Multiple Nonlinear Constraints Research Articles

Optimizing Patient Recruitment in Global Clinical Trials using Nature-Inspired Metaheuristics

A common problem seen in the ineffective execution of global multicenter trials is the frequent inability to recruit a sufficient number of patients. A myriad of barriers to recruiting enough patients timely exist and may include practical limits on the number of recruiting sites imposed by the various countries, as well as administrative, cost and unanticipated issues. The Poisson-gamma recruitment model is a widely accepted statistical tool used to predict and track recruitment at different levels of a trial and make inference. An optimal recruitment plan can be designed using mathematical optimization, however, the search is complicated with multiple nonlinear objectives and constraints that arise from regulatory and budgetary considerations. We review nature-inspired metaheuristic algorithms which are powerful, flexible, general purpose optimization tools and demonstrate their capability and utility in optimizing complex recruitment designs for global clinical trials using the Poisson-gamma model as an exemplary model. This research opens up new avenues for improving the efficiency and effectiveness of patient recruitment in clinical trials, thereby potentially accelerating the development of new medical treatments.

Trajectory planning for satellite swarms with nonlinear terminal constraints using penalty concave relaxation

Convex programming (CP) is widely used in trajectory planning for space missions due to its high efficiency. Existing CP methods may not handle well with trajectory planning problem for satellite swarms involving multiple nonlinear terminal constraints, leading to solution chattering and initial infeasibility. To overcome this issue, this paper introduces a penalty concave relaxation (PCR) treatment to the conventional CP methods. First, the nonconvex terms in multiple nonlinear terminal constraints are equivalently concentrated into a unique equality constraint to reduce the number of relaxation variables. This equality constraint is then relaxed along the concave direction to guarantee the initial feasibility of the subsequent iterative solution process. Meanwhile, only one penalty coefficient is added on the objective. Finally, the relaxed concave constraint is linearized so that chattering can be avoided in an iterative strategy to solve the original swarm trajectory planning problem. Numerical examples are set up for two swarm trajectory planning missions with nonlinear terminal constraints. The performance of avoiding chattering and ensuring initial feasibility is verified.

Six-Degree-of-Freedom Rocket Landing Optimization via Augmented Convex–Concave Decomposition

In this paper an augmented convex–concave decomposition (ACCD) method for treating nonlinear equality constraints in an otherwise convex problem is proposed. This augmentation improves greatly the feasibility of the problem when compared to the original convex–concave decomposition approach. The effectiveness of the ACCD is demonstrated by solving a fuel-optimal six-degree-of-freedom rocket landing problem in atmosphere, subject to multiple nonlinear equality constraints. Compared with known approaches such as sequential convex programming, where conventional linearization (with or without slack variables) is employed to treat those nonlinear equality constraints, it is shown that the proposed ACCD leads to more robust convergence of the solution process and a more interpretable behavior of the sequential convex algorithm. The methodology can also be applied effectively in the case where the determination of discrete controls or decision-making variables needs to be made, such as the on–off use of reaction control system thrusters in the rocket landing problem, without the need for a mixed integer solver. Numerical results are shown for a representative, reusable rocket benchmark problem.

Multi-objective drilling trajectory optimization using decomposition method with minimum fuzzy entropy-based comprehensive evaluation

This paper is concerned with the optimization problem of drilling trajectory design, which plays a vital role in ensuring the safety and enhancing the efficiency of an industrial drilling process. Due to complex geological environment and limited capacity of equipment, the problem to be addressed features multi-objective and multi-constraint, and consists of two challenges: (1) how to formulate a proper optimization scheme and efficiently solve it for a group of solutions; and (2) how to pick out a desired result from the obtained Pareto solutions according to certain requirements. In this paper, to meet drilling practice, three objective functions are introduced regarding trajectory length, well-profile energy, and target error, respectively. Constraints are the range of decision parameters, non-negative constraints and bound of the target area. As a result, a comprehensive optimization model for the design of drilling trajectory is established. A novel optimization algorithm is devised to deal with the contradictory objectives and multiple nonlinear constraints, which combines an adaptive penalty function with multi-objective evolutionary algorithm based on decomposition. A fuzzy-entropy-based evaluation approach is further employed to determine a satisfactory solution from the group of obtained ones. A case study illustrates that (1) our optimization solution is indeed beneficial to the optimization of drilling trajectory; and (2) the optimization algorithm and the decision method therein outperform some existing ones, which shows both practical and theoretical significance.

