Nonlinear Constrained Optimization Problem Research Articles

The Proximal Alternating Direction Method of Multipliers for a Class of Nonlinear Constrained Optimization Problems

This paper presents a class of proximal alternating direction multiplier methods for solving nonconvex, nonsmooth optimization problems with nonlinear coupled constraints. The key feature of the proposed algorithm is that we use a linearized proximal technique to update the primary variables, followed by updating the dual variables using a discounting approach. This approach eliminates the requirement for an additional proxy function and then simplifies the optimization process. In addition, the algorithm maintains fixed parameter selection throughout the update process, removing the requirement to adjust parameters to ensure the decreasing nature of the generated sequence. Building on this framework, we establish a Lyapunov function with sufficient decrease and a lower bound, which is essential for analyzing the convergence properties of the algorithm. We rigorously prove both the subsequence convergence and the global convergence of the algorithm and ensure its robustness and effectiveness in solving complex optimization problems. Our paper provides a solid theoretical foundation for the practical application of this method in solving nonconvex optimization problems with nonlinear coupled constraints.

Perturbation Method and Regularization of the Lagrange Principle in Nonlinear Constrained Optimization Problems

Perturbation Method and Regularization of the Lagrange Principle in Nonlinear Constrained Optimization Problems

Mechanism and Control Strategies for Current Sharing in Multi-Chip Parallel Automotive Power Modules

Multi-chip parallel power modules are highly favored in applications requiring high capacity and high switching frequency. However, the dynamic current imbalance between parallel chips caused by asymmetric layouts limits the available capacity. This paper presents a method to optimize dynamic current distribution by adjusting the lengths and connection points of bond wires. For the first time, a response surface model and nonlinear constraint optimization algorithm are introduced, along with parameter analysis based on finite element methods, to establish the response surface models for the parasitic inductance of bond wires and DBC (direct bonded copper). By leveraging the optimization goals for parasitic inductance and the analytical expressions of all response surfaces, the dynamic current sharing issue was transformed into a nonlinear constrained optimization problem. The solution to this optimization problem identified the optimal connection points for the bond wires, enhancing dynamic current sharing performance. Simulations and experiments were conducted, revealing that the optimized automotive-grade module exhibited a significant reduction in current differences between parallel branches, from 41.7% to 5.03% compared with the original design. This indicated that the proposed optimization scheme for adjusting bond wire connection points could significantly mitigate current disparities, thereby markedly improving current distribution uniformity.

A Novel Path-Following Method for Time-Varying Optimizations with Optimal Parametric Functions

We address a broad category of nonlinear constrained optimization problems. We then reformulate it as a time-varying optimization using continuous-time parametric functions and derive a dynamical system to track the respective optimal solution. We then re-parameterize the dynamical system according to a linear combination of parametric functions. By applying the calculus of variations, we optimize these parametric functions to minimize the optimality distance. Consequently, we develop an iterative dynamic algorithm, termed as OP-TVO, to achieve efficient convergence to the solution. We compare the performance of the proposed algorithm with the prediction correction method (PCM) in terms of optimality and computational complexity. The results demonstrate that OP-TVO effectively competes with PCM for the given class of optimization problems, suggesting a promising alternative to PCM. This work also introduces a novel paradigm for solving parametric dynamic systems.

A new optimal strategy for energy minimization in wireless sensor networks

In recent years, evolutionary and metaheuristic algorithms have emerged as crucial tools for optimization in the field of artificial intelligence. These algorithms have the potential to revolutionize various aspects of our lives by leveraging the multidisciplinary nature of wireless sensor networks (WSNs). This study aims to introduce genetic and simulated annealing algorithms as effective solutions for enhancing WSN performance. Our contribution entails two main phases. Firstly, we establish mathematical models and formulate objectives as a nonlinear constrained optimization problem. Secondly, we develop two algorithmic solutions to address the formulated optimization problem. The obtained results from multiple simulations demonstrate the positive impact of the proposed strategies on improving network performance in terms of energy consumption.

