Constrained Multi-objective Problem Research Articles

Evolution-guided Bayesian optimization for constrained multi-objective optimization in self-driving labs

The development of automated high-throughput experimental platforms has enabled fast sampling of high-dimensional decision spaces. To reach target properties efficiently, these platforms are increasingly paired with intelligent experimental design. However, current optimizers show limitations in maintaining sufficient exploration/exploitation balance for problems dealing with multiple conflicting objectives and complex constraints. Here, we devise an Evolution-Guided Bayesian Optimization (EGBO) algorithm that integrates selection pressure in parallel with a q-Noisy Expected Hypervolume Improvement (qNEHVI) optimizer; this not only solves for the Pareto Front (PF) efficiently but also achieves better coverage of the PF while limiting sampling in the infeasible space. The algorithm is developed together with a custom self-driving lab for seed-mediated silver nanoparticle synthesis, targeting 3 objectives (1) optical properties, (2) fast reaction, and (3) minimal seed usage alongside complex constraints. We demonstrate that, with appropriate constraint handling, EGBO performance improves upon state-of-the-art qNEHVI. Furthermore, across various synthetic multi-objective problems, EGBO shows significative hypervolume improvement, revealing the synergy between selection pressure and the qNEHVI optimizer. We also demonstrate EGBO’s good coverage of the PF as well as comparatively better ability to propose feasible solutions. We thus propose EGBO as a general framework for efficiently solving constrained multi-objective problems in high-throughput experimentation platforms.

Dual-stage and dual-population cooperative evolutionary algorithm for solving constrained multiobjective problems

During the search process, the characteristics of the feasible regions encountered by the population continually change in Constrained Multiobjective Optimization Problems (CMOPs). This variability poses a challenge for traditional evolutionary algorithms, which often struggle to adapt to the diverse problem characteristics of the encountered feasible regions. To overcome this limitation, we propose a Dual-Stage and Dual-Population Cooperative Evolutionary Algorithm (DDCEA) to address CMOPs characterized by diverse feasible regions. DDCEA employs a dual-stage mechanism to adapt the offspring generation strategy and establishes two distinct populations to evaluate offspring using constraint-sensitive and constraint-free strategies. Comparative analyses reveal that DDCEA surpasses chosen state-of-the-art CMOEAs in adapting to the changing feasible regions and then approximating the constrained Pareto fronts.

Process Knowledge-Guided Autonomous Evolutionary Optimization for Constrained Multiobjective Problems

Various real-world problems can be attributed to constrained multi-objective optimization problems. Although there are various solution methods, it is still very challenging to automatically select efficient solving strategies for constrained multi-objective optimization problems. Given this, a process knowledge-guided constrained multi-objective autonomous evolutionary optimization method is proposed. Firstly, the effects of different solving strategies on population states are evaluated in the early evolutionary stage. Then, the mapping model of population states and solving strategies is established. Finally, the model recommends subsequent solving strategies based on the current population state. This method can be embedded into existing evolutionary algorithms, which can improve their performances to different degrees. The proposed method is applied to 41 benchmarks and 30 dispatch optimization problems of the integrated coal mine energy system. Experimental results verify the effectiveness and superiority of the proposed method in solving constrained multi-objective optimization problems.

Fitness Function with Two Rankings Embedded in a Push and Pull Search Framework for Constrained Multi-objective Optimisation Problems

Fitness Function with Two Rankings Embedded in a Push and Pull Search Framework for Constrained Multi-objective Optimisation Problems

Linear Regression-based Autonomous Intelligent Optimization for Constrained Multi-objective Problems

Linear Regression-based Autonomous Intelligent Optimization for Constrained Multi-objective Problems

A constrained multi-objective evolutionary algorithm with clustering based weight vector adaptation