Topology optimization via sequential integer programming and Canonical relaxation algorithm

The mathematical essence of structural topology optimization is large-scale nonlinear integer programming. To overcome its huge computational burden, a popular way is to relax the 0–1 variable constraints and transform the integer programming problem into a continuous variable programming problem which can be solved by using gradient based mathematical programming methods. To cope with the variable transformation, the well-known SIMP (Solid Isotropic Material with Penalty) method introduces interpolation schemes for the material properties versus design variables with penalty and achieves great success and popularity. However, there is no doubt that directly tackling the large-scale nonlinear integer programming is very important. This paper solves the structural topology optimization problems with single or multiple constraints by applying the Canonical Dual Theory (CDT) by Gao and Ruan (2010) together with Sequential Approximate Programming approach under the classic structural topology optimization formulations.Following the Sequential Approximate Programming (SAP) frame from the structural optimization, the present paper firstly utilizes sensitivity information to construct the explicit and separable approximate Sequential Quadratic Integer Programming (SQIP) or Sequential Linear Integer Programming (SLIP) subproblems for the topology optimization. And then, the subproblems are solved by applying the Canonical relaxation algorithm based on CDT theory. Their special mathematical structures are exploited to develop analytic solution of Kuhn–Tucker condition of the dual programming. Numerical experiments of two linear and quadratic integer programming problems with random coefficients assert that the Canonical relaxation algorithm can obtain approximate solutions with good properties very efficiently and the dual gap is negligible when the number of design variables increases.Because move limit strategies play a key role in many search algorithms of structural optimization, this paper combines one of two different move limit strategies within the new method. The new method first solves a set of classic topology optimization problems with only material usage constraint, including minimum structural compliance design under constant load, maximum heat transfer efficiency for the heat conduction problem. And then we apply the method to the topology optimization problems with multiple constraints, including minimum structural compliance design under an additional local displacement constraint and minimum structural compliance design under infill constraints. The results of these problems demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and systematic way. Beyond that, the new method can achieve integer solutions when combined with the move limit strategy of controlling volume fraction parameter. It can deal with much more design variables than the general branch-and-bound method and makes no use of any sensitivity threshold or heuristic stabilization scheme during the iterative process in comparison with BESO method. Finally, the new method in this paper can be further developed as a general solver for these large-scale discrete variable structural topology optimization problems.

Bio-inspired computing: Algorithms review, deep analysis, and the scope of applications

Bio-inspired computing represents the umbrella of different studies of computer science, mathematics, and biology in the last years. Bio-inspired computing optimization algorithms is an emerging approach which is based on the principles and inspiration of the biological evolution of nature to develop new and robust competing techniques. In the last years, the bio-inspired optimization algorithms are recognized in machine learning to address the optimal solutions of complex problems in science and engineering. However, these problems are usually nonlinear and restricted to multiple nonlinear constraints which propose many problems such as time requirements and high dimensionality to find the optimal solution. To tackle the problems of the traditional optimization algorithms, the recent trends tend to apply bio-inspired optimization algorithms which represent a promising approach for solving complex optimization problems. This paper presents state-of-art of nine of recent bio-inspired algorithms, gap analysis, and its applications namely; Genetic Bee Colony (GBC) Algorithm, Fish Swarm Algorithm (FSA), Cat Swarm Optimization (CSO), Whale Optimization Algorithm (WOA), Artificial Algae Algorithm (AAA), Elephant Search Algorithm (ESA), Chicken Swarm Optimization Algorithm (CSOA), Moth flame optimization (MFO), and Grey Wolf Optimization (GWO) algorithm. The previous related works are collected from Scopus databases are presented. Also, we explore some key issues in optimization and some applications for further research. We also analyze in-depth discussions the essence of these algorithms and their connections to self-organization and its applications in different areas of research are presented. As a result, the proposed analysis of these algorithms leads to some key problems that have to be addressed in the future.

