Nonlinear Constrained Optimization Problem Research Articles

Research on The Optimal Solution of Lagrangian Multiplier Function Method in Nonlinear Programming

For the general nonlinear constrained optimization model, this article will propose a new nonlinear Lagrange function, discuss the properties of the function at the KKT point, and prove that under appropriate conditions, the iterative point generated by the dual algorithm based on the function The column is locally convergent, and then an error estimate of the solution related to the penalty parameter is given. This provides a new way to solve nonlinear constrained optimization problems.

A mixture varying-gain dynamic learning network for solving nonlinear and nonconvex constrained optimization problems

Nonlinear and nonconvex optimization problem (NNOP) is a challenging problem in control theory and applications. In this paper, a novel mixture varying-gain dynamic learning network (MVG-DLN) is proposed to solve NNOP with inequality constraints. To do so, first, this NNOP is transformed into some equations through Karush–Kuhn–Tucker (KKT) conditions and projection theorem, and the neuro-dynamics function can be obtained. Second, the time varying convergence parameter is utilized to obtain a faster convergence speed. Third, an integral term is used to strengthen the robustness. Theoretical analysis proves that the proposed MVG-DLN has global convergence and good robustness. Three numerical simulation comparisons between FT-FP-CDNN and MVG-DLN substantiate the faster convergence performance and greater robustness of the MVG-DLN in solving the nonlinear and nonconvex optimization problems.

A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.

Optimization of a deteriorated two-warehouse inventory problem with all-unit discount and shortages via tournament differential evolution

Tournamenting process plays an important role in the selection of best alternative in a knockout system. This concept is applied in the development of a hybrid algorithm based on differential evolution known as tournament differential evolution. In this work, a hybrid tournament differential evolution algorithm is applied on a two-warehouse inventory control problem of deteriorating item with the objective for determining the lot-size, maximum shortage level and cycle length of the concerned system. The corresponding inventory model is formulated with two separate warehouse facilities, partially backlogged shortages and all-unit discount. Here the demand of the product increases with time. The corresponding optimization problem of the developed inventory model is highly non-linear constrained optimization problem which cannot be solved analytically. In this context, to solve the optimization problem, the said hybrid algorithm is developed with the help of six different options of tournamenting procedure and differential evolution. Then, to examine the validity of the proposed model and also to test the performance of the hybrid algorithm, numerical experiments are performed by solving a numerical example and nonparametric statistical test. Finally, to investigate the effects of changes of different parameters, sensitivity analyses are carried out on the best found policy and a fruitful conclusion is drawn along with the future scope of research.

Optimized time step for electric vehicle charging optimization considering cost and temperature

An optimal decentralized scheduling strategy for charging one Electric Vehicle (EV) is proposed to minimize the customer charging cost. Moreover, the EVs can offers more profit when considering the vehicle to grid feature, by discharging the EV in the grid at high peak demand the EV’ owner can earn money and reduce his charging bill. Compared to existing methods, the main advantages of the proposed strategy is the considerations of an optimized time step. By doing so, the optimization problem uses a minimum number of decision variables and constraints. Then, the problem can be solved by all optimization method to reach the global optimum in reduced time. To formulate and solve a non-linear constrained optimization problem, the scheduling process takes into consideration: the time of arrival and time departure of the EV, the daily energy prices, the initial State of Charge (SoC) and the final SoC desired by the customer, the power limitations, and the temperature. The results obtained show a high impact of the optimal scheduling strategy and significant charging cost reduction compared to the uncontrolled charging and fixed time step algorithms. Moreover, the charging strategy only requires that each EV solves its optimization problem locally, therefore, its deployment requires a low computing capacity.

Nonlinear MPC for collision-free and deadlock-free navigation of multiple nonholonomic mobile robots

In this paper, we present an online nonlinear Model Predictive Control (MPC) method for collision-free, deadlock-free navigation by multiple autonomous nonholonomic Wheeled Mobile Robots (WMRs). Our proposed method solves a nonlinear constrained optimization problem at each time step over a specified horizon to compute a sequence of optimal control inputs that drive the robots to target poses along collision-free trajectories, where the robots’ future states are predicted according to a unicycle kinematic model. To reduce the computational complexity of the optimization problem, we formulate it without stabilizing terminal constraints or terminal costs. We describe a computationally efficient approach to programming and solving the optimization problem, using open-source software tools for fast nonlinear optimization and applying the multiple-shooting method. We also provide rigorous proofs of the feasibility of the optimization problem and the stability of the proposed method. To validate the performance of our MPC method, we implement it in both 3D robot simulations and experiments with real nonholonomic WMRs for different multi-robot navigation scenarios with up to six robots. In all scenarios, the robots successfully navigate to their goal poses without colliding with one another or becoming trapped in a deadlock.

