Dynamic Constrained Optimization Problems Research Articles

Performance Analysis of TBR Tire Curing Process

Numerical algorithms and computer programs have been developed to determine optimal cure steps in a tire curing process. A dynamic constrained optimization problem was formulated with the following ingredients: (1) an objective function that measures product quality in terms of final state of cure and temperature history at selected points in a tire; (2) constraints that consist of a process model and temperature limits imposed on cure media; (3) B-spine representation of a time-varying profile of cure media temperature. The optimization problem was solved using the complex algorithm along with a finite element model solver. Numerical simulations were carried out to demonstrate the procedure of determining optimal cure steps for a truck/bus radial tire. © 1999 John Wiley & Sons, Inc. J Apple Polymer Sic 74: 2063–2071, 1999.

Dynamic constrained evolutionary optimization based on deep Q-network

Dynamic constrained optimization problems (DCOPs) are common and important optimization problems in real-world, which have great difficulty to solve. Dynamic constrained evolutionary algorithms (DCEAs) are widely used methods for solving DCOPs. However, existing DCEAs often struggle with convergence, particularly for DCOPs with drastic dynamic changes or intricate constraints. To address this issue, this paper proposes a novel DCEA called DCEA-DQN, which leverages the powerful perception and decision-making capabilities of Deep Q-Network (DQN). DCEA-DQN integrates two DQNs to enhance its performance. The first DQN is designed to adaptively respond to dynamic changes, enabling effective handling of DCOPs with various types and degrees of changes. It provides a high-quality re-initialized population for subsequent static optimization, resulting in faster and improved convergence. The second DQN is introduced to guide the mutation direction during offspring generation. It steers the population towards better feasible regions or directs it towards the optimal individual within the current feasible region. Moreover, a penalty mechanism is employed to handle constraints during offspring generation. To evaluate the performance of DCEA-DQN, comprehensive empirical studies are conducted using a new test suite called C-GMPB and a dynamic flexible job-shop scheduling problem. The experimental results, using two commonly used metrics EB and EO in the field of DCOPs, demonstrate that DCEA-DQN outperforms six state-of-the-art DCEAs and achieved optimal performance on 80% and 75% of all 24 test problems, respectively. The source code for DCEA-DQN is available at https://github.com/CIA-SZU/YRT.

Experiment Analysis of TBR Tire Curing Process

The optimization problem was solved using the complex algorithm along with a finit element model solver. Numerical simulations were carried out to demonstrate the procedure of determining optimal cure steps for a truck/bursa dial tire. © 1999 John Wiley & Sons, Inc. J App Poly m Sci 74: 2063–2071, 1999 dynamic constrained optimization problem was formulated with the following ingredients: (1) an objective function that measures product quality in terms of Final state of cure and temperature history a at selected points in a tire; (2) constraints that consist of a process model and temperature limits imposed on cure media; (3) B-splines representation of a time-varying Profile of cure Media temperature

A dynamic constrained multiobjective evolutionary algorithm based on decision variable classification

In dynamic constrained multiobjective optimization problems (DCMOPs), dynamics may arise from time-varying objective functions or/and constraints. To solve these problems, maintaining a good balance among feasibility, convergence and diversity of a population under dynamic environments is a critical challenge. Although appropriately utilizing the characteristic of decision variables can promote algorithms to better track the Pareto optima under dynamic environments, their sensitivity to constraints is neglected. Therefore, a dynamic constrained multiobjective evolutionary algorithm based on decision variable classification (DC-MOEA-DVC) is proposed. Under each environment, decision variables are classified into four types in terms of their influence on convergence, distribution, and constraint violation. Based on them, a new offspring generation method is developed, decision variables with different characteristics are rationally combined to generate offspring, with the purpose of accelerating the convergence of the population. Once an environmental change appears, a hybrid strategy consisting of four change response techniques is introduced for the corresponding types of decision variables, producing a new initial population. The experimental results show that DC-MOEA-DVC is superior to the other five state-of-the-art algorithms.