In constrained multi-objective optimization, balancing convergence, diversity, and feasibility of solutions in the population are challenging. Most existing approaches focus on convergence and feasibility but fail to enforce the diversity and uniformity of solutions in the approximated Pareto Front (PF). In decomposition-based approaches, the diversity and uniformity of the obtained solution set depend on the set of weight vectors employed, which are generally fixed and initialized uniformly. However, Constrained Multi-objective Optimization Problems (CMOPs) are characterized by discontinuous Pareto-optimal fronts due to the presence of constraints. Therefore, to maintain better diversity and uniformity in the population, the weight vector set employed in the framework should consider the characteristics of PF, which is not known in advance. However, during the evolution, the characteristics of PF can be estimated and the associated weight vectors can be appropriately distributed. Motivated by this issue, we propose a novel Constrained Multi-Objective Evolutionary Algorithm with a Clustering-based Weight vector Adaptation strategy called CMOEA-CWA. The proposed CMOEA-CWA uses the basic framework of CMOEA-DPMS. After exploring the search space, CMOEA-CWA reinitializes the weight vectors if there exist any discontinuities in the approximated pareto front (PF). In CMOEA-CWA, the PF is approximated using the feasible solutions stored in an archive and novel parameter-free clustering techniques. During the clustering process, if there exist any discontinuities in the approximated PF, one population will exploit the feasible regions with a re-initialized set of weight vectors while the other population will explore the feasible regions with uniformly distributed weight vectors. Empirical results obtained on four popular and widely used benchmarks when compared with nine latest state-of-the-art algorithms (ARMOEA, CCMO, TiGE-2, ToP, MOEA/D-DAE, PPS, CTAEA, MOEA/D-2WA and CMOEA-DPMS) suggest the superiority of the proposed algorithm, especially on 47 complicated constrained multi-objective problems. The proposed algorithm performs statistically better than or comparable (in terms of HV) to ARMOEA, CCMO, TiGE-2, ToP, MOEA/D-DAE, PPS, CTAEA, MOEA/D-2WA and CMOEA-DPMS in 91.48 %, 78.72 %, 100 %, 70.21 %, 85.11 %, 78.72 %, 91.49 % and 82.97 % of test instances, respectively. The performance of the proposed algorithm is also demonstrated on a real-world Economic Environmental Dispatch (EED) problem.

Robust Constrained Multi-Objective Guidance of Supersonic Transport Landing Using Evolutionary Algorithm and Polynomial Chaos

Landing of supersonic transport (SST) suffers from a large uncertainty due to its highly sensitive aerodynamic properties in the subsonic domain, as well as the wind gusts around runways. At the vehicle design stage, a landing trajectory optimization under wind uncertainty in a multi-objective solution space is desired to explore the possible trade-off in its key flight performance metrics. The proposed algorithm solves this robust constrained multi-objective optimal control problem by integrating non-intrusive polynomial chaos expansion into a constrained evolutionary algorithm. The computationally tractable optimization is made possible through the conversion of a probabilistic problem into an equivalent deterministic representation while maintaining a form of the multi-objective problem. The generated guidance trajectories achieve a significant reduction of the uncertainty in their terminal states with a marginal modification in the control history of the deterministic solutions, validating the importance of the consideration of robustness in trajectory optimization.

A coevolutionary constrained multi-objective algorithm with a learning constraint boundary

When solving constrained multi-objective optimization problems, the balance of convergence, diversity, and feasibility plays a pivotal role. To address this issue, this paper proposes a coevolutionary constrained multi-objective algorithm with learning constraint boundary (CCMOLCB). Firstly, the constrained multi-objective problems are transformed by adding an additional objective using the constraint violation degree. Then, the transformed problem is solved by an improved coevolutionary framework which employs two populations. The main population explores the objective space and repairs infeasible solutions to maintain the feasibility of population. Meanwhile, the feasibility and diversity of solutions are balanced by using a dynamic weight coefficient during the evolution, it changes as the number of iterations increases. The subordinate population selects solutions by taking into consideration the learning constraint boundary (LCB). This boundary guarantees convergence of solutions by constraining the search range of the main population, thereby enhancing the environmental selection pressure. The performance of CCMOLCB is compared with seven state-of-the-art constrained multi-objective evolutionary algorithms on five test suites. The experimental results illustrate that CCMOLCB exhibits competitive performance when dealing with this family of problems.

Multistate-Constrained Multiobjective Differential Evolution Algorithm With Variable Neighborhood Strategy.