Robust design of ball bearings for an improved performance using genetic algorithm

The design of rolling bearings must be optimized to get an improved fatigue life. As the fatigue life is directly affected by the dynamic capacity, therefore the later is considered as an objective function, which is to be maximized using a robust design optimization. Such design procedures enhance the qualities and reliability of products by ensuring that they are insensitive to the source of variabilities or uncertainties present in them, without eliminating these sources and at the same time maintaining the design feasibility with probabilistic constraints. In the present work, deep groove ball bearings (DGBBs) are designed in such a way that it will have minimum variation in the objective function due to the variation in design variables. A nonlinear constrained optimization problem is formulated with one objective function, ten design variables and a set of total thirty-five constraints. The real-coded genetic algorithm (GA) is used as the optimization tool. The result obtained shows the optimized bearings have improved fatigue life. A convergence study is performed to ensure global optima in the design space and further a sensitivity analysis is performed. The analysis shows that the dynamic capacity is the most sensitive to the variation in the inner groove curvature radius and least sensitive to the bearing pitch diameter. This paper allows the designers to design the rolling bearings by optimizing the fatigue life along with the minimization of their variations, which are due to variations or having tolerances in input design variables during its manufacturing. Moreover, the proposed design methodology is not limited to the bearings only, but can also be enforced in the design of any machine components (like clutches, gears, cams, levers, springs, etc.) having substantial uncertainties in their geometry or material, such as the residual strain, modulus, thickness, density, etc.

Computationally Efficient Approximations Using Adaptive Weighting Coefficients for Solving Structural Optimization Problems

With rapid development of advanced manufacturing technologies and high demands for innovative lightweight constructions to mitigate the environmental and economic impacts, design optimization has attracted increasing attention in many engineering subjects, such as civil, structural, aerospace, automotive, and energy engineering. For nonconvex nonlinear constrained optimization problems with continuous variables, evaluations of the fitness and constraint functions by means of finite element simulations can be extremely expensive. To address this problem by algorithms with sufficient accuracy as well as less computational cost, an extended multipoint approximation method (EMAM) and an adaptive weighting-coefficient strategy are proposed to efficiently seek the optimum by the integration of metamodels with sequential quadratic programming (SQP). The developed EMAM stems from the principle of the polynomial approximation and assimilates the advantages of Taylor’s expansion for improving the suboptimal continuous solution. Results demonstrate the superiority of the proposed EMAM over other evolutionary algorithms (e.g., particle swarm optimization technique, firefly algorithm, genetic algorithm, metaheuristic methods, and other metamodeling techniques) in terms of the computational efficiency and accuracy by four well-established engineering problems. The developed EMAM reduces the number of simulations during the design phase and provides wealth of information for designers to effectively tailor the parameters for optimal solutions with computational efficiency in the simulation-based engineering optimization problems.

Optimum Design of Reinforced Concrete Box Culvert by Using Deterministic Approach

An optimum safe design is achieved for a reinforced concrete box culvert considering hydraulic, geotechnical and structural requirements and prices. To reach this intention, an optimization technique methodology is applied in estimation a safe design with a minimum price. The non-linear constrained optimization problem is generally expressed in terms of an objective function and constraints. Eleven design variables and forty constraint equations were employed in this work, with the total price of reinforced concrete box culvert as the objective function. The solution is obtained by means of developed computer software written in Visual BASIC. The computer software considers the Sequential Unconstrained Minimization Technique (SUMT), which is one of non-linear optimization techniques. In order to evaluate the influence of the input parameters on the box culvert design, the optimization program has been modified. The input design parameters are the discharge, head loss, length, entrance loss coefficient, Manning’s roughness coefficient, angle of internal friction of soil, compressive strength of concrete, yield strength of steel, unit price of concrete, unit price of excavation, unit price of filling and unit price of steel. A parametric study is formed to examine the influence and significance of each input design parameter on the total price of box culvert.