EvoDCMMO: Benchmarking and solving dynamic constrained multimodal optimization problems

In the field of evolutionary computation, many dynamic optimization problems whose fitness values and/or constraints are changed over time have been studied. As the research progresses, complex dynamic optimization problems draw the attention of researchers, such as the dynamic constrained optimization problems with constraints and the dynamic multimodal optimization problems with multiple optima in each environment. Dynamic, constrained, and multimodal properties frequently appear in real-world applications, but the dynamic constrained multimodal optimization problems (DCMMOPs) that simultaneously possess all these properties have not been studied yet. This paper is the extended version of our conference work, aiming to investigate the effect of different techniques for solving DCMMOP. Compared with the previous work, we proposed a more challenging benchmark suite and designed a framework called EvoDCMMO to make further research on DCMMOPs. The new challenge of the proposed benchmark problems arises from the additional constraint based on the fitness values. More techniques, including the optimizers, niching methods, constraint handling techniques and dynamic response strategies are implemented in EvoDCMMO to carry out the comparative studies for solving DCMMOPs and discussed in our work.

Evolutionary approach for dynamic constrained optimization problems

The number of research works on dynamic constrained optimization problems has been increasing rapidly over the past two decades. In this domain, many real-life decision problems need to be solved repeatedly with changing data and parameters. However, no research on dynamic problems with changes in the coefficients of the constraint functions has been reported. In this paper, to deal with such problems, a new evolutionary framework with multiple novel mechanisms is proposed. The new mechanisms are for (1) dealing with both linear and non-linear components in the constraint functions, (2) identifying the rate of change in the coefficients of the variables and (3) updating the population efficiently after every change occurs in the problem. To evaluate the performance of the proposed algorithm, we designed a new set of 13 dynamic benchmark problems, each of which consists of 20 dynamic changes and 3 different scenarios. The results demonstrate that the proposed algorithm significantly contributes in achieving good quality solutions, high feasibility rates and fast convergence in rapidly changing environments. In addition, the framework shows its capability of using different meta-heuristics to solve dynamic problems.

Dynamic Optimization in Fast-Changing Environments via Offline Evolutionary Search

Dynamic optimization, for which the objective functions change over time, has attracted intensive investigations due to the inherent uncertainty associated with many real-world problems. For its robustness with respect to noise, Evolutionary Algorithms (EAs) have been expected to have great potential for dynamic optimization. Many dynamic optimization methods such as diversity-driven methods, memory methods, and prediction methods have been proposed based on EAs to deal with environmental changes. However, they face difficulties in adapting to fast changes in dynamic optimization as EAs normally need quite a few fitness evaluations to find a near-optimum solution. To address this issue, this paper proposes a new framework of applying EAs in the context of dynamic optimization to deal with fast changing environments. We suggest that, instead of online evolving (searching) solutions for the ever-changing objective function, EAs are more suitable for acquiring an archive of solutions in an offline way, which could be adopted to construct a system to provide high-quality solutions efficiently in a dynamic environment. To be specific, we formulate the offline search as a static set-oriented optimization problem. Then, a set of solutions is obtained by an EA for this set-oriented optimization problem. After this, the obtained solution set is adopted to do fast adaptation to the corresponding dynamic optimization problem. The general framework is instantiated for continuous dynamic constrained optimization problems, and the empirical results show the potential of the proposed framework. The superiority of the framework is also verified on a dynamic vehicle routing problem with changing demands.

Self-organizing migrating algorithm using covariance matrix adaptation evolution strategy for dynamic constrained optimization

The dynamic constrained optimization problems can be a challenge for the optimization algorithms. They must tackle global optimum detection, as well as the change of the environment. Recently, a novel test suite for dynamic constrained optimization was introduced. Furthermore, three well-performed evolutionary algorithms were compared based on it. The experimental results show that each algorithm performed best for a different type of optimization problem. The objective of our work was to develop an algorithm reflecting requirements arising from the novel test suite and regarding the results provided by the tested algorithms. In this work, we present a novel evolutionary algorithm for dynamic constrained optimization. The algorithm hybridizes the self-organizing migrating algorithm and the covariance matrix adaptation evolution strategy with constraints handling approach. To avoid premature convergence, the best solutions representing feasible regions do not affect the rest of the population. Two clustering methods, exclusion radius, and quantum particles are used to preserve population diversity. The performance is evaluated on the recently published test suite and compared to the three state-of-the-art algorithms. The presented algorithm outperformed these algorithms in most test cases, which indicates the efficiency of the utilized mechanisms.

