Exterior Penalty Function Method Research Articles

A recurrent neural network approach for nonconvex interval quadratic programming

In the paper, a new approach is presented to tackle an interval optimization problem with nonconvex interval objective function and affine interval equality and inequality constraints utilizing a recurrent neural network. The approach involves converting the nonconvex interval quadratic programming into a cubic optimization via convex combinations and considering its weight as a decision variable. The recurrent neural network is then established to solve the cubic optimization through subgradient techniques and an exterior penalty function method. Moreover, it is proven that there is a unique global solution to the recurrent neural network, and its trajectory can reach the feasible region in a finite time and remain there. Additionally, it is demonstrated that the trajectory of recurrent neural network converges exponentially or in finite time towards a singleton belonging to the set of critical points of the cubic optimization. This approach is beneficial for interval quadratic programming problems where the weight is unknown or challenging to determine in practical applications. Finally, the effectiveness of this approach is verified through two numerical examples.

Optimization of Rice (Oryza sativa) Crop in Simalungun District using Quadratic Programming Wolfe Model and Exterior Penalty Function Method

Optimization is a way of optimizing the objective function while still paying attention to existing constraints. The aim of this research is to form a mathematical model to optimize the average production of lowland and field rice in Simalungun Regency and to complete the model using Wolfe model quadratic programming and the exterior penalty function method. The mathematical model in this research is a nonlinear model created using the least squares method. Quadratic programming solves nonlinear problems by turning them into linear problems using the Kuhn Tucker condition and linear problems are solved using the Wolfe model. Meanwhile, the exterior penalty function method solves limited (constrained) nonlinear problems into unlimited (unconstrained) nonlinear problems. Based on the calculations of the two methods, optimal results were obtained for both the Wolfe quadratic programming model and the exterior penalty function method, where the average production of wetland and field rice in Simalungun Regency was 203.1925 tonnes with a wetland rice harvest area of 90.077 hectares and a field rice harvest area of 13.092. hectare. Optimization testing using this algorithm can be used as a basis for related parties such as the community and government in developing productivity. The implication of this research is an increase in harvested area and average rice production, in this case maintaining productivity stability.

A Two-Phase Constraint-Handling Technique for Constrained Optimization

A two-phase constraint-handling technique is integrated into the evolutionary algorithms to solve constrained optimization problems (called TPDE) in this article. In phase one, denoted as the exploration phase, an exterior penalty function method with the dynamic penalty coefficients is developed to compare any two candidate solutions, which aims to push the population into the feasible region. To reduce the computational burden, in phase two, denoted as the exploitation phase, an interior penalty function method with the dynamic penalty coefficients is developed, which enhances the search ability by using the information of constraints in feasible solutions. During the optimization process, differential evolution is adopted as the search algorithm to produce the offspring population. Experiment results on four benchmark test suites, namely, IEEE CEC 2006, IEEE CEC 2010, IEEE CEC 2017, and IEEE CEC 2020, indicate that TPDE is competitive with other popular algorithms.

Numerical Optimization of CNT Distribution in Functionally Graded CNT-Reinforced Composite Beams.

This paper is concerned with the numerical optimization of the thickness-wise CNT (carbon nanotube) distribution in functionally graded CNT-reinforced composite (FG-CNTRC) beams to secure the structural safety. The FG-CNTRC in which CNTs are inserted according to the specific thickness-wise distribution pattern are extensively investigated for high-performance engineering applications. The mechanical behaviors of FG-CNTRC structures are definitely affected by the distribution pattern of CNTs through the thickness. Hence, the tailoring of suitable CNT distribution pattern is an essential subject in the design of FG-CNTRC structure for a given boundary and loading conditions. Nevertheless, the thickness-wise CNT distribution pattern has been assumed by several linear functions so that these assumed primitive patterns cannot appropriately respond to arbitrary loading and boundary conditions. In this context, this paper aims to introduce a numerical method for optimally tailoring the CNT distribution pattern of FG-CNTRC beams. As a preliminary stage, the effective stress is defined as the objective function and the layer-wise CNT volume fractions are chosen as the design variables. The exterior penalty-function method and golden section method are adopted for the optimization formulation, together with finite difference scheme for the design sensitivity analysis. The proposed optimization method is illustrated and validated through the benchmark experiments, such that it successfully provides an optimum CNT distribution which can significantly minimize the effective stress, with a stable and rapid convergence in the iterative optimization process.