Analysis of Solution Quality of Multiobjective Evolutionary Algorithms for Service Restoration in Distribution Systems

This paper presents a comparative study of six multiobjective evolutionary algorithms (MOEAs) with the node-depth encoding (NDE) which have been used to solve the service restoration in distribution systems. The study has been divided into three steps: (1) the MOEAs have been evaluated taking into account the switching operations necessary to find adequate restoration plans considering multiple nonlinear constraints and objective functions; (2) the MOEAs have been employed to solve four different datasets with 3860, 7720, 15,440 and 30,880 buses, respectively; (3) comparisons have been performed using the hypervolume indicator and the results obtained with each approach are statistically compared using Kruskal–Wallis nonparametric tests and multiple comparisons. In addition, this paper provides a comprehensive evaluation of six combinations of MOEAs based on NDE and our objective is to identify the features of each approach that consistently produce best results applied to network reconfiguration for service restoration in distribution systems. Simulations results have shown that MOEA based on NDE with crowding distance and strength pareto found good configurations with low switching operations and explored the search space better than others approaches used in this paper, approximating better the pareto-optimal front.

Optimization techniques in respiratory control system models

One of the most complex physiological systems whose modeling is still an open study is the respiratory control system where different models have been proposed based on the criterion of minimizing the work of breathing (WOB). The aim of this study is twofold: to compare two known models of the respiratory control system which set the breathing pattern based on quantifying the respiratory work; and to assess the influence of using direct-search or evolutionary optimization algorithms on adjustment of model parameters. This study was carried out using experimental data from a group of healthy volunteers under CO2 incremental inhalation, which were used to adjust the model parameters and to evaluate how much the equations of WOB follow a real breathing pattern. This breathing pattern was characterized by the following variables: tidal volume, inspiratory and expiratory time duration and total minute ventilation. Different optimization algorithms were considered to determine the most appropriate model from physiological viewpoint. Algorithms were used for a double optimization: firstly, to minimize the WOB and secondly to adjust model parameters. The performance of optimization algorithms was also evaluated in terms of convergence rate, solution accuracy and precision. Results showed strong differences in the performance of optimization algorithms according to constraints and topological features of the function to be optimized. In breathing pattern optimization, the sequential quadratic programming technique (SQP) showed the best performance and convergence speed when respiratory work was low. In addition, SQP allowed to implement multiple non-linear constraints through mathematical expressions in the easiest way. Regarding parameter adjustment of the model to experimental data, the evolutionary strategy with covariance matrix and adaptation (CMA-ES) provided the best quality solutions with fast convergence and the best accuracy and precision in both models. CMAES reached the best adjustment because of its good performance on noise and multi-peaked fitness functions. Although one of the studied models has been much more commonly used to simulate respiratory response to CO2 inhalation, results showed that an alternative model has a more appropriate cost function to minimize WOB from a physiological viewpoint according to experimental data.