Transient analysis of systems exhibiting inverse response and their control with CSTR as a case study

<span>In this paper the phenomenon of inverse response from systems was investigated and its transient response thoroughly analyzed. Inverse response is shown by non-minimum phase systems and some minimum phase systems. Transient analysis of these systems is lacking in literature A case study was done for a non-linear, non-minimum phase CSTR (Continuous Stirred Tank Reactor), which was identified and a novel optimized trajectory for temperature feed was synthesized, which is a non-linear dynamic constrained optimization problem solved using ACADO for a particular chemical reaction. Different control schemes were also implemented on other systems as well, which exibhit inverse responses, including model reference adaptive control. The finding shows that PID-ZN goes with the inverse response, is unable to suppress it and requires large control effort which can have serious hardware limitations. Robustness is another area where PID is lacking with these systems. MRAC shemes were able to overcome all these issues. For CSTR also, these findings hold true thus points at using advance stratigies in process control industries for maximzing product yield.</span>

A Novel Evolutionary Algorithm for Dynamic Constrained Multiobjective Optimization Problems

To promote research on dynamic constrained multiobjective optimization, we first propose a group of generic test problems with challenging characteristics, including different modes of the true Pareto front (e.g., convexity–concavity and connectedness–disconnectedness) and the changing feasible region. Subsequently, motivated by the challenges presented by dynamism and constraints, we design a dynamic constrained multiobjective optimization algorithm with a nondominated solution selection operator, a mating selection strategy, a population selection operator, a change detection method, and a change response strategy. The designed nondominated solution selection operator can obtain a nondominated population with diversity when the environment changes. The mating selection strategy and population selection operator can adaptively handle infeasible solutions. If a change is detected, the proposed change response strategy reuses some portion of the old solutions in combination with randomly generated solutions to reinitialize the population, and a steady-state update method is designed to improve the retained previous solutions. The experimental results show that the proposed test problems can be used to clearly distinguish the performance of algorithms, and that the proposed algorithm is very competitive for solving dynamic constrained multiobjective optimization problems in comparison with state-of-the-art algorithms.

Evolutionary dynamic constrained optimization: Test suite construction and algorithm comparisons

Many real-world applications can be modelled as dynamic constrained optimization problems (DCOPs). Due to the fact that objective function and/or constraints change over time, solving DCOPs is a challenging task. Although solving DCOPs by evolutionary algorithms has attracted increasing interest in the community of evolutionary computation, the design of benchmark test functions of DCOPs is still insufficient. Therefore, we propose a test suite for DCOPs. A dynamic unconstrained optimization benchmark with good time-varying characteristics, called moving peaks benchmark, is chosen to be the objective function of our test suite. In addition, we design adjustable dynamic constraints, by which the size, number, and change severity of the feasible regions can be flexibly controlled. Furthermore, the performance of three dynamic constrained optimization evolutionary algorithms is tested on the proposed test suite, one of which is presented in this paper, named dynamic constrained optimization differential evolution (DyCODE). DyCODE includes three main phases: 1) the first phase intends to enter the feasible region from different directions promptly via a multi-population search strategy; 2) in the second phase, some excellent individuals chosen from the first phase form a new population to search for the optimal solution of the current environment; and 3) the third phase combines the memory individuals of the first two phases with some randomly generated individuals to re-initialize the population for the next environment. From the experiments, one can understand the strengths and weaknesses of the three compared algorithms for solving DCOPs in depth. Moreover, we also give some suggestions for researchers to apply these three algorithms on different occasions.