Multiobjective differential evolution (DE) algorithm (MODE) has been widely used in multiobjective optimization problems. However, due to the complex feasible regions, the optimization efficiency of MODE may decrease when solving constrained multiobjective problems. It is challenging to promote the evolution of population with few feasible solutions. In this article, a multistate-constrained MODE with variable neighborhood strategy (MSCMODE-VNS) is proposed to enhance the optimization effectiveness with complex feasible regions. First, a variable neighborhood DE strategy, based on a specially designed convergence indicator, is designed to accelerate the generation of feasible solutions. Second, a multistate population updating strategy with a comprehensive solution evaluation mechanism is devised to update the population of the next generation to improve the performance of solutions. Third, the convergence analysis, based on the probability theory, is derived to verify the effectiveness of the proposed MSCMODE-VNS algorithm. Finally, experimental results indicate that MSCMODE-VNS can achieve a satisfactory performance on three benchmark test suites and two real-world-constrained multiobjective problems.

Cognitive Diagnosis-Based Personalized Exercise Group Assembly via a Multi-Objective Evolutionary Algorithm

Exercise group recommendation plays an important role in many intelligent education tasks. However, existing approaches make recommendations based on the intrinsic features of exercises without considering students' learning abilities, or make selections from several pre-built exercise groups at the expense of flexibility. Furthermore, although many cognitive diagnosis approaches have successfully revealed students' abilities, how to leverage the diagnosis results for exercise group recommendation is hardly explored. To flexibly recommend suitable exercise groups to students, this paper proposes to assemble personalized exercises as a group based on students' abilities, called Personalized Exercise Group Assembly (PEGA). To solve the PEGA task, we first formulate it as a constrained multi-objective problem (CMOP), where three objectives are designed for enabling the assembled exercises to enhance students' abilities on both weak and new knowledge concepts. Then, we devise an extended neural cognitive diagnosis model to learn student's ability/proficiency on all knowledge concepts to compute the weakness consolidation objective. Besides, we propose a dual-encoding and dual-population based co-evolutionary algorithm to tackle the CMOP, where the main population with binary encoding is used to search which exercises are selected, and the auxiliary population with integer encoding is responsible for accelerating the convergence of the main population via guiding offspring generation of the main population. Experiments on three popular datasets demonstrate the effectiveness of exercises assembled by the proposed algorithm compared to state-of-the-art exercise group recommendation approaches, where our assembled exercises can enhance students' proficiency on both poorly mastered and new knowledge concepts.

Bayesian optimization with active learning of design constraints using an entropy-based approach

The design of alloys for use in gas turbine engine blades is a complex task that involves balancing multiple objectives and constraints. Candidate alloys must be ductile at room temperature and retain their yield strength at high temperatures, as well as possess low density, high thermal conductivity, narrow solidification range, high solidus temperature, and a small linear thermal expansion coefficient. Traditional Integrated Computational Materials Engineering (ICME) methods are not sufficient for exploring combinatorially-vast alloy design spaces, optimizing for multiple objectives, nor ensuring that multiple constraints are met. In this work, we propose an approach for solving a constrained multi-objective materials design problem over a large composition space, specifically focusing on the Mo-Nb-Ti-V-W system as a representative Multi-Principal Element Alloy (MPEA) for potential use in next-generation gas turbine blades. Our approach is able to learn and adapt to unknown constraints in the design space, making decisions about the best course of action at each stage of the process. As a result, we identify 21 Pareto-optimal alloys that satisfy all constraints. Our proposed framework is significantly more efficient and faster than a brute force approach.

A flexible two-stage constrained multi-objective evolutionary algorithm based on automatic regulation

The core element in solving constrained multi-objective problems (CMOPs) with evolutionary algorithms is simultaneously balancing objective optimization and constraint satisfaction. Maintaining this balance becomes more challenging for existing algorithms when dealing with complex CMOPs, as various complex feasible regions often result in CMOPs with very different characteristics. To address this issue, we propose a more flexible two-stage evolutionary algorithm based on automatic regulation (ARCMO), which can effectively control evolutionary trends to adapt to complex CMOPs. Specifically, the first stage performs a fast global search and passes the population information to the second stage. The second stage consists of two dynamically complementary sub-processes: the exploration subprocess and the convergence subprocess. The ratio of these two subprocesses is dynamically adjusted based on information from the first stage, allowing ARCMO to adapt to CMOPs with different complexities. Experiments on several recently proposed benchmark suites and real-world application problems show that ARCMO is more adaptable than the contender algorithms when solving different complex CMOPs.