Joint Resource Allocation for Full-Duplex Ambient Backscatter Communication: A Difference Convex Algorithm

Nowadays, Ambient Backscatter Communication (AmBC) systems have emerged as a green communication technology to enable massive self-sustainable wireless networks by leveraging Radio Frequency (RF) Energy Harvesting (EH) capability. A Full-duplex Ambient Backscatter Communication (FAmBC) network with a Full-duplex Access Point (AP), a dedicated Legacy User (LU), and several Backscatter Devices (BDs) is considered in this study. The AP with two antennas transfers downlink Orthogonal Frequency Division Multiplexing (OFDM) information and energy to the dedicated LU and several BDs, respectively, while receiving uplink backscattered information from BDs at the same time. One of the key aims in AmBC networks is to ensure fairness among BDs. To address this, we propose the Multi-objective Lexicographical Optimization Problem (MLOP), which aims to maximize the minimum BD’s throughput while enhancing overall BDs’ throughput, subject to the AP’s subcarrier power, BDs’ reflection coefficients, and backscatter time allocation. Owe to the MLOP is non-convex, we propose Difference Convex Algorithm (DCA) using Exterior Penalty Function Method (EPFM)—an inventive non-convex optimization method— to reach the optimal solution. The most critical advantage of applying this proposed approach is finding the globally optimal solution. The effectiveness of the proposed method supported by theoretical analysis confirms its superiority compared to some of the investigated suboptimal algorithms with the same computational complexity.

The joint eigenvalue decomposition algorithm based on first-order Taylor expansion via the exterior penalty function method

The Joint Eigenvalue Decomposition (JEVD) problem has many applications, especially in the field of signal processing. In this paper, we establish two new algorithms to solve the JEVD for the real or complex case. As the JEVD algorithm requires to inverse a matrix, it can be usefully derived by using a first order Taylor expansion under a reasonable condition.In order to possibly enforce the condition, we propose to take advantage of the Penalty Function Method. By adding a quadratic penalty to the regularized criterion, we turn the constrained problem into a single unconstrained problem. Consequently, we show that this regularized criterion leads to a regularized optimal solution. Thus, we propose the Joint Eigenvalue Decomposition Algorithm Based on First-order Taylor Expansion via the Exterior Penalty Function Method (JDTE-EP) and Sweeping JDTE-EP (SJDTE-EP) algorithms based on this idea. And the two algorithms turn out to enjoy low complexity and good accuracy in all numerical experiments.

Ensemble-based constrained optimization using an exterior penalty method

In science and engineering, non-linear constrained optimization has been a useful mathematical technique for many practical applications. Of interest to us is its applicability in the modeling and prediction of hydrocarbon reservoir production. In this paper, a new efficient, robust, and accurate optimal solution strategy based on the exterior penalty function (EPF) method and the adaptive ensemble-based optimization (EnOpt) approach (with backtracking line-search technique) for non-linear constrained optimization problems is presented. The purpose of this work is to provide a better user-friendly strategy which mitigates the problem often faced with the current constraints handling technique utilized when using the EnOpt method to solve constrained problems of water or EOR flooding. This study notes that the problem contributes to uncertainties in the gradient computation of the objective function and hence leads to the poor convergence rate of the standard EnOpt method. In this work, we used the EPF method to transform a given constrained optimization problem to a sequence of unconstrained subproblems and then sequentially solve the subproblems by unconstrained EnOpt procedure until convergence to the solution of the original problem. To demonstrate the advantage of the proposed methodology, we used it to solve analytical 2D bound constrained Rosenbrock’s problem and a practical high dimensional bound constrained water flooding optimization problem associated with a 2D 5Spot field and a 3D Reek reservoir field. The numerical results are compared with EnOpt using classical Lagrangian approach, as well as the traditional EnOpt. Our findings showed that the proposed solution method has a fast convergence rate and is more accurate and robust.