Explicit Solution for Constrained Scalar-State Stochastic Linear-Quadratic Control With Multiplicative Noise

We study in this paper, a class of constrained linear-quadratic (LQ) optimal control problem formulations for the scalar-state stochastic system with multiplicative noise, which has various applications, especially in the financial risk management. The linear constraint on both the control and state variables considered in our model destroys the elegant structure of the conventional LQ formulation and has blocked the derivation of an explicit control policy so far in the literature. We successfully derive in this paper, the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure. We reveal that the optimal control policy is a piecewise affine function of the state and can be computed offline efficiently by solving two coupled Riccati equations. Under some mild conditions, we also obtain the stationary control policy for an infinite time horizon. We demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems.

Antenna design using dynamic multi‐objective evolutionary algorithm

To enhance the robustness and stability of the antenna design, a novel methodology is presented for modelling the antenna design problem as the constrained optimisation problem (COP). Then a framework is introduced for solving the COP. The framework converts a COP into an equivalent dynamic constrained multi-objective optimisation problem (DCMOP). A dynamic constrained multi-objective evolutionary algorithm is applied to address the transformed DCMOP, which means the COP as well as the antenna design problem can be solved. Then it is applied to design three classes of antennas: a wire antenna, a patch antenna and an antenna array. Obtained results of these three kinds of antenna suggest that the proposed methods can well satisfy the antenna design requirements.

Continuous Dynamic Constrained Optimization With Ensemble of Locating and Tracking Feasible Regions Strategies

Dynamic constrained optimization problems (DCOPs) are difficult to solve because both objective function and constraints can vary with time. Although DCOPs have drawn attention in recent years, little work has been performed to solve DCOPs with multiple dynamic feasible regions from the perspective of locating and tracking multiple feasible regions in parallel. Moreover, few benchmarks have been proposed to simulate the dynamics of multiple disconnected feasible regions. In this paper, first, the idea of tracking multiple feasible regions, originally proposed by Nguyen and Yao, is enhanced by specifically adopting multiple subpopulations. To this end, the dynamic species-based particle swam optimization (DSPSO), a representative multipopulation algorithm, is adopted. Second, an ensemble of locating and tracking feasible regions strategies is proposed to handle different types of dynamics in constraints. Third, two benchmarks are designed to simulate the DCOPs with dynamic constraints. The first benchmark, including two variants of G24 (called G24v and G24w), could control the size of feasible regions. The second benchmark, named moving feasible regions benchmark (MFRB), is highly configurable. The global optimum of MFRB is calculated mathematically for experimental comparisons. Experimental results on G24, G24v, G24w, and MFRB show that the DSPSO with the ensemble of strategies performs significantly better than the original DSPSO and other typical algorithms.

An effective simultaneous approach with variable time nodes for dynamic optimization problems

ABSTRACTAn effective simultaneous approach with variable time nodes is proposed to solve the dynamic optimization problems with multiple control components, where the variable time nodes for each control component are considered as parameters directly and the interval between the neighbouring variable nodes is further refined uniformly to ensure accuracy. Consequently, the method does not treat all the nodes as parameters to ensure efficiency. The gradient formulae and the sensitivities of the states with respect to the controls and the variable time nodes are further derived to solve the nonlinear programming problem transformed from the original dynamic optimization problem. The complete framework and detailed steps of the proposed method are also given. Two classic constrained dynamic optimization problems have been tested as an illustration, and detailed comparisons of the reported literature methods are carried out. The research results show the characteristics and the effectiveness of the proposed approach.

A General Framework of Dynamic Constrained Multiobjective Evolutionary Algorithms for Constrained Optimization

A novel multiobjective technique is proposed for solving constrained optimization problems (COPs) in this paper. The method highlights three different perspectives: 1) a COP is converted into an equivalent dynamic constrained multiobjective optimization problem (DCMOP) with three objectives: a) the original objective; b) a constraint-violation objective; and c) a niche-count objective; 2) a method of gradually reducing the constraint boundary aims to handle the constraint difficulty; and 3) a method of gradually reducing the niche size aims to handle the multimodal difficulty. A general framework of the design of dynamic constrained multiobjective evolutionary algorithms is proposed for solving DCMOPs. Three popular types of multiobjective evolutionary algorithms, i.e., Pareto ranking-based, decomposition-based, and hype-volume indicator-based, are employed to instantiate the framework. The three instantiations are tested on two benchmark suites. Experimental results show that they perform better than or competitive to a set of state-of-the-art constraint optimizers, especially on problems with a large number of dimensions.