Energy-Optimized Partial Computation Offloading in Mobile-Edge Computing With Genetic Simulated-Annealing-Based Particle Swarm Optimization

Smart mobile devices (SMDs) can meet users’ high expectations by executing computational intensive applications but they only have limited resources, including CPU, memory, battery power, and wireless medium. To tackle this limitation, partial computation offloading can be used as a promising method to schedule some tasks of applications from resource-limited SMDs to high-performance edge servers. However, it brings communication overhead issues caused by limited bandwidth and inevitably increases the latency of tasks offloaded to edge servers. Therefore, it is highly challenging to achieve a balance between high-resource consumption in SMDs and high communication cost for providing energy-efficient and latency-low services to users. This work proposes a partial computation offloading method to minimize the total energy consumed by SMDs and edge servers by jointly optimizing the offloading ratio of tasks, CPU speeds of SMDs, allocated bandwidth of available channels, and transmission power of each SMD in each time slot. It jointly considers the execution time of tasks performed in SMDs and edge servers, and transmission time of data. It also jointly considers latency limits, CPU speeds, transmission power limits, available energy of SMDs, and the maximum number of CPU cycles and memories in edge servers. Considering these factors, a nonlinear constrained optimization problem is formulated and solved by a novel hybrid metaheuristic algorithm named genetic simulated annealing-based particle swarm optimization (GSP) to produce a close-to-optimal solution. GSP achieves joint optimization of computation offloading between a cloud data center and the edge, and resource allocation in the data center. Real-life data-based experimental results prove that it achieves lower energy consumption in less convergence time than its three typical peers.

A Fuzzy BWM Method for Evaluating Supplier Selection Factors in a SME Paper Manufacturer

In today’s competitive world, companies are required to increase productivity and produce high-quality products while managing their operational efficiency. This leads a firm to involve its performing suppliers by taking into account multiple factors. For small and medium-sized enterprises (SMEs), despite its importance, supplier selection is less considered. In this study, a case study for evaluating supplier selection factors in a small-medium scale paper manufacturer in Indonesia is demonstrated using the fuzzy best-worst method (fuzzy BWM). The BWM method that provides a structurally consistent factors comparison through the best and the worst criteria is extended using a fuzzy set to address the ambiguity from decision-makers’ judgement. The evaluation includes five supplier criteria: service, flexibility and delivery, reputation, quality, and purchasing cost. To determine the fuzzy criteria’ weights, the nonlinearly constrained optimization problem was introduced. The result indicated that the best and the worst criteria were respectively service and purchasing cost, while the service criterion was weakly crucial than quality. Besides, the Fuzzy BWM has also provided high consistency in the comparison. The optimal weights determined in this study can be applied for ranking suppliers for further attempts.

Towards Human-like Bimanual Movements in Anthropomorphic Robots: A Nonlinear Optimization Approach

Previously we have presented a model for generating human-like arm and hand movements on an unimanual anthropomorphic robot involved in human-robot collaboration tasks. The present paper aims to extend our model in order to address the generation of human-like bimanual movement sequences which are challenged by scenarios cluttered with obstacles. Movement planning involves large scale nonlinear constrained optimization problems which are solved using the IPOPT solver. Simulation studies show that the model generates feasible and realistic hand trajectories for action sequences involving the two hands. The computational costs involved in the planning allow for real-time human robot-interaction. A qualitative analysis reveals that the mo vements of the robot exhibit basic characteristics of human movements.