Monarch butterfly optimization-based genetic algorithm operators for nonlinear constrained optimization and design of engineering problems

Abstract This paper aims to present a hybrid method to solve nonlinear constrained optimization problems and engineering design problems (EDPs). The hybrid method is a combination of monarch butterfly optimization (MBO) with the cross-over and mutation operators of the genetic algorithm (GA). It is called a hybrid monarch butterfly optimization with genetic algorithm operators (MBO-GAO). Combining MBO and GA operators is meant to overcome the drawbacks of both algorithms while merging their advantages. The self-adaptive cross-over and the real-valued mutation are the GA operators that are used in MBO-GAO. These operators are merged in a distinctive way within MBO processes to improve the variety of solutions in the later stages of the search process, speed up the convergence process, keep the search from getting stuck in local optima, and achieve a balance between the tendencies of exploration and exploitation. In addition, the greedy approach is presented in both the migration operator and the butterfly adjusting operator, which can only accept offspring of the monarch butterfly groups who are fitter than their parents. Finally, popular test problems, including a set of 19 benchmark problems, are used to test the proposed hybrid algorithm, MBO-GAO. The findings obtained provide evidence supporting the higher performance of MBO-GAO compared with other search techniques. Additionally, the performance of the MBO-GAO is examined for several EDPs. The computational results show that the MBO-GAO method exhibits competitiveness and superiority over other optimization algorithms employed for the resolution of EDPs.

Combined emission economic dispatch using quantum-inspired particle swarm optimization and its variants

The ever-increasing electricity demand, its dependency on fossil fuels, and the consequent environmental degradation are major concerns of this era. The worldwide domination of fossil fuels in bulk electricity generation is rapidly increasing the emissions of CO2 and other environmentally dangerous gases that are contributing to climate change. The economic and emission dispatch are two important problems in thermal power generation whose combination produces a complex highly constrained nonlinear optimization problem known as combined economic and emission dispatch. The optimization of combined economic and emission dispatch aims to allocate the generation of committed units to minimize fuel cost and emissions, simultaneously while honoring all equality and inequality constraints. Therefore, in this article, we investigate a solution of the combined economic and emission dispatch problem using quantum particle swarm optimization and its two modified versions, that is, enhanced quantum particle swarm optimization and quantum particle swarm optimization integrated with weighted mean personal best and adaptive local attractor. The enhanced quantum particle swarm optimization algorithm achieves particles’ diversification at early stages and shows good performance in local search at later stages. The quantum particle swarm optimization integrated with weighted mean personal best and adaptive local attractor boosts search performance of quantum particle swarm optimization and attains better global optimality. The suggested methods are employed to achieve solution for the combined economic and emission dispatch in four distinct systems, encompassing two scenarios with 6 units each, one with a 10-unit configuration, and another with an 11-unit setup. A comparative analysis with methodologies documented in existing literature reveals that the proposed approach outperforms others, demonstrating superior computational performance and robust efficiency.

Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems

Abstract Motion planning and controller design are challenging tasks for highly coupled and nonlinear dynamical systems such as autonomous vehicles and robotic applications. Nonlinear model predictive control (NMPC) is an emerging technique in which sampling-based methods are used to synthesize the control and trajectories for complex systems. In this study, we have developed the sampling-based motion planning algorithm with NMPC through Bayesian estimation to solve the online nonlinear constrained optimization problem. In the literature, different filtration techniques have been applied to extract knowledge of states in the presence of noise. Due to the detrimental effects of linearization, the Kalman filter with NMPC only achieves modest effectiveness. Moving horizon estimation (MHE), on the other hand, frequently relies on simplifying assumptions and lacks an effective recursive construction. Additionally, it adds another optimization challenge to the regulation problem that has to be solved online. To address this problem, particle filtering is implemented for Bayesian filtering in nonlinear and highly coupled dynamical systems. It is a sequential Monte Carlo method that involves representing the posterior distribution of the state of the system using a set of weighted particles that are propagated through time using a recursive algorithm. For nonlinear and strongly coupled dynamical systems, the novel sampling-based NMPC technique is effective and simple to use. The efficiency of the suggested method has been assessed using simulated studies.

