Penalty Function Method Research Articles

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

An exact l1 penalty function method for a multitime control optimization problem with data uncertainty

SummaryIn this article, we study the robust solution set of a multitime first‐order PDE constrained control optimization problem with data uncertainty (MCOPU) and an unconstrained multitime control optimization problem (MCOPU)ρ. We establish the equivalence between a robust optimal solution of (MCOPU) and a robust minimizer of (MCOPU)ρ under the appropriate assumptions. Furthermore, we define the uncertain Lagrange functional for (MCOPU) and show the relationship between the robust solution set of (MCOPU) and (MCOPU)ρ under the convexity hypothesis of uncertain Lagrange functional. Moreover, we present some nontrivial examples to illustrate the established results.

Task Planning of Space-Robot Clusters Based on Modified Differential Evolution Algorithm

This study studies the problem of on-orbit maintenance task planning for space-robot clusters. Aiming at the problem of low maintenance efficiency of space-robot cluster task-planning, this study proposes a cluster-task-planning method based on energy and path optimization. First, by introducing the penalty-function method, the task planning problem of the space-robot cluster under limited energy is analyzed, and the optimal-path model for task planning with comprehensive optimization of revenue and energy consumption are constructed; then, the maintenance task points are clustered to reduce the scale of the problem, thus reducing the difficulty of solving the problem; finally, a modified differential evolution algorithm is proposed to solve the problem of space-robot cluster task-planning, improve the performance of space-robot cluster task-assignment and path planning. Simulation results show that the proposed optimal-path model of space-robot cluster and the modified differential evolution algorithm can effectively solve the task-planning problem of spatial robot clusters.

Initialization-Free Distributed Fixed-Time Convergent Algorithms for Optimal Resource Allocation

This article considers the distributed fixed-time resource allocation problem subject to global equality and local inequality constraints. In the case that the optimal solutions are strictly feasible, a projection-gradient-based continuous-time algorithm is first proposed to minimize a team of cost functions in the fixed time. As some optimal solutions locate on the boundary of local constraints, we further provide an alternative suboptimal solution based on the <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-exact penalty function method. Different from the existing distributed optimal allocation results, the (sub)optimal solutions can be obtained in the fixed time and the optimization protocols can be implemented in an initialization-free manner. Thus, the convergence time can be offline preassigned and the initial states can be chosen arbitrarily such that a better robust performance is achieved for the new algorithms. Case studies of the widely discussed economic dispatch problems are performed to validate the effectiveness of the obtained results.

DEVELOPMENT OF SOFTWARE FOR SOLVING PROBLEMS WITH THE PENALTY FUNCTION METHOD

The penalty function method is one of the most popular and universal methods of convex programming and belongs to the group of indirect methods for solving nonlinear programming problems. Thе article discusses the algorithm for solving problems by the penalty function method, provides an example of a solution. A complete definition of the concepts used in the theoretical material of the method, and examples of its application are also given. It is worth noting that these methods are widely used to solve technical and economic problems. Also they are quite often used both in theoretical research and in the development of algorithms. The result of the work is the development of software for solving problems using the penalty function method.

Underwater TDOA/FDOA joint localisation method based on cross‐ambiguity function

In the underwater sound source localisation systems, the sound wave signal is accompanied by time delay and Doppler shift, where the cross-ambiguity function (CAF) is commonly applied as a method of estimating them. This study proposes a novel method for calculating the exact value of the CAF,based on the theory of curved surface interpolation. Firstly, two optimisation problems with equality and inequality constraints are developed, where the penalty function method is employed to turn the equality constraint problem into an unconstrained problem. Secondly, the time delay and Doppler shift of the CAF solution are processed to obtain the measurement values required for accurate localisation. Furthermore, the authors propose an improved time difference of arrival and frequency difference of arrival (TDOA/FDOA)-based joint localisation algorithm, which changes the structure of traditional localisation equations and eliminates the localisation imprecision caused by neglecting the square term of the noise. The performance of the proposed algorithm is verified by extensive simulations and comparisons with several established methods. Remarkably, the proposed localisation method is confirmed to be noticeably superior and effective for the considered application.

Element-Free Galerkin method for dynamic boundary flow problems

This paper focuses on the study of dynamic boundary flow problems based on the Element-Free Galerkin method. First, Navier–Stokes equation is discretized with the Galerkin method. The inertial term in the equation is discretized with the method of the speed term and direct deduction, respectively. The penalty function method is used to deal with the pressure and the essential boundary condition in the equation, and the discretization of two-dimensional N–S equation based on the EFG method is established. However, irregular changes in boundary conditions are often encountered in practical fluid problems. For example, the motion of flapping-foil is not uniform relative to the flow. In this paper, numerical experiments are carried out for the flow problems with non-uniform boundary motions. The problem which a plate with non-uniform drag movement above a rectangular tank filled with water is studied with the EFG method. The feasibility of the proposed algorithm is verified by comparison with the FEM method. Then, the procession of the water in the tank is stimulated. In the end, the influence of different calculation time steps on the accuracy of the solution is discussed.