A Constraint Method for Supersaturation in a Variational Data Assimilation System

Abstract The reproducibility of precipitation in the early stages of forecasts, often called a spindown or spinup problem, has been a significant issue in numerical weather prediction. This problem is caused by moisture imbalance in the analysis data, and in the case of the Japan Meteorological Agency’s (JMA’s) mesoscale data assimilation system, JNoVA, we found that the imbalance stems from the existence of unrealistic supersaturated states in the minimal solution of the cost function in JNoVA. Based on the theory of constrained optimization problems, we implemented an exterior penalty function method for the mixing ratio within JNoVA to suppress unrealistic supersaturated states. The advantage of this method is the simplicity of its theory and implementation. The results of twin data assimilation cycle experiments conducted for the heavy rain event of July 2018 over Japan show that—with the new method—unrealistic supersaturated states are reduced successfully, negative temperature bias to the observations is alleviated, and a sharper distribution of the mixing ratio is obtained. These changes help to initiate the development of convection at the proper location and improve the fractions skill score (FSS) of precipitation in the early stages of the forecast. From these results, we conclude that the initial shock caused by moisture imbalance is mitigated by implementing the penalty function method, and the new moisture balance has a positive impact on the reproducibility of precipitation in the early stages of forecasts.

On the Bridge Between Exterior and Interior Penalty Method

In this paper, we present methods of penalty functions for solving constrained optimization problems. The methods that we characterize presently, attempt to approximate a constrained optimization problem with an unconstrained, one and then apply standard search techniques such as the exterior penalty function method and the interior penalty method to get solutions. The paper that follows assure exterior penalty methods recognizing that interior penalty function methods incarnate the same principles.

First-order reliability method based on Harris Hawks Optimization for high-dimensional reliability analysis

The first-order reliability method (FORM) is a prevalent method in the structural reliability community. However, when solving the high-dimensional problem with a highly nonlinear limit state function, FORM usually encounters non-convergence or divergence. In this study, an improved FORM combining Harris Hawks Optimization (HHO-FORM) is presented for high-dimensional reliability analysis. HHO is a meta-heuristic algorithm mimicking the predatory behavior of Harris hawks, and efficient in finding the global optimum of high-dimensional problems. In HHO-FORM, the reliability index is firstly formulated as the solution of a constrained optimization problem according to the FORM theory. Then, the constraints are handled with the exterior penalty function method. In addition, the optimal reliability index is determined by the Harris Hawks Optimization that accelerates the convergence by the population-based mechanism and the strategy of Levy Flight. The HHO-FORM does not require the derivatives of the limit state functions that reduce the computational burden for high-dimensional problems. So the simplicity of HHO-FORM greatly improves the efficiency in solving high-dimensional reliability problems. The HHO-FORM is firstly tested on three challenging numerical high-dimensional problems and then applied to two high-dimensional engineering problems to verify its performance. Four gradient-based FORM algorithms and four heuristic-based FORM algorithms are also compared with the proposed method. The experimental results demonstrate that HHO-FORM provides good accuracy and efficiency for high-dimensional reliability problems.

Structural reliability assessment by salp swarm algorithm–based FORM

AbstractThe first‐order reliability method (FORM) is well recognized as an efficient approach for reliability analysis. Rooted in considering the reliability problem as a constrained optimization of a function, the traditional FORM makes use of gradient‐based optimization techniques to solve it. However, the gradient‐based optimization techniques may result in local convergence or even divergence for the highly nonlinear or high‐dimensional performance function. In this paper, a hybrid method combining the Salp Swarm Algorithm (SSA) and FORM is presented. In the proposed method, a Lagrangian objective function is constructed by the exterior penalty function method to facilitate meta‐heuristic optimization strategies. Then, SSA with strong global optimization ability for highly nonlinear and high‐dimensional problems is utilized to solve the Lagrangian objective function. In this regard, the proposed SSA‐FORM is able to overcome the limitations of FORM including local convergence and divergence. Finally, the accuracy and efficiency of the proposed SSA‐FORM are compared with two gradient‐based FORMs and several heuristic‐based FORMs through eight numerical examples. The results show that the proposed SSA‐FORM can be generally applied for reliability analysis involving low‐dimensional, high‐dimensional, and implicit performance functions.