Wind Power Bidding Based on an Ensemble Differential Evolution Algorithm with a Problem-Specific Constraint-Handling Technique

The intermittent nature of wind power generation induces great challenges for power bidding in the electricity market. The deployment of battery energy storage can improve flexibility for power bidding. This paper investigates an optimal power bidding strategy for a wind–storage hybrid power plant in the day-ahead electricity market. To handle the challenges of the uncertainties of wind power generation and electricity prices, the optimal bidding problem is formulated as a risk-aware scenario-based stochastic programming, in which a number of scenarios are generated using a copula-based approach to represent the uncertainties. These scenarios consider the temporal correlation of wind power generation and electricity prices between consecutive time intervals. In the stochastic programming, a more practical but nonlinear battery operation cost function is considered, which leads to a nonlinear constrained optimization problem. To solve the nonlinear constrained optimization problem, an ensemble differential evolution (EDE) algorithm is proposed, which makes use of the merits of an ensemble of mutant operators to generate mutant vectors. Moreover, a problem-specific constraint-handling technique is developed. To validate the effectiveness of the proposed EDE algorithm, it is compared with state-of-the-art DE-based algorithms for constrained optimization problems, including a constrained composite DE (C2oDE) algorithm and a novel DE (NDE) algorithm. The experimental results demonstrate that the EDE algorithm is much more reliable and much faster in finding a better bidding strategy against benchmarking algorithms. More precisely, the average values of the success rate are 0.893, 0.667, and 0.96 for C2oDE, NDE, and EDE, respectively. Compared to C2oDE and NDE, the average value of the mean number of function evaluations to succeed with EDE is reduced by 76% and 59%, respectively.

A Novel Fuzzy Best-Worst Multicriteria Decision-Making Method Based on the Dual Interval Algorithm for Environmental Decision Support Systems

Considering the double uncertainty caused by the ambiguity of statistical data and the ambiguity produced by the subjective assessment of decision makers, the crisp values of criteria may be insufficient to model the multi-criteria decision-making (MCDM) problem in the real world. This paper proposes a fuzzy best-worst method (FBWM) based on a dual-interval solution algorithm to extend the best-worst method (the most recent MCDM method) to fuzzy environments. The reference comparisons for the best criteria and for the worst criteria are represented by fuzzy numbers. Then, according to the BWM method, a nonlinear constrained optimization problem with fuzzy parameters is formulated. We decompose the membership function in fuzzy numbers into several interval numbers of special form and solve the aforementioned fuzzy BWM problem by compound interval algorithm to obtain fuzzy weights of different criteria. Meanwhile, an integral type-reduced method is proposed for determining the fuzzy consistency ratio in order to assess the reliability of the FBWM results. The viability of the new algorithm to expand the BWM method into fuzzy environments has been validated through three numerical examples. In contrast to the existing FBWM method, the proposed method avoids the arithmetic operation between fuzzy numbers during the solution process, directly transfers the uncertain information in the membership function corresponding to the fuzzy comparison vector to the result, and generates fuzzy weight value, which indicates that the proposed algorithm is able to obtain accurate BWM results in fuzzy environments. The results of the study provide new solution ideas for multi-criteria optimization problems under uncertainty.

