomposition algorithm for the solution of the multi-trip time-dependent vehicle routing problem with time windows

The logistics companies executing the last mile delivery of goods in urban areas deal every day with the problem of routing their vehicles, while taking into account multiple trips per vehicle, time-dependent travel time, customers’ time windows and loading time at the depot simultaneously. This paper addresses this problem, known as Multi-Trip Time-Dependent Vehicle Routing Problem with Time Windows, aiming at its exact solution, minimizing the total travelled distance of a company's fleet. Based on a literature model, a new reformulation with reduced size is suggested. This reformulation is decomposed by applying the Benders method in an effective way, resulting in a subproblem with no duality gap. By exploiting the special features of the problem and the particular structure of the decomposition made, several novel valid inequalities are introduced, in order to both tighten the non-decomposed formulations and warm start the relaxed master problem to achieve less infeasible solutions and higher lower bounds. For the solution of the problem, an innovative algorithm is proposed, including suboptimal master solutions and a multi-cut generation procedure, which is based on the careful observation of the values of the Benders dual subproblem variables. The impact of the valid inequalities as well as two variants of the suggested algorithm are tested on benchmark data and they are compared with the non-decomposed models and a heuristic introduced in the literature. The computational results indicate improved efficiency and stronger bounds for the proposed algorithm.

Maximizing Net Present Value for Resource Constraint Project Scheduling Problems with Payments at Event Occurrences Using Approximate Dynamic Programming

Resource Constraint Project Scheduling Problems with Discounted Cash Flows (RCPSPDC) focuses on maximizing the net present value by summing the discounted cash flows of project activities. An extension of this problem is the Payment at Event Occurrences (PEO) scheme, where the client makes multiple payments to the contractor upon completion of predefined activities, with additional final settlement at project completion. Numerous approximation methods such as metaheuristics have been proposed to solve this NP-hard problem. However, these methods suffer from parameter control and/or the computational cost of correcting infeasible solutions. Alternatively, approximate dynamic programming (ADP) sequentially generates a schedule based on strategies computed via Monte Carlo (MC) simulations. This saves the computations required for solution corrections, but its performance is highly dependent on its strategy. In this study, we propose the hybridization of ADP with three different metaheuristics to take advantage of their combined strengths, resulting in six different models. The Estimation of Distribution Algorithm (EDA) and Ant Colony Optimization (ACO) were used to recommend policies for ADP. A Discrete cCuckoo Search (DCS) further improved the schedules generated by ADP. Our experimental analysis performed on the j30, j60, and j90 datasets of PSPLIB has shown that ADP–DCS is better than ADP alone. Implementing the EDA and ACO as prioritization strategies for Monte Carlo simulations greatly improved the solutions with high statistical significance. In addition, models with the EDA showed better performance than those with ACO and random priority, especially when the number of events increased.

Mixed Integer Nonlinear Programming-Based Unit Commitment of Conventional Thermal Generators Using Hybrid Evolutionary Algorithms

Unit Commitment (UC) discusses the optimized generation resources (to turn on economical generators and turn off expensive generators),which are subjected to satisfy all the operational constraints. The operational constraints such as load balancing, security maximization, minimum up and down time, spinning reserve, and ramp up and down constraints are difficult to satisfy. Although, UC is a cost minimization problem that is realized by committing less expensive units while satisfying the corresponding constraints, and dispatching the committed units economically. The UC problem is an np-hard Mixed Integer Nonlinear Problem (MINLP). Therefore, in this paper, hybrid EA based on a Genetic Algorithm (GA) has been applied to find the optimal solution to the UC problem. Moreover, during the search process, it is very difficult to discard infeasible solutions in EAs. Hence, the Genetic Algorithm (GA) is integrated with the feasibility rule constraint handling technique to emphasize feasible solutions. IEEE RTS Eleven Thermal Generator Standard Test system is used to validate the performance of proposed methods. For the validation and the superiority of the proposed algorithm, simulation results are compared with the classical Lagrangian Relaxation (LR) methods. Results show that the proposed method can find the global optimal solution to the UC problem which is subjected to satisfy all the operational constraints.

Effect of constraint relaxation on the minimum vertex cover problem in random graphs.