Dynamic differential evolution with combined variants and a repair method to solve dynamic constrained optimization problems: an empirical study

An empirical study of the algorithm dynamic differential evolution with combined variants with a repair method (DDECV $$+$$+ Repair) in the solution of dynamic constrained optimization problems is presented. Unexplored aspects of the algorithm are of particular interest in this work: (1) the role of each one of its elements, (2) its sensitivity to different change frequencies and change severities in the objective function and the constraints, (3) its ability to detect a change and recover after it, besides its diversity handling (percentage of feasible and infeasible solutions) during the search, and (4) its performance with dynamism present in different parts of the problem. Seven performance measures, eighteen recently proposed test problems and eight algorithms found in the specialized literature are considered in four experiments. The statistically validated results indicate that DDECV $$+$$+ Repair is robust to change frequency and severity variations, and that it is particularly fast to recover after a change in the environment, but highly depends on its repair method and its memory population to provide competitive results. DDECV $$+$$+ Repair shows evidence on the convenience of keeping a proportion of infeasible solutions in the population when solving dynamic constrained optimization problems. Finally, DDECV $$+$$+ Repair is highly competitive particularly when dynamism is present in both, objective function and constraints.

An Adaptive Pseudospectral Method for Constrained Dynamic Optimization Problems in Chemical Engineering

AbstractAn adaptive pseudospectral method with two novel strategies is proposed to solve constrained dynamic optimization problems in chemical engineering. The first strategy is to introduce slope information in designing appropriate subintervals, so that the approach becomes more efficient to track the optimal control profiles. The second strategy is to redistribute the collocation points based on the approximation error, thus ensuring the accuracy of the method. Two constrained dynamic optimization problems with multiple control variables are tested as illustrations, and the proposed approach is compared with other methods. The research results reveal the effectiveness of the proposed method in improving the solution accuracy of dynamic optimization problems.

Many-objective optimization with dynamic constraint handling for constrained optimization problems

In real-world applications, the optimization problems are usually subject to various constraints. To solve constrained optimization problems (COPs), this paper presents a new methodology, which incorporates a dynamic constraint handling mechanism into many-objective evolutionary optimization. Firstly we convert a COP into a dynamic constrained many-objective optimization problem (DCMaOP), which is equivalent to the COP, then the proposed many-objective optimization evolutionary algorithm with dynamic constraint handling, called MaDC, is realized to solve the DCMaOP. MaDC uses the differential evolution (DE) to generate individuals, and a reference-point-based nondominated sorting approach to select individuals. The effectiveness of MaDC is verified on 22 test instances. The experimental results show that MaDC is competitive to several state-of-the-art algorithms, and it has better global search ability than its peer algorithms.

Solving chemical dynamic optimization problems with ranking-based differential evolution algorithms

Dynamic optimization problems (DOPs) described by differential equations are often encountered in chemical engineering. Deterministic techniques based on mathematic programming become invalid when the models are non-differentiable or explicit mathematical descriptions do not exist. Recently, evolutionary algorithms are gaining popularity for DOPs as they can be used as robust alternatives when the deterministic techniques are invalid. In this article, a technology named ranking-based mutation operator (RMO) is presented to enhance the previous differential evolution (DE) algorithms to solve DOPs using control vector parameterization. In the RMO, better individuals have higher probabilities to produce offspring, which is helpful for the performance enhancement of DE algorithms. Three DE-RMO algorithms are designed by incorporating the RMO. The three DE-RMO algorithms and their three original DE algorithms are applied to solve four constrained DOPs from the literature. Our simulation results indicate that DE-RMO algorithms exhibit better performance than previous non-ranking DE algorithms and other four evolutionary algorithms.