A machine learning-based stochastic optimal energy management framework for a renewable energy-assisted isolated microgrid system

ABSTRACT This paper proposes a cost-based stochastic optimal energy management framework for a renewable energy-assisted isolated microgrid system. These microgrids encourage the integration of multiple distributed energy sources, including the penetration of renewable energy. For this purpose, the optimal day-ahead dispatch of the connected energy resources is obtained for an economically viable system by solving a nonlinear constrained optimization problem. The renewable energy and the load demand data forecasting are accomplished using the Gaussian process regression learning model in the MATLAB/SIMULINK® environment for obtaining the day-ahead dispatch. The optimal problem is solved through sequential quadratic programming and a hybrid function approach incorporating particle swarm optimization for a comprehensive techno-economical analysis. A comparative assessment of the results is accomplished to obtain a more feasible and economical system operation corresponding to different time horizons and other critical factors such as fast iterations, computational accuracy, solution feasibility, convergence rate, etc.

ToRoS: A Topology Optimization Approach for Designing Robotic Skins

Soft robotics offers unique advantages in manipulating fragile or deformable objects, human-robot interaction, and exploring inaccessible terrain. However, designing soft robots that produce large, targeted deformations is challenging. In this paper, we propose a new methodology for designing soft robots that combines optimization-based design with a simple and cost-efficient manufacturing process. Our approach is centered around the concept of robotic skins---thin fabrics with 3D-printed reinforcement patterns that augment and control plain silicone actuators. By decoupling shape control and actuation, our approach enables a simpler and cost-efficient manufacturing process. Unlike previous methods that rely on empirical design heuristics for generating desired deformations, our approach automatically discovers complex reinforcement patterns without any need for domain knowledge or human intervention. This is achieved by casting reinforcement design as a nonlinear constrained optimization problem and using a novel, three-field topology optimization approach tailored to fabrics with 3D-printed reinforcements. We demonstrate the potential of our approach by designing soft robotic actuators capable of various motions such as bending, contraction, twist, and combinations thereof. We also demonstrate applications of our robotic skins to robotic grasping with a soft three-finger gripper and locomotion tasks for a soft quadrupedal robot.

Adaptive Beamforming with Sidelobe Level Control for Multiband Sparse Linear Array

Multiband antenna arrays have the capability of effectively working in multiple frequency bands and thus significantly simplify the antenna system. To further reduce the system overhead, this paper discusses the joint design of antenna selection and adaptive beamforming for multiband antenna arrays, where the sidelobe level is also controlled so as to alleviate the effect of unknown sporadic interference. Based on the maximum signal-to-interference-plus-noise ratio (SINR) criterion and sidelobe level constraints, the proposed multiband sparse array design is formulated into a nonconvex constrained nonlinear optimization problem with an l0,2-mixed norm regularization. This problem ensures that the same antenna positions are selected at all operating frequencies while the beamformer weights of each frequency are optimized independently. By exploiting the semi-definite relaxation and the reweighted l1,∞-norm approximation, the problem is converted into a series of convex subproblems and is then effectively solved by the proposed iterative reweighted method. Numerical results show that the proposed multiband sparse array significantly reduces the sidelobe levels in all operating frequencies while maintaining the maximum SINR, so it provides superior performance of interference suppression.

Phase retrieval via Lagrange programming neural network

In this paper, the problem of phase retrieval is addressed. Its solution is based on the Lagrange programming neural network (LPNN), which is an analog neural computational technique for solving nonlinear constrained optimization problems according to the Lagrange multiplier theory. The local stability of the proposed algorithm is also investigated. Furthermore, we extend the LPNN based approach to more challenging array signal processing applications, namely, when multiple vector signals with full column rank or other constraints are required to be recovered from measured magnitudes. One of the key difficulties for these challenges is pairing the separately estimated parameters, while the parameters estimated by the extended method are automatically paired. The performance of the developed algorithms is demonstrated via computer simulations.