An adaptive multi-path computation framework for centrally controlled networks

The Adaptive Dynamic Multi-Path Computation Framework (ADMPCF) is to provide an integrated resource control and management platform with an adequate set of management applications for better routing and resource allocation in centrally controlled or loosely coupled distributed software defined networking (SDN), especially for large network systems. As an open and extensible solution framework, it can provide the necessary infrastructure and integrates data collection and analytics, network performance evaluation, and various optimization algorithms. ADMPCF utilizes a set of complementary algorithms that work together in an adaptive and intelligent fashion that enable global routing and resource allocation optimization. It can also be easily extended to incorporate new algorithms through some open APIs. Such an approach would be able to efficiently and effectively adapt to the rapid changes in network topology, states, and most importantly application traffic, while it is often infeasible for a single optimization algorithm to get satisfactory solution for multiple nonlinear optimization objectives and constraints for a large and centrally controlled network. As it would be costly for centrally controlled global optimization algorithms to calculate good routes dynamically with adequate response time, the proposed ADMPCF framework takes advantage of many hidden patterns of the network information fragments in the combinations of network topology, states, and traffic flows. Therefore, it can lead to a much improved data structure for fast search and match that avoids the expensive re-optimization whenever possible.

Hidden Markov model parameters estimation with independent multiple observations and inequality constraints

In this study, the authors focus on hidden Markov model (HMM) parameters estimation with independent multiple observations and non-linear inequality constraints. The parameters estimation process is divided into four steps: initialisation, parameters pre-estimation, parameters re-estimation and termination. The pre-estimation results are used to approximate non-linear inequality constraints to linear inequality constraints. In parameters re-estimation step, the active-set optimisation is combined with the expectation maximisation (EM) algorithm in M-step and the active set-based EM algorithm is proposed to re-estimate HMM parameters when inequality constraints are not satisfied in pre-estimation. An auxiliary function is devised for reconstructing the optimisation objective function and the convergence of the proposed algorithm is also demonstrated. Simulation results indicate that the proposed algorithm provides better performance by modifying the random error of observation data appropriately and it is powerful for industry process fault diagnosis.

Multiobjective evolutionary algorithm with a discrete differential mutation operator developed for service restoration in distribution systems

Network reconfiguration for service restoration in distribution systems is a combinatorial complex optimization problem that usually involves multiple non-linear constraints and objective functions. For large scale distribution systems no exact algorithm has found adequate restoration plans in real-time. On the other hand, the combination of Multi-Objective Evolutionary Algorithms (MOEAs) with the Node-Depth Encoding (NDE) has been able to efficiently generate adequate restoration plans for relatively large distribution systems (with thousands of buses and switches). The approach called MEAN results from the combination of NDE with a technique of MOEA based on subpopulation tables. In order to improve the capacity of MEAN to explore both the search and objective spaces, this paper proposes a new approach that results from the combination of MEAN with characteristics from the mutation operator of the Differential Evolution (DE) algorithm. Simulation results have shown that the proposed approach, called MEAN-DE, is able to find adequate restoration plans for distribution systems from 3860 to 30,880 switches. Comparisons have been performed using the Hypervolume metric and the Wilcoxon rank-sum test. In addition, a MOEA using subproblem Decomposition and NDE (MOEA/D-NDE) was investigated. MEAN-DE has shown the best average results in relation to MEAN and MOEA/D-NDE.

Massively Parallel Industry-strength Design of Aerodynamic Wings

The paper addresses an application of the software product OPTIMENGA_AERO to a real-life design of aerodynamic shapes. The optimization method employs Genetic Algorithms which make use of a full Navier-Stokes solver for accurate estimation of major aerodynamic coefficients in the evaluation of the objective function. Thereby, the implementation of the method requires huge computational resources, which makes efficient parallelization crucial for successful promotion of the method to an engineering environment. The algorithm is based on a multilevel embedded parallelization approach, which includes five levels of parallelization. Multi-point aerodynamic shape design (implemented on a medium-size distributed memory cluster) deals with transonic wing optimization for minimum drag in the presence of multiple nonlinear constraints. The software has showed its robustness in the demanding industrial environment when applied to real-life industrial design. The results demonstrate high accuracy of optimization combined with high parallel efficiency. The results indicate that the described computational technology may allow for a technological breakthrough in the industrial aerodynamic design of air vehicles in aircraft industry.