A Heuristic Approach to Identify the Steel Grid Direction of R/C Slabs Using the Yield-Line Method for Analysis

In the last few years, nonregular reinforced concrete (R/C) slabs have become more popular in buildings and bridges due to architectural or functional requirements. In these cases, an optimum design method to obtain the ultimate load capacity and the minimum reinforcement amount should be used. For simple R/C slabs, the yield-line method is extensively used in engineering practice. In addition to strength, the “true” failure mechanism is also obtained by identifying the parameters that define it and minimizing the collapse load. Unfortunately, when the mechanism is too complicated to be described or defined by several parameters (e.g., in slabs with complicated geometry), the method becomes more difficult because the system of nonlinear equations becomes harder to solve through traditional methods. In this case, an efficient and robust algorithm becomes necessary. In this paper, a structural analysis of R/C slabs is performed by using the yield-line method in association with a zero-th order optimization algorithm (the sequential simplex method) to avoid calculating gradients as well as any derivatives. The constraints that often limit these parameters are taken into account through the exterior penalty function method, leading to a successful solution of the problem. Considering that the direction of each yield-line is sought by minimizing the ultimate load and finding the parameters defining the collapse mechanism, another parameter concerned with the direction of an orthotropic reinforcement grid is introduced. In this way, the number of unknown parameters increases, but aside from obtaining the ultimate load and the parameters defining the collapse mechanism, the solution also finds both best and worst reinforcement orientations.

An innovative hybrid multi-objective particle swarm optimization with or without constraints handling

This paper presents a new hybrid optimizer in which an innovative optimal particles local search strategy on basis of bound optimization by quadratic approximation (BOBYQA) algorithm and exterior penalty function method is integrated into particle swarm optimization (PSO). The main goal of the approach is to improve the convergence performance of PSO, and preserve the diversity of non-dominated set. Our algorithm selects some non-dominated solutions lied in less-crowded region of external archive based upon crowding distance value to construct a leader particles set, and make full use of optimal particles method to guide leader particles approach the Pareto front quickly. Meanwhile, a local optimal particles search strategy is proposed after particular analysis on disadvantage of global optimal particle search method, and names our algorithm as LOPMOPSO. Furthermore, the multi-dimensional uniform mutation operator is performed to prevent algorithm from trapping into local optimum, and a dynamic archive maintenance strategy is applied to improve the diversity of solutions. For coping with the constrained conditions consists in objective functions, we adopt an efficient infeasibility degree evaluation criterion to deal with these complex problems. Simulation results of various kinds of benchmark functions show that our approach is highly competitive in convergence speed and generates a well distributed and accurate set of non-dominated solutions easily. The solving of a 2-D aerodynamic optimization problem further validates its speed and effectiveness.

A Hybrid Global Optimization Method Based on Genetic Algorithm and Shrinking Box

<p class="zhengwen">This paper proposes a hybrid genetic algorithm method for optimizing constrained black box functions utilizing shrinking box and exterior penalty function methods (SBPGA). The constraints of the problem were incorporated in the fitness function of the genetic algorithm through the penalty function. The hybrid method used the proposed Variance-based crossover (VBC) and Arithmetic-based mutation (ABM) operators; moreover, immigration operator was also used. The box constraints constituted a hyperrectangle that kept shrinking adaptively in the light of the revealed information from the genetic algorithm about the optimal solution. The performance of the proposed algorithm was assessed using 11 problems which are used as benchmark problems in constrained optimization literatures. ANOVA along with a success rate performance index were used to analyze the model.</p>Based on the results, we believe that the proposed method is fairly robust and efficient global optimization method for Constrained Optimization Problems whether they are continuous or discrete.