Global optimization method with dual Lipschitz constant estimates for problems with non-convex constraints

This paper considers the constrained global optimization problems, in which the functions are of the “black-box” type and satisfy the Lipschitz condition. The algorithms for solving the problems of this class require the use of adequate estimates of the a priori unknown Lipschitz constants for the problem functions. A novel approach presented in this paper is based on a simultaneous use of two estimates of the Lipschitz constant: an overestimated and an underestimated one. The upper estimate provides the global convergence, whereas the lower one reduces the number of trials necessary to find the global optimizer with the required accuracy. The considered algorithm for solving the constrained problems does not use the ideas of the penalty function method; each constraint of the problem is accounted for separately. The convergence conditions of the proposed algorithm are formulated in the corresponding theorem. The results of the numerical experiments on a series of multiextremal problems with non-convex constraints demonstrating the efficiency of the proposed scheme of dual Lipschitz constant estimates are presented.

Predefined-Time Distributed Optimal Allocation of Resources: A Time-Base Generator Scheme

In this article, we present a novel predefined-time distributed optimization protocol for the resource allocation problem (RAP). Different from the existing finite/fixed-time distributed optimal algorithms, the proposed protocol is built on the time-varying time-based-generator scheme. The settling time is independent of the initial values of the agents and any designed parameters, which implies that the settling time can be predesigned by the user. To address the inequality constraints, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -exact penalty function method is introduced to solve the RAP within a predefined-time. In addition, the designed optimization protocol can be implemented in an initialization-free manner, which makes the algorithms feasible under arbitrary initial states. Finally, several numerical simulation results are provided to demonstrate the effectiveness of the proposed algorithms.

Penalty function method for imposing nonlinear multi freedom and multi node constraints in finite element analysis of frame systems

This paper focuses on the treatment of nonlinear multi-freedom and multi-point boundary condition in finite element analysis of frame systems. The treatment of boundary constraints is required to produce modified system of equations based on master stiffness equations considering nonlinear constraints. For imposing nonlinear multi freedom and multi point constraints, the Penalty Augmentation method and Lagrange Multiplier Adjunction method are better in many applications. In present paper, the Penalty Augmentation method is using for implementation of nonlinear boundary constraints. The nonlinear relationship considerably increases the difficulty in constructing and solving the modified system of equations. The Newton Raphson method, have been the most powerful and popular method for solution of nonlinear equations, is used for solving studied problem. Using the Newton Raphson technique, this paper develops the incremental-iterative algorithm to solve the nonlinear modified system of equations. Based on the presented algorithm, the paper proposed calculation procedure and established programs for determining internal forces and displacements of frames having nonlinear multi freedom and multi point constrains boundary. The numerical test results using the proposed method show the efficiency and reliability of proposed algorithm.

Optimization design of neutrally buoyant hull for underwater gliders

The net buoyancy, driving force of underwater gliders, decreases gradually when the gliders dive or climb in the seawater, which increases the energy consumption and reduces the gliding range. The neutrally buoyant hull is an effective measure to reduce this buoyancy loss of underwater gliders. In this paper, a multiple intersecting spheres (MIS) pressure hull is designed to provide neutral buoyancy. A mechanical model of the hull is established with the thin shell theory, based on which the optimization is carried out by combining the penalty function method (PFM) and multiple population genetic algorithm (MPGA). Thickness of the spherical shell, reinforcing rib, width of the reinforcing rib, and intersection angle of the spherical shell are optimized for minimizing the buoyancy factor which is defined as the ratio of pressure hull mass to the undeformed displacement. Additionally, the mechanical model is verified by the finite element analysis and the pressure test. The results show that the MIS pressure hull can increase the buoyancy compensation by 69.69% and increase battery capacity of the glider by 12.89% as compared with conventional cylindrical hull. This will improve the navigation range and duration of Petrel-L glider more obviously when its hotel load is lower.

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.