A statistical-mechanical study of the effect of constraint relaxation on the minimum vertex cover problem in Erdős-Rényi random graphs is presented. Using a penalty-method formulation for constraint relaxation, typical properties of solutions, including infeasible solutions that violate the constraints, are analyzed by means of the replica method and cavity method. The problem involves a competition between reducing the number of vertices to be covered and satisfying the edge constraints. The analysis under the replica-symmetric (RS) ansatz clarifies that the competition leads to degeneracies in the vertex and edge states, which determine the quantitative properties of the system, such as the cover and penalty ratios. A precise analysis of these effects improves the accuracy of RS approximation for the minimum cover ratio in the replica symmetry-breaking (RSB) region. Furthermore, the analysis based on the RS cavity method indicates that the RS/RSB boundary of the ground states with respect to the mean degree of the graphs is expanded, and the critical temperature is lowered by constraint relaxation.

Allocating the fixed cost as a complementary input in two-stage system: A DEA approach

The existing research on fixed cost allocation (FCA) in two-stage system data envelopment analysis (DEA) models typically regards the fixed cost as an additional input for each decision-making unit (DMU). However, these models overlook the prevalent real-world scenario where fixed costs act as complementary inputs in actual production processes. This study proposes a general two-stage network DEA model that incorporates the fixed cost as a complementary input to optimize the allocation scheme. First, we construct a functional relationship between the efficiency scores of DMUs and their allocated fixed costsusing a modified super-efficiency DEA model, which effectively classifies DMUs and solves infeasible solutions within the variable returns to scale (VRS) framework. Then, we propose a fair and unique allocation model based on fairness and efficiency maximization principles. Specifically, we allocate fixed costs based on the operational scale for inelastic DMUs while equitably increasing the efficiency scores for elastic DMUs. Finally, a numerical example and an empirical study on subsidy allocation among 30 provinces demonstrate the rationality and acceptability of our approach.

Non-invasive calibration of a Stewart platform by photogrammetry

AbstractAccurate calibration of a Stewart platform is important for their precise and efficient operation. However, the calibration of these platforms using forward kinematics is a challenge for researchers because forward kinematics normally generates multiple feasible and unfeasible solutions for any pose of the moving platform. The complex kinematic relations among the six actuator paths connecting the fixed base to the moving platform further compound the difficulty in establishing a straightforward and efficient calibration method. The authors developed a new forward kinematics-based calibration method using Denavit-Hartenberg (DH) convention and used the Stewart platform “Tiger 66.1” developed in their lab for experimenting with the photogrammetry-based calibration strategies described in this paper. This system became operational upon completion of construction, marking its inaugural use. The authors used their calibration model for estimating the errors in the system and adopted three compensation options or strategies as per least-square method to improve the accuracy of the system. These strategies leveraged a high-resolution digital camera and off-the-shelf software to capture the poses of the moving platform’s center. This process is non-invasive and does not need any additional equipment to be attached to the hexapod or any alteration of the hexapod hardware. This photogrammetry-based calibration process involves multiple high-resolution images from different angles to measure the position and orientation of the platform center in the three-dimensional space. The target poses and actual poses are then compared, and the error compensations are estimated using the least-square methods to calculate the predicted poses. Results from each of the three compensation approaches demonstrated noticeable enhancements in platform pose accuracies, suggesting room for further improvements. Given that “Tiger 66.1” is based on the general Stewart platform structure, the proposed calibration method holds promise for extension to machines operating on similar principles where non-invasive calibration is desirable. This study contributes to advancing the field of Stewart platform calibration, paving the way for more precise and efficient applications in various domains.

A novel tri-stage with reward-switching mechanism for constrained multiobjective optimization problems

The effective exploitation of infeasible solutions plays a crucial role in addressing constrained multiobjective optimization problems (CMOPs). However, existing constrained multiobjective optimization evolutionary algorithms (CMOEAs) encounter challenges in effectively balancing objective optimization and constraint satisfaction, particularly when tackling problems with complex infeasible regions. Subsequent to the prior exploration, this paper proposes a novel tri-stage with reward-switching mechanism framework (TSRSM), including the push, pull, and repush stages. Each stage consists of two coevolutionary populations, namely Pop1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_1$$\\end{document} and Pop2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_2$$\\end{document}. Throughout the three stages, Pop1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_1$$\\end{document} is tasked with converging to the constrained Pareto front (CPF). However, Pop2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_2$$\\end{document} is assigned with distinct tasks: (i) converging to the unconstrained Pareto front (UPF) in the push stage; (ii) utilizing constraint relaxation technique to discover the CPF in the pull stage; and (iii) revisiting the search for the UPF through knowledge transfer in the repush stage. Additionally, a novel reward-switching mechanism (RSM) is employed to transition between different stages, considering the extent of changes in the convergence and diversity of populations. Finally, the experimental results on three benchmark test sets and 30 real-world CMOPs demonstrate that TSRSM achieves competitive performance when compared with nine state-of-the-art CMOEAs. The source code is available at https://github.com/Qu-jq/TSRSM.