Optimization-Based Energy Disaggregation: A Constrained Multi-Objective Approach

Recently, optimization-based energy disaggregation (ED) algorithms have been gaining significance due to their capability to perform disaggregation with minimal information compared to the pattern-based ED algorithms, which demand large amounts of data for training. However, the performances of optimization-based ED algorithms depend on the problem formulation that includes an objective function(s) and/or constraints. In the literature, ED has been formulated as a constrained single-objective problem or an unconstrained multi-objective problem considering disaggregation error, sparsity of state switching, on/off switching, etc. In this work, the ED problem is formulated as a constrained multi-objective problem (CMOP), where the constraints related to the operational characteristics of the devices are included. In addition, the formulated CMOP is solved using a constrained multi-objective evolutionary algorithm (CMOEA). The performance of the proposed formulation is compared with those of three high-performing ED formulations in the literature based on the appliance-level and overall indicators. The results show that the proposed formulation improves both appliance-level and overall ED results.

Error estimates for halfspace regularization of state constrained multiobjective elliptic control problems

This paper generalizes the halfspace regularization given in (Jadamba et al. in Syst Control Lett 61(6):707–713, 2012) from scalar to multiobjective optimization for an abstract linearly constrained multiobjective optimization problem. We establish quantitative error estimates for the regularization error in terms of the Hausdorff distance of the efficient set of the original problem to the regularized efficient set. We apply these results to state-constrained multiobjective optimal control problems. In particular, we consider two different situations: distributed and finite-dimensional controls, and for each one, we define two different halfspace regularization schemes given in terms of a family of triangulations of the domain. This leads to two regularized problems corresponding to finitely many pointwise and integral constraints. By applying the abstract results, we get regularization error estimates for the four control problems under general conditions.

Solution of constrained mixed‐integer multi‐objective optimal power flow problem considering the hybrid multi‐objective evolutionary algorithm

An Optimal power flow (OPF) is non-linear and constrained multi-objective problem. OPF problems are expensive and evolutionary algorithms (EAs) are computationally complex to obtain uniformly distributed and global Pareto front (PF). Therefore, here, hybrid two-phase algorithm integrated with parameter less constraint technique is applied to solve OPF problem. Proposed technique combines single and multi-objective EAs to find better convergence and evenly distributed PF. For the validation and effectiveness of proposed algorithm, various conflicting objective functions are formulated and implemented on IEEE 30 and 300-bus network. Each case is independently run twenty times. Hyper volume indicator technique is employed to find the best PF, and the best-compromised solution is obtained by using fuzzy decision-making technique. Recently, maximum integration of wind and solar power is highly encouraged. Complexity of OPF is increased with the integration of uncertain renewable energy resources. Hence, 30-bus test system is modified by replacing some conventional generators with the wind and solar generation. Uncertainties in wind, solar and load demand are modelled by appropriate probability distribution functions. Simulation results show that the proposed method can find the near global PF of highly complex problems subject to satisfying all the operational constraints.

A hybrid granular-evolutionary computing method for cooperative scheduling optimization on integrated energy system in steel industry

Integrated Energy System (IES) in steel industry, typically involving gas, heat and electricity, exhibits distributed nature and complex coupling relationship, which demands effective scheduling optimization to cooperatively utilize the multiple energy media. For this purpose, a Granular-Evolutionary Hybrid Computing (GEHC) framework is proposed in this study to solve the scheduling of IES as a Constrained Multi-Objective Problem (CMOP). More specifically, a Collaborative Fuzzy C-Means (CFCM) clustering model is firstly established, where the horizontal and vertical structures are well constructed to obtain information granules regarding the cooperative relationship among the units in different energy subsystems. Then, a Probability-based Granular Computing (PGrC) approach is proposed for the initial optimization of the IES scheduling, which is guided by the expert knowledge compacted in the information granules. To determine the collaborative parameters in CFCM that essentially controls the cooperation among the multiple energy media, as well as satisfying the complex practical constraints, a Multi-Objective Differential Evolution (MODE)-based method is proposed for further scheduling optimization. To evaluate the performance on cooperativity, convergency and diversity, the proposed GEHC and the compared methods that either has partial collaborations or use other optimization frameworks are applied to a real-world optimization problem. The experimental results demonstrate that the proposed GEHC performs superior to the other methods considering both the objective functions and performance indicators, which can be potentially helpful to achieve well-performed unmanned IES scheduling in steel industry.