Numerical fact-finding of different functions impact on the fuzzy preference programming optimality

The fuzzy analytic hierarchy process (FAHP) is a widely implemented approach for determining the weights of criteria and alternatives from pairwise comparison matrices. FAHP methodology uses triangular fuzzy as a tool for the decision-maker to arrange judgments expressed as the pairwise comparison matrix elements. FAHP is then modeled into a nonlinear constrained optimization problem to increase the decision maker’s satisfaction. However, in some cases, this model would yield multiple optimal solutions. Logarithmic fuzzy preference programming (LFPP) is offered to overcome this unfavorable situation. Nevertheless, LFPP includes a parameter in its model for which no specific method is available to determine. Meanwhile, the differences in parameter values will affect the solution. In the worst case, the weights obtained will be inaccurate. As a result, the decision taken based on the weights will be biased and unreliable. This issue is particularly important when it requires high accuracy. It includes the decisions on the portfolio investment, economic, human capital measurements, and shipping registry. Another crucial thing is that the use of the natural logarithm function in the LFPP method has the potential for an overflow effect at the computational stage when it is implemented to derive the weights. Therefore, in the application context, it is significant to improve the shortcomings of the LFPP method in terms of parameter involvement and the use of the logarithmic function. For this purpose, this paper proposes a non-parameter transcendental fuzzy preference programming (TFPP) model for deriving the optimal weights from fuzzy pairwise comparison matrices. The TFPP is obtained by investigating different functions’ effects on fuzzy preference programming (FPP). The proposed method has a more general form. One of the specific forms of the TFPP is chosen in a computational implementation. The numerical results confirm that the TFPP optimization model is reliable in obtaining the optimum global weights. The comparison with the LFPP method also shows that the TFPP is more efficient.

A Nonlinear Optimization Design Algorithm for Nearly Linear-Phase 2D IIR Digital Filters

In this paper, a new optimization method for the design of nearly linear-phase two-dimensional infinite impulse (2D IIR) digital filters with a separable denominator is proposed. A design framework for 2D IIR digital filters is formulated as a nonlinear constrained optimization problem where the group delay deviation in the passband is minimized under prescribed soft magnitude constraints and hard stability requirements. To achieve this goal, sub-level sets of the group delay deviations are utilized to generate a sequence of filters, from which the one with the best performance is selected. The quality of the obtained filter is evaluated using three quality factors, namely, the passband magnitude quality factor Qh and the group delay deviation quality factor Qτ, while the third one is a new quality factor Qs that assesses the performance in the stopband relative to the minimum filter gain in the passband. The proposed framework is implemented using the interior-point (IP) method in a MATLAB environment, and the experimental results show that filters designed using the proposed method have good magnitude response and low group delay deviation. The performance of the resulting filters is compared with the results of other methods.

A sustainable two-echelon green supply chain coordination model under fuzziness incorporating carbon pricing policies.

Noxious effect on environmental due to carbon emissions is being addressed worldwide by governments through carbon pricing instruments. Two of prevalent instruments adopted by governments are simple carbon tax and cap-and-trade policy. Effectiveness of carbon pricing instruments towards achieving reduction in carbon emissions is a matter of study on one hand. Whereas, the planning of supply chain operations under imposition of such financial instruments is a challenge for small and medium scale business enterprises. Addressing this situation, the present study develops a supply chain model for coordinated planning of production and inventory replenishment schedules along with decision on economic amounts of expenditure for green resources. This decision model is formulated separately under the enforcement of each of two carbon pricing policies. The study focuses a manufacturer-retailer duo which works by adopting certain sustainability and conservation practices. Manufacturer reworks on rectifiable proportion of defective units, while retailer launches discounts-based sales of partially damaged units. The decision-making model developed in this study incorporates such activities. Furthermore, practical aspects like a slowdown in production due to unforeseeable disruptions and the effect of the quality of product and advertisement campaigns on demand rates are included in the proposed model. Under the purview of each carbon pricing policy, decision-making model is formulated as a nonlinear constrained optimization problem with objective towards profit maximization. A novel conception of fuzziness has been suggested for tackling imprecision in the assessment of certain parameters involved in the model. A numerical study on an appropriate case of a manufacturer-retailer duo system is presented. Empirical results of numerical study evince that a substantial reduction in carbon emission is achieved, even with an escalation in the profit through appropriate green expenditure. This trend is observed under the imposition of each of the carbon pricing policies, thereby substantiating an encouragement to supply chain partners for making expenditures on green resources. Thereby, the hypothesis of getting desired response on curbing emissions by incentivization through carbon pricing is satisfied in the studied case. Furthermore, a sensitivity analysis concludes the stability of formulated model against most of the parameters involved.