An evolutionary approach for worst-case tolerance design

A sliding-level orthogonal differential evolution algorithm with a two-level orthogonal array (SLODEA2OA) is proposed for solving worst-case tolerance design problems. Tolerance affects system performance and leads to violate design constraints. By including a two-level orthogonal array, the proposed SLODEA2OA obtains robust optimal solutions that minimize the impact of parameter variations and that maintain compliance with a comprehensive constraint set. Two design examples are used for performance evaluation of the SLODEA2OA. The first is a 10-variable function, which includes linear, non-linear, quadratic, and polynomial forms to illustrate its general robustness and computational efficiency. The second example is a speed reducer design that involves seven variables and multiple non-linear engineering constraints. The SLODEA2OA is also compared with sliding-level orthogonal differential evolution algorithms with either three-level orthogonal array or two-level full-factorial design. Additionally, performance comparisons confirm that the proposed SLODEA2OA outperforms nature-inspired methods presented in the literature.

Optimization of Ships’ Diagonal Ballast Water Exchange Sequence Using a Multiobjective Genetic Algorithm

Ballast water management and the method chosen to achieve it is a key issue and concerns key technologies in ship design. If the sequential exchange method is the chosen method, the sequence chosen to perform the exchange is very important and affects many aspects including ships' stability, structural strength, maneuverability, operational expenses and building cost, and so forth. In this paper, based on the multiple risk assessment criteria of the sequential method, the problem of finding a feasible and optimum exchange sequence is boiled down to a multiobjective combinatorial optimization problem with multiple nonlinear constraints. The diagonal exchange strategy was adopted, and the diagonal exchange mathematical model was built, taking into consideration the ship's intact stability, structural strength, trims, draughts, and bridge vision. In order to find a set of Pareto solutions, a multiobjective genetic algorithm (MOGA) was used. In the algorithm, a constraint-domination principle was adopted to handle the multiple constraints, and a nondominated sorting method was used to perform the selection of the Pareto solutions. Using the proposed mathematic model and the MOGA, a Pareto solution set that met all the design criteria could be efficiently and accurately obtained for the engineers to choose from within short running times. Compared with the traditional symmetrical exchanging method, the simulation results showed that the proposed method can produce more and better solutions with smaller trims, smaller bridge blind vision range, and better structural strength performances.

Optimization of ship’s subdivision arrangement for offshore sequential ballast water exchange using a non-dominated sorting genetic algorithm

Ship’s subdivision arrangement is a multi-objective combinatorial optimization problem with multiple nonlinear constraints. This study focuses on finding a methodology for ship’s subdivision arrangement that can guarantee ship’s offshore sequential ballast water exchanging (SBWE) performances in the preliminary design stage. A mathematical model is built using minimizing trims and hull girder longitudinal bending moments and shearing forces occurred in the SBWE as the objectives, and the multiple safety criteria of the SBWE as the constraints. The longitudinal lengths of the ballast water tanks (BWTs) are taken as design variables that will alter within a reasonable length range. An elitist nondominated sorting genetic algorithm (NSGA-II) is utilized to perform the optimization, in which the nondominated sorting mechanism is employed to handle the multiple objectives, and the constraint-domination principle is utilized to handle the multiple constraints. A special crossover operator called the dispersion apportioned allelic (DAA) crossover is introduced to perform the reproduction of the special problem. A real case study of the subdivision arrangement based on the SBWE of a 50,000DWT double hull product tanker is conducted to demonstrate the feasibility and effectiveness of the proposed approach.