Containing Small Hydropower and Wind Power System Optimal Spinning Reserve

When larger scale of small hydropower and wind power exist in power system, the traditional reserve capacity algorithm is no longer applicable at the same time of reducing the carbon emission. Aiming at the volatility and non-regulatory of small hydropower and wind power, considering the forecast error of small hydropower, wind power and load,the unit outage rate and other factors, combined with the actual operation parameters of one regional power system in Southwest of China, the wind-hydro – thermal power coordinated operation of multiple objective function optimization calculation model of reserve capacity is established. Using the weighted coefficient method of unified objective method, it transforms the multi-objective optimization problem into a single objective optimization one. Applying the exterior penalty function method, the constrained optimization problem is changed into a non-constrained optimization one. The improved particle swarm optimization algorithm is got by introducing the particle concentration cognition into traditional particle swarm optimization algorithm. The simulation results verify the validity of the model. To Compare them under different strategies, this method can get less fuel expenses of thermal power units in the case of lower loss of load probability (LOLP).With the small hydropower and wind power output size to optimize the spinning reserve capacity. The model and the algorithm are helpful for drawing up optimization strategies of the reserve capacity in those regions in which larger scale small hydropower and wind power exists.

Study of Computation for Structural Reliability Index Based on Penalty Function Method

According to the geometric meaning of the structural reliability index, an unconstrained optimization model with structural reliability index and design point is obtained by exterior penalty function method. The Powell method, golden section method and extrapolation method are used to solve the unconstrained optimization problem. The proposed method not has to deal with the any derivative of the limited state function, and can been used to obtain structural reliability index and design point of the strong nonlinear limit state function, which first-order reliability method (FORM) may fail to converge. Three examples are given to compare penalty function method with the difference methods. The results show that the given method is simply, effective and precise enough.

Optimal coordination of wind-hydro-thermal based on water complementing wind

The increasing integration of wind power into the existing power system demands for effective strategies to deal with wind intermittency and uncertainty. Relying solely on thermal power to cover wind uncertainty will sacrifice the operating efficiency and economy of thermal generators. In view of this, the adjustable hydropower is preferred for complementing wind fluctuation and uncertainty and the coordinated dispatch problem of wind-hydro-thermal power is established. Based on a newly designed water supplementing wind strategy, the original complex problem is decomposed into wind-hydro subproblem and thermal subproblem. A novel stochastic constraint related to wind power uncertainty is proposed and handled according to stochastic programming theory. By introducing the concept of expected breed rate and elitist preservation strategy, the particle swarm optimization (PSO) algorithm is improved and combined with the exterior penalty function method for solving the complete optimization problem. Optimal generation scheduling schemes that can make full use of wind energy and ensure efficient and economic operating of thermal generators are obtained by the proposed approach. Meanwhile the coordinating operation of wind, hydro and thermal power under different water resources and wind penetrations respectively are revealed and discussed.

Numerical optimization on plant noise control by gradient method

Noise control is important and essential in a process plant, where the noise level is restricted by the Occupational Safety and Health Act. However, noise control for individual equipment is the only concern in most situations. When considering the overall noise control within the plant boundary, the inadequate noise simulation, identification, and strategy often cause deprived plant arrangement. These improper abatements often result in significant cost and ineffectiveness of noise reduction. In this paper, noise control on a process plant with one to five equipment is introduced. To minimize the influence of machine noise along the plant boundary, the studies of optimum allocation planning and appropriate sound transmission loss of acoustic hood added to machine are then presented. Through various multi-station sound monitoring systems and gradient numerical method—exterior penalty function method (EPFM), the numerical assessment is consequently developed. Besides, an accuracy check of the optimization model with single noise is pre-run and found good before EPFM optimization. It is shown that the optimum solution can be reached exactly through the proposed technique. A numerical technique to adjust the optimum allocations and to search for a suitable acoustic hood to the machine is first demonstrated; thereafter, the effect of using different monitoring systems with respect to the same equipment is discussed. This study definitely contributes to the effective control of plant noise, being an economical technique.