Health-considered energy management strategy for fuel cell hybrid electric vehicle based on improved soft actor critic algorithm adopted with Beta policy

Deep reinforcement learning-based energy management strategy (EMS) is essential for fuel cell hybrid electric vehicles to reduce hydrogen consumption, improve health performance and maintain charge. This is a complex nonlinear constrained optimization problem. In order to solve the problem of high bias caused by the inconsistency between the infinite support of stochastic policy and the bounded physics constraints of application scenarios, this paper proposes the Beta policy to improve standard soft actor critic (SAC) algorithm. This work takes hydrogen consumption, health degradation of both fuel cell system and power battery, and charge margin into consideration to design an EMS based on the improved SAC algorithm. Specifically, an appropriate tradeoff between money cost during driving and charge margin is firstly determined. Then, optimization performance differences between the Beta policy and the standard Gaussian policy are presented. Thirdly, ablation experiments of health constraints are conducted to show the validity of health management. Finally, comparison experiments indicate that, the proposed strategy has a 5.12% performance gap with dynamic programming-based EMS with respect to money cost, but is 4.72% better regarding to equivalent hydrogen consumption. Moreover, similar performances in validation cycle demonstrate good adaptability of the proposed EMS.

New structure to design interval observers for linear continuous-time systems

In this contribution, a new approach to design interval observer for continuous-time linear systems is introduced. First, a more general structure than the classical Luenberger observer is proposed and extended to the interval case based on an explicit set integration method. Then, to obtain tight state enclosures, the computation of the design matrices involved in this new structure is formulated as a nonlinear constrained optimisation problem. In addition, using the symmetric property of interval vectors, a less complex form of the proposed interval observer is presented. On the other hand, compared to the conventional interval observers, no preserving order property on the dynamics of the estimation error is required to apply the introduced method. A numerical case study example is considered to illustrate the effectiveness of the proposed method and to compare its performance to that of a former method.

Dynamic Optimal Power Dispatch in Unbalanced Distribution Networks with Single-Phase Solar PV Units and BESS

Battery energy storage systems (BESSs) are a promising solution for increasing efficiency and flexibility of distribution networks (DNs) with a significant penetration level of photovoltaic (PV) systems. There are various issues related to the optimal operation of DNs with integrated PV systems and BESS that need to be addressed to maximize DN performance. This paper deals with day-ahead optimal active–reactive power dispatching in unbalanced DNs with integrated single-phase PV generation and BESS. The objectives are the minimization of cost for electricity, energy losses in the DN, and voltage unbalance at three-phase load buses by optimal management of active and reactive power flows. To solve this highly constrained non-linear optimization problem, a hybrid particle swarm optimization with sigmoid-based acceleration coefficients (PSOS) and a chaotic gravitational search algorithm (CGSA)called the PSOS-CGSA algorithm is proposed. A scenario-based approach encompassing the Monte Carlo simulation (MCS) method with a simultaneous backward reduction algorithm is used for the probabilistic assessment of the uncertainty of PV generation and power of loads. The effectiveness of the proposed procedure is evaluated through aseries test cases in a modified IEEE 13-bus feeder. The simulation results show that the proposed approach enables a large reduction in daily costs for electricity, as well as a reduction in expected daily energy losses in the DN by 22% compared to the base case without BESS while ensuring that the phase voltage unbalance rate (PVUR) is below the maximum limit of 2% for all three-phase buses in the DN.