A self-adaptive Gaussian mutation-based arithmetic optimiser algorithm for integrated production scheduling and vehicle routing problem in the distributed manufacturing environment

ABSTRACT In response to the challenges posed by economic globalisation and increasing customer demands, enterprises are compelled to adapt and refine their production models and operational objectives, which motivates this paper to address the integrated production scheduling and vehicle routing problem in the distributed manufacturing environment (IPSVRP-DME). The objective is to simultaneously minimise both total energy consumption and total earliness/tardiness. Initially, a mixed integer programming model is proposed to address small-scale problems using Gurobi. Considering the NP-hardness of the problem, a novel Self-adapted Gaussian Mutation-based Arithmetic Optimiser Algorithm (SGMAOA) is developed to handle medium-scale and large-scale instances. To address the complexity of decision-making, an innovative two-level encoding method that encompasses four decision dimensions is introduced. Additionally, an incremental repair strategy is devised to rectify infeasible solutions caused by unreasonable delivery batching, while a time relaxation strategy is proposed to further enhance service levels without compromising energy consumption. Comparative experiments are conducted to demonstrate the effectiveness of SGMAOA, which is benchmarked against five prominent metaheuristics. In addition, a specific example is applied for the discussion of managerial applications and to illustrate the practicality of the proposed model and solution method.

Constrained Bayesian Optimization under Partial Observations: Balanced Improvements and Provable Convergence

The partially observable constrained optimization problems (POCOPs) impede data-driven optimization techniques since an infeasible solution of POCOPs can provide little information about the objective as well as the constraints. We endeavor to design an efficient and provable method for expensive POCOPs under the framework of constrained Bayesian optimization. Our method consists of two key components. Firstly, we present an improved design of the acquisition functions that introduce balanced exploration during optimization. We rigorously study the convergence properties of this design to demonstrate its effectiveness. Secondly, we propose Gaussian processes embedding different likelihoods as the surrogate model for partially observable constraints. This model leads to a more accurate representation of the feasible regions compared to traditional classification-based models. Our proposed method is empirically studied on both synthetic and real-world problems. The results demonstrate the competitiveness of our method for solving POCOPs.

Constraint subsets-based evolutionary multitasking for constrained multiobjective optimization

Constrained multiobjective optimization problems (CMOPs) are challenging because they need to optimize multiple conflicting objectives and satisfy various constraints simultaneously. To solve CMOPs, various constrained multiobjective evolutionary algorithms have been proposed in recent years. However, most of them tackle constraints by considering all constraints or zero constraint scenarios, these extreme treatment methods may be unable to utilize the information of multiple constraint subsets composed of partial constraints to maintain more promising infeasible solutions. To remedy this issue, a constraint subsets-based evolutionary multitasking method is developed, where a CMOP is transformed into a multitasking optimization problem by creating multiple simple auxiliary CMOPs. Particularly, the original CMOP is the main task, while the newly created CMOPs with different constraint subsets are the auxiliary tasks. Each task will be evolved by one specific population, and the knowledge transfer is conducted among tasks to realize the collaborative search. Meanwhile, the updating of the auxiliary task involves two stages. In the first stage, each auxiliary population will evolve to approach the pareto front of the constraint subset (s-CPF), promoting crossing the infeasible regions. While in the second stage, a feasible region-based search mechanism is proposed to approach the pareto front of the main task from each s-CPF, by utilizing unique and promising infeasible solutions. In addition, systematic experiments are carried out on three benchmark test suites and four real-world CMOPs. The experimental results fully demonstrate that the proposed algorithm is highly competitive with other state-of-the-art constrained multiobjective evolutionary algorithms.

Research on Dynamic Economic Dispatch Optimization Problem Based on Improved Grey Wolf Algorithm

The dynamic economic dispatch (DED) problem is a typical complex constrained optimization problem with non-smooth, nonlinear, and nonconvex characteristics, especially considering practical situations such as valve point effects and transmission losses, and its objective is to minimize the total fuel costs and total carbon emissions of generating units during the dispatch cycle while satisfying a series of equality and inequality constraints. For the challenging DED problem, a model of a dynamic economic dispatch problem considering fuel costs is first established, and then an improved grey wolf optimization algorithm (IGWO) is proposed, in which the exploitation and exploration capability of the original grey wolf optimization algorithm (GWO) is enhanced by initializing the population with a chaotic algorithm and introducing a nonlinear convergence factor to improve weights. Furthermore, a simple and effective constraint-handling method is proposed for the infeasible solutions. The performance of the IGWO is tested with eight benchmark functions selected and compared with other commonly used algorithms. Finally, the IGWO is utilized for three different scales of DED cases, and compared with existing methods in the literature. The results show that the proposed IGWO has a faster convergence rate and better global optimization capabilities, and effectively reduces the fuel costs of the units, thus proving the effectiveness of IGWO.