Designing cutouts for optimum residual strength in plane structural elements

The paper presents a new approach for shape optimisation of structures with residual strength as the design objective. It must be emphasized that flaws are inevitably present in most structures, and hence the influence of cracks on optimised shapes needs to be investigated. Numerical simulation of cracks using the finite element method requires a very fine mesh to model the singularity at crack tip. This makes fracture calculations computationally intensive. Furthermore, for a damage tolerance based optimisation numerous cracks are to be considered along the structural boundary, and fracture analysis needs to be repeated for each crack at every iteration, thus making the whole process extremely computationally expensive for practical purpose. Moreover, the lack of information concerning crack size, orientation, and location makes the formulation of the optimisation problem difficult. As a result, little attention has been paid to date to consider fracture parameters in the optimisation objective. To address this, the paper presents a methodology for the shape optimisation of structures with strength and durability as the design objectives. In particular, the damage tolerance optimisation is illustrated via the problem of optimal design of a ‘cutout in a rectangular block under biaxial loading’. A parametric shape representation has been used to describe the problem geometry. Damage tolerance based optimisation was performed using nonlinear programming algorithms, and NE-NASTRAN was used for finite element analysis. The first order mathematical programming algorithms, viz: the Broydon–Fletcher–Goldfarb–Shanno (BFGS) and the Fletcher–Reeves (Conjugate Direction) Methods were evaluated as the potential algorithms for optimising the residual strength in the presence of flaws. Another recently developed Sequential Unconstrained Minimisation Technique (SUMT), based on an exterior penalty function method and especially suited for large problems, was also investigated for fracture based optimisation. The effects of the orientation and the number of boundary cracks on the optimal solutions were also studied. It has been shown that the residual strength optimised shapes can be different from the corresponding and commonly adopted stress optimised solution. This emphasises the need to explicitly consider residual strength as the design objective. In all cases a significant reduction in the maximum stress intensity factor was achieved with the generation of a ‘near’ uniform fracture critical surface. The design space near the optimal region was found to be relatively flat. This is beneficial as a significant structural performance enhancement is important rather than precise identification of the local/global optimum solution. The optimal solutions obtained using the nonlinear programming algorithms were compared against those obtained in the literature using a heuristic optimisation method (Biological algorithm). The results obtained using the two methods, employing inherently different (gradient-based and gradient-less) algorithms, were found to agree very well.

On Clamping Planning in Workpiece-Fixture Systems

Deformations of contacts between the workpiece and locators/clamps resulting from large contact forces cause overall workpiece displacement, and affect the localization accuracy of the workpiece. An important characteristic of a workpiece-flxture system is that locators are passive elements and can only react to clamping forces and external loads, whereas clamps are active elements and apply a predetermined normal load to the surface of workpiece to prevent it from losing contact with the locators. Clamping forces play an important role in determining the final workpiece quality. This paper presents a general method for determining the optimal clamping forces including their magnitudes and positions. First, we derive a set of compatibility equations that describe the relationship between the displacement of the workpiece and the deformations at contacts. Further, we develop a locally elastic contact model to characterize the nonlinear coupling between the contact force and elastic deformation at the individual contact. We define the minimum norm of the elastic deformations at contacts as the objective function, then formulate the problem of determining the optimal clamping forces as a constrained nonlinear programming problem which guarantees that the fixturing of the workpiece is force closure. Using the exterior penalty function method, we transform the constrained nonlinear programming into an unconstrained nonlinear programming which is, in fact, the nonlinear least square. Consequently, the optimal magnitudes and positions of clamping forces are obtained by using the Levenberg-Marquardt method which is globally convergent. The proposed planning method of optimal clamping forces, which may also have an application to other passive, indeterminate problems such as power grasps in robotics, is illustrated with numerical example.