Optimal disc cutters plane layout design of the full-face rock tunnel boring machine (tbm) based on a multi-objective genetic algorithm

Improving of the quality of the disc cutters’ plane layout design of the full-face rock tunnel boring machine (TBM) is the most effective way to improve the global performance of a TBM. The plane layout design of disc cutters contains multiple complex engineering technical requirements and belongs to a multi-objective optimization problem with multiple nonlinear constraints. Based on analysis of the technical requirements of the plane layout problem, an optimizing mathematical model was built. To obtain a set of design schemes for engineers to choose from, a multi-objective genetic algorithm (MOGA) was applied to carry out the optimization of the mathematical model. A constraint-domination principle was utilized to handle the constraints, and a nondominated sorting method was adopted to obtain Pareto solutions. Simulation results showed that the proposed method was efficient and accurate in obtaining the Pareto layout solutions.

On multiple constraint skewed models

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.

Optimal ballast scheme design for cargo ships using an improved genetic algorithm

Loading design is a key stage in ships' design process, and directly affects ships' stability, strength, and attitude at sea. The problem of loading design is a multi-objective combinatorial optimisation problem with multiple nonlinear constraints. In this paper, based on the complex technical requirements of ships' loading design, a mathematical model was constructed, taking the overall longitudinal strength and the intact stability performances as objectives, and the limits of ship's attitude, stability and strength, etc. as constraints. A strategy of tanks being fully-loaded-first was adopted and the corresponding improved genetic algorithm was presented where a novel heuristic dynamic adjusting rule was proposed to dynamically adjust the normalising coefficients of the non-commensurable objectives. Finally, a real case study on the loading design of a 50,000 DWT double hull product tanker was conducted to illustrate the effectiveness of the proposed method.

Design and optimization of interplanetary low‐thrust trajectory with planetary aerogravity‐assist maneuver

PurposeThe purpose of this paper is to study the application of the planetary aerogravity‐assist (AGA) technique to the interplanetary transfer mission with low‐thrust engine, and the design and optimization approach of low‐thrust AGA trajectory.Design/methodology/approachIn the research, the transfer trajectory with planetary AGA maneuver is analyzed first, the maximum atmospheric turn angle and the matching condition for AGA trajectory is derived out, which is the significant principle for AGA trajectory design and studies. Then, a design and optimization approach for interplanetary low‐thrust trajectory with AGA maneuver is developed. The complicated design problem is transformed into a parameter optimization problem with multiple nonlinear constraints by using calculus of variations and the matching condition associated with AGA trajectory. Furthermore, since the optimization problem is very sensitive to the launch date and AGA maneuver parameters, three ordinal sub‐problems are reformulated to reduce the sensitivity. Finally, a direct/indirect hybrid approach is utilized to solve these sub‐problems.FindingsThe planetary AGA maneuver is feasible and effective in decreasing the propellant consumption and flight time for interplanetary low‐thrust mission and provides better performance than pure planetary gravity assist. Moreover, the proposed approach is effective to design and optimize the low‐thrust transfer mission with AGA maneuver.Research limitations/implicationsIn further research, some simple preliminary design approaches for interplanetary low‐thrust trajectory with AGA maneuver are required to developed, which can provide a good initial conjecture for a hybrid optimization algorithm.Originality/valueThe paper provides the matching condition for interplanetary AGA transfer trajectory by analyzing some characteristics of planetary AGA maneuver, and presents an effective approach to design and optimize interplanetary low‐thrust AGA trajectory.

Developing Centrifugal Compressor Train Optimization Models for Performance Evaluation

Recent advances in numerical optimization software now allow problems arising in the evaluation of complex centrifugal compressor train performance to be solved readily by machinery users. Analyses of compressor operations often require consideration of multiple nonlinear constraints involving pressure, temperature, flow rate, composition, and power. Based on these constraints, the operational boundaries of the machinery must be evaluated. Frequently operating points of greatest interest exist where one of the operational variables, such as flow rate, is maximized subject to operational constraints. A methodology is proposed that can be used to apply commercially available optimization software to problems of this type. A review of gas equation of state relationships needed to determine thermodynamic properties is provided, as well as methods for calculating compressor discharge pressure based on polytropic head and efficiency. Recycle control is considered and a model is proposed.