A grasshopper optimization algorithm for optimal short-term hydrothermal scheduling

The optimal generation for short-term hydrothermal scheduling (OGStHS) with the deliberation of various purposes is a complex non-linear constrained optimization problem. There exist numerous constraints, which make the OGStHS optimization problem more complicated. The considered constraints for this problem are mostly related to energy performance, operational conditions, water, and power infrastructure. All these constraints would generally influence the cost of fuel. In this study, a multi-objective optimization form of OGStHS is suggested to estimate the minimum cost of fuel, which mainly influences industrial operation. The water transfer delays among multi-related reservoirs and the thermal plants’ valve-point influences are considered for the accurate formulation of the OGStHS problem. Meantime, a grasshopper optimization algorithm (GOA) is performed to handle the OGStHS problem by getting optimized for both objectives concurrently. A modern approach is shown in this study to get a solution to the OGStHS problem. Furthermore, to deal with the complex restraints efficiently, modern heuristic restriction treatment processes with no drawback impact frames have been offered in this study. Two hydrothermal power systems have illustrated the suggested GOA technique’s utility and performance. Compared with other available approaches, the analytical results are admitted that GOA can provide a better understanding by decreasing fuel cost and emission concurrently.

Optimum design of 3R manipulator using hybrid PSOGSA algorithm

Larger workspace volume is one of the most important objectives in the optimum design of robot manipulator. In the present study, the workspace volume of a 3R manipulator has been maximised. The nonlinear constrained optimisation problem has been solved using the particle swarm optimisation (PSO) and a hybrid PSO and gravitational search algorithm (GSA). The proposed algorithm combines the search method of PSO and GSA with enhanced exploration ability to achieve higher workspace volume as compared to the established available results. Further, the total void cross section area has been estimated and a quantitative analysis of the results has been made to identify the key influencing kinematic parameters and prioritise the constraints. Applications of the hybrid algorithm on two industrial manipulators are presented as case studies. An important implication of this article is that the proposed hybrid algorithm provides encouraging result in terms of objective function values and CPU time.

Višekriterijumski optimalni izbor hidrauličkog cilindra za pogonske mehanizme

The paper reports the optimization synthesis of a hydraulically actuated drive mechanism. A mathematical model of the mechanism using vector closure equations is developed. Based on the functional purpose of the mechanism, a set of geometric and force/moment requirements are defined which must be met by a proper selection of a standardized hydraulic cylinder and its points of attachment. A multiobjective design optimization task is defined with three objective functions whose minimum is searched - the mass of the hydraulic cylinder, the squared total deviation of the developed by the hydraulic cylinder moments from the predefined values of the external moments and the force in the hydraulic cylinder. The defined multiobjective optimization task is considered as a mixed variable nonlinear constrained optimization problem containing 5 continuous and 2 discrete variables and the multistage Monte Carlo method is used for its solution. Using different weighting schemes several Pareto-optimal compromise solutions are obtained.

Easy Particle Swarm Optimization for Nonlinear Constrained Optimization Problems

Particle swarm optimization (PSO) is a popular stochastic approach for solving practical optimal problems from industries due to its effective performance and few hyperparameters. Nonlinear constrained optimization (NCO) problems frequently cause multiple optimal regions and can cause many infeasible regions in the search space. The state-of-the-art approaches for handling the infeasible regions generated by problems’ constraints either block particles’ paths or penalize NCO problems’ objective values based on the standard updating velocity formula. The standard updating velocity formula introduces difficulties for particles in searching the undiscovered optimal solutions separated by infeasible regions and being mutually restrained on directions by social and cognitive factors. Afterward, the particles cause premature convergence and difficulty searching the undiscovered optimal regions to improve their solutions. Observing the biological ant colony and inspired by lazy ant behavior, this study proposes an easy particle that simulates the lazy ant to diversify the moving direction. Finally, this study integrates the proposed easy particles with referenced PSO-based approaches for solving NCO problems. The experiment results show that the proposed easy particles can effectively reinforce exploration abilities and improve the performances of all referenced PSO-based algorithms to reduce the status of premature convergence in solving NCO problems.

Solving optimal power flow problem with stochastic wind–solar–small hydro power using barnacles mating optimizer