A Forward–Backward Simheuristic for the Stochastic Capacitated Dispersion Problem

In an effort to balance the distribution of services across a given territory, dispersion and diversity models typically aim to maximize the minimum distance between any pair of facilities. Specifically, in the capacitated dispersion problem (CDP), each facility has an associated capacity or level of service, and the objective is to select a set of facilities so that the minimum distance between any pair of them (dispersion) is maximized, while ensuring a user-defined level of service. This problem can be formulated as a linear integer model, where the sum of the capacities of the selected facilities must match or exceed the total demand in the network. Real-life applications often necessitate considering the levels of uncertainty affecting the capacity of the nodes. Failure to account for this uncertainty could lead to low-quality or infeasible solutions in practical scenarios. However, research addressing the stochastic version of the CDP is scarce. This paper introduces two models for the CDP with stochastic capacities, incorporating soft constraints and penalty costs for violating the total capacity constraint. The first model includes a probabilistic constraint to ensure the required level of service with a certain probability, while the second model introduces a soft constraint with penalty costs for violations. To solve both variants of the model, a forward–backward simheuristic algorithm is proposed. Our approach combines a metaheuristic algorithm with Monte Carlo simulation, enabling the efficient handling of the random behavior of node capacities and obtaining reliable solutions regardless of their probability distribution.

A novel solution to optimal power flow problems using composite differential evolution integrating effective constrained handling techniques

Optimal power flow is a complex and highly non-linear problem in which steady-state parameters are needed to find a network’s efficient and economical operation. In addition, the difficulty of the Optimal power flow problem becomes enlarged when new constraints are added, and it is also a challenging task for the power system operator to solve the constrained Optimal power flow problems efficiently. Therefore, this paper presents a constrained composite differential evolution optimization algorithm to search for the optimum solution to Optimal power flow problems. In the last few decades, numerous evolutionary algorithm implementations have emerged due to their superiority in solving Optimal power flow problems while considering various objectives such as cost, emission, power loss, etc. evolutionary algorithms effectively explore the solution space unconstrainedly, often employing the static penalty function approach to address the constraints and find solutions for constrained Optimal power flow problems. It is a drawback that combining evolutionary algorithms and the penalty function approach requires several penalty parameters to search the feasible space and discard the infeasible solutions. The proposed a constrained composite differential evolution algorithm combines two effective constraint handling techniques, such as feasibility rule and ɛ constraint methods, to search in the feasible space. The proposed approaches are recognized on IEEE 30, 57, and 118-bus standard test systems considering 16 study events of single and multi-objective optimization functions. Ultimately, simulation results are examined and compared with the many recently published techniques of Optimal power flow solutions owing to show the usefulness and performance of the proposed a constrained composite differential evolution algorithm.

The Importance of Being Constrained: Dealing with Infeasible Solutions in Differential Evolution and Beyond.

We argue that results produced by a heuristic optimisation algorithm cannot be considered reproducible unless the algorithm fully specifies what should be done with solutions generated outside the domain, even in the case of simple bound constraints. Currently, in the field of heuristic optimisation, such specification is rarely mentioned or investigated due to the assumed triviality or insignificance of this question. Here, we demonstrate that, at least in algorithms based on Differential Evolution, this choice induces notably different behaviours in terms of performance, disruptiveness, and population diversity. This is shown theoretically (where possible) for standard Differential Evolution in the absence of selection pressure and experimentally for the standard and state-of-the-art Differential Evolution variants, on a special test function and the BBOB benchmarking suite, respectively. Moreover, we demonstrate that the importance of this choice quickly grows with problem dimensionality. Differential Evolution is not at all special in this regard-there is no reason to presume that other heuristic optimisers are not equally affected by the aforementioned algorithmic choice. Thus, we urge the heuristic optimisation community to formalise and adopt the idea of a new algorithmic component in heuristic optimisers, which we refer to as the strategy of dealing with infeasible solutions. This component needs to be consistently: (a) specified in algorithmic descriptions to guarantee reproducibility of results, (b) studied to better understand its impact on an algorithm's performance in a wider sense (i.e., convergence time, robustness, etc.), and (c) included in the (automatic) design of algorithms. All of these should be done even for problems with bound constraints.