Model Predictive Control for Automotive Climate Control Systems via Value Function Approximation

Among the auxiliary loads in light-duty vehicles, the air conditioning system is the single largest energy consumer. For electrified vehicles, the impact of heating and cooling loads becomes even more significant, as they compete with the powertrain for battery energy use and can significantly reduce the range or performance. While considerable work has been made in the field of optimal energy management for electrified vehicles and optimization of vehicle velocity for eco-driving, few contributions have addressed the application of energy-optimal control for heating and cooling loads. This letter proposes an energy management strategy for the thermal management system of an electrified powertrain, based on Model Predictive Control. Starting from a nonlinear model of the vapor compression refrigeration system that captures the dynamics of the refrigerant in the heat exchangers and the power consumption of the system, a constrained multi-objective optimal control problem is formulated to reduce energy consumption while tracking a desired thermal set point. An efficient implementation of MPC is proposed for real-time applications by introducing a terminal cost obtained from the approximation of the global optimal solution.

A novel method for multi-objective design optimization based on fuzzy systems

A novel strategy to design optimization is expressed using the fuzzy preference function concept. This method efficiently uses the designer’s experiences by preference functions and it is also able to transform a constrained multi-objective optimization problem into an unconstrained single-objective optimization problem. These two issues are the most important features of the proposed method which using them, you can achieve a more practical solution in less time. To implement the proposed method, two design optimizations of an unmanned aerial vehicle are considered which are: deterministic and non-deterministic optimizations. The optimization problem in this paper is a constrained multiobjective problem that with attention to the ability of genetic algorithm, this algorithm is selected as the optimizer. Uncertainties are considered and the Monte Carlo simulation (MCS) method is used for uncertainties modeling. The obtained results show a good performance of this technique in achieving optimal and robust solutions.

Integer Search Algorithm: A New Discrete Multi-Objective Algorithm for Pavement Maintenance Management Optimization

Optimization techniques keep road performance at a good level using a cost-effective maintenance strategy. Thus, the trade-off between cost and road performance is a multi-objective function. This paper offers a new multi-objective stochastic algorithm for discrete variables, which is called the integer search algorithm (ISA). This algorithm is applied to an optimal pavement maintenance management system (PMMS), where the variables are discrete. The PMMS optimization can be achieved by maximizing the condition of pavement with a minimum cost at specified constraints, so the PMMS is a constrained multi-objective problem. The ISA and genetic algorithm (GA) are applied to improve the performance condition rating (PCR) of the pavement in developing countries, where the annual budget is limited, so a minimum cost for three years’ maintenance is scheduled. Study results revealed that the ISA produced an optimal solution for multi-function objectives better than the optimal solution of GA.

(2011-6286) A novel method for multi-objective design optimization based on fuzzy systems

A novel strategy to design optimization is expressedusing the fuzzy preference function concept. This method efficientlyuses the designer's experiences by preference functions and it isalso able to transform a constrained multi-objective optimizationproblem into an unconstrained single-objective optimization problem.These two issues are the most important features of the proposedmethod which using them, you can achieve a more practical solutionin less time. To implement the proposed method, two designoptimizations of an unmanned aerial vehicle are considered whichare: deterministic and non-deterministic optimizations. Theoptimization problem in this paper is a constrained multi-objectiveproblem that with attention to the ability of genetic algorithm,this algorithm is selected as the optimizer. Uncertainties areconsidered and the Monte Carlo simulation (MCS) method is used foruncertainties modeling. The obtained results show a good performanceof this technique in achieving optimal and robust solutions.