Optimal Power Flow (OPF) is a complex and challenging problem in power system that includes non-convex and non-linear constrained optimization problems. Due to these features, solving the OPF problem is becoming a well-known area to be solved by researchers for the past decades especially involving with the proper optimizing of the control variables. This issue is vital to be solved in achieving the objectives while maintaining of the stability of the system. In this paper, recent metaheuristic algorithm namely Barnacles Mating Optimizer (BMO) will be used to solve three objective functions of OPF problem viz. (1) cost minimization of the power generation that consists of thermal and stochastic wind–solar–small hydro power generations, (2) power loss minimization, and (3) combined cost and emission minimization of mentioned power generations. To assess the performance of BMO into OPF problem, modified IEEE 30-bus and IEEE 57-bus systems that incorporate the stochastic wind–solar–small hydro power generators will be employed. Statistical studies are performed to show the feasible and effectiveness of BMO compared with other selected metaheuristic algorithms. Based on the obtained results, the BMO has shown the best results for all cases of simulation study. For example, the cost of power generations obtained by BMO for IEEE 30-bus and IEEE 57-bus systems are 789.1248 $/h and 5300.457 $/hr, respectively which 1.45% and 2.2% cost saving per hour compared to the worst results obtained from compared algorithms. The results suggest that BMO performs better compared to the rest of algorithms and demonstrate as effective alternative for the OPF problem solution.

A general two-phase Markov chain Monte Carlo approach for constrained design optimization: Application to stochastic structural optimization

This contribution presents a general approach for solving structural design problems formulated as a class of nonlinear constrained optimization problems. A Two-Phase approach based on Bayesian model updating is considered for obtaining the optimal designs. Phase I generates samples (designs) uniformly distributed over the feasible design space, while Phase II obtains a set of designs lying in the vicinity of the optimal solution set. The equivalent model updating problem is solved by the transitional Markov chain Monte Carlo method. The proposed constraint-handling approach is direct and does not require special constraint-handling techniques. The population-based stochastic optimization algorithm generates a set of nearly optimal solutions uniformly distributed over the vicinity of the optimal solution set. The set of optimal solutions provides valuable sensitivity information. In addition, the proposed scheme is a useful tool for exploration of complex feasible design spaces. The general approach is applied to an important class of problems. Specifically, reliability-based design optimization of structural dynamical systems under stochastic excitation. Numerical examples are presented to evaluate the effectiveness of the proposed design scheme.

Optimal Command Shaping Design for a Liquid Slosh Suppression in Overhead Crane Systems

Abstract In many industries, liquid container transport is carried out by an overhead traveling crane. The operation of crane transferring liquid slosh containers required both operator experience and an automated control system. The goal of this research is to move the liquid inside a container in a short time and less spill for process effectiveness and safety. For controller design, a nonlinear mathematical model is developed to represent the actual system. A cost-effective, smooth continuous command shaper is presented to suppress sloshing vibration. The designed shaper is a multisine-wave function with adjustable and independent time maneuvering used to design the acceleration profile. The coefficients that control the shaper profile are obtained by solving a nonlinear constrained optimization problem using particle swarm algorithm. Simulation and experimental comparative results proved that the proposed command shaper can reduce transient peak slosh amplitudes. Moreover, it can simultaneously cancel both residual sloshing vibrations and container oscillations at the end of the transportation process which cannot be achieved using conventional zero-vibration (ZV), zero-vibration derivative (ZVD), and jerk-limited shaper. Furthermore, sensitivity analysis demonstrates that the proposed command shaper is robust to model parameters variation such as liquid depth, suspension length, or moving distance of the trolley.

Optimal power flow incorporating stochastic wind and solar generation by metaheuristic optimizers

Optimal power flow (OPF) is one of the complex problems in power system operation that includes multi-modal, large-scale, non-convex and non-linear constrained optimization problems. Due to these features, solving the OPF problem is becoming an active topic to be solved by power engineers and researchers. In this paper, recent metaheuristic algorithms namely Grasshopper Optimization Algorithm (GOA), Black Widow Optimization Algorithm, Grey Wolves Optimizer, Ant Lion Optimizer, Particles Swarm Optimization, Gravitational Search Algorithm, Moth-Flame Optimization and Barnacles Mating Optimizer (BMO) will be used to solve three objective functions of OPF problem viz. (1) cost minimization of the power generation that consists of thermal, stochastic wind and solar power generations, (2) power loss minimization, and (3) combined cost and emission minimization of power generations. To assess the performance of these selected metaheuristic algorithms on OPF, a modified IEEE 30-bus system that incorporate the stochastic wind and solar power generators will be used. Statistical studies are performed to identify the effectiveness of algorithms under consideration. Test results suggest that BMO performs better compared to the rest of algorithms and demonstrate that it can be effective alternative for the OPF problem solution.