Robust saddle-point criteria for multi-dimensional control optimisation problems with data uncertainty

This paper deals with some new outcomes on the multi-dimensional control optimisation problem with data uncertainty (UMCOP). By employing the exact penalty function method, we formulate an equivalent unconstrained control optimisation problem and obtain the robust saddle-point criteria for (UMCOP). After that, we show the equivalences between the robust saddle-point of the considered control problem (UMCOP) and a robust minimiser of its associated penalised unconstrained control optimisation problem (UMCOP). We also provide some numerical illustrations to verify the developed results in this paper.

Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimization Methods (Steepest Descent and Newton’s Method)

Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.

Application of modified pigeon-inspired optimization algorithm and constraint-objective sorting rule on multi-objective optimal power flow problem

To solve the non-differentiable optimal power flow (OPF) problems with multiple contradictory objectives, a modified pigeon-inspired optimization algorithm (MPIO) is put forward in this paper. Combining with the common-used penalty function method (PFM), the MPIO-PFM algorithm is proposed and applied to optimize the active power loss, emission and fuel cost (with valve-point loadings) of power system. Eight simulation trials carried out on MATLAB software validate MPIO-PFM algorithm can obtain superior Pareto Frontier (PF) comparing with the typical NSGA-II algorithm. Nevertheless, some Pareto solutions obtained by MPIO-PFM algorithm cannot satisfy all system constraints due to the difficulty in choosing the proper penalty coefficients. Thus, an innovative approach named as constraint-objective sorting rule (COSR) is presented in this paper. The bi-objective and tri-objective trials implemented on IEEE 30-node, 57-node and 118-node systems demonstrate that the Pareto optimal set (POS) obtained by MPIO-COSR algorithm realizes zero-violation of various system constraints. Furthermore, the generational-distance and hyper-volume indexes quantitatively illustrate that in contrast to NSGA-II and MPIO-PFM methods, the MPIO-COSR algorithm can determine the evenly-distributed PFs with satisfactory-diversity. The intelligent MPIO-COSR algorithm provides an effective way to handle the non-convex MOOPF problems.

Unveiling controlling factors of the S0/S1 minimum energy conical intersection (2): Application to penalty function method.

Minimum-energy conical intersection (MECI) geometries play an important role in photophysics, photochemistry, and photobiology. In a previous study [Nakai et al., J. Phys. Chem. A 122, 8905 (2018)], frozen orbital analysis at the MECI geometries between the ground and first electronic excited states (S0/S1 MECI), which considers the main configurations contributing to the excitation, inductively clarified two controlling factors. First, the exchange integral between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) approximately becomes zero. Second, the HOMO-LUMO gap becomes close to the HOMO-LUMO Coulomb integral. This study applies the controlling factors to the penalty function method, which is the standard MECI optimization technique, and minimizes the energy average of the two states with the constraint that the energy gap between the states vanishes. Numerical assessments clarified that the present method could obtain the S0/S1 MECI geometries more efficiently than the conventional one.

Heterogeneous cellular network optimization for green access of IoT traffics

The Internet of Things(IoT) has become an essential supporting platform for the present and future cyber-enabled services. Cellular networks is considered as the main channel of the data access for IoT terminals distributed in the region of interest, and they have an irreplaceable value, especially in wide-area coverage. Thus, it has a significant application value to reduce the downlink transmit power consumption of base stations under the restrictions of the coverage requirements for the green communication in heterogeneous cellular networks. A gradient descent algorithm was proposed based on smooth approximation and root mean square propagation. The algorithm could minimize the total downlink power consumption of base stations while satisfying the IoT service coverage. First, the penalty function method was used to simplify such an optimization problem with complicated constraints to a new one with simple constraints. Then, the non-derivative objective function was transformed by an approximation method into a derivable form. We also presented the close-form of the gradient of the objective function with respect to both the azimuths of the antennas installed in the base stations and the downlink transmit power levels related to these antennas. Finally, the gradient descent algorithm with root mean square propagation was used to execute the optimization of the newly approximated but smoothed version of the original objective function. Simulation experiments were conducted, and the results show that the proposed algorithm can significantly reduce the total power consumption of the downlink radio frequency transmit under the restrictions of the coverage ratio requirements in the region of interest. Furthermore, not only is the convergence speed of the proposed algorithm very fast, but also the oscillation phenomenon that occurs during the iterative procedure steps of the optimization is greatly suppressed by the proposed algorithm compared with the meta-heuristic algorithms and ordinary gradient descent method.

Computing shadow prices with multiple Lagrange multipliers

There is a wide consensus that the shadow prices of certain resources in an economic system are equal to Lagrange multipliers. However, this is misleading with respect to multiple Lagrange multipliers. In this paper, we propose a new type of Lagrange multiplier, the weighted minimum norm Lagrange multiplier, which is a type of shadow price. An attractive aspect of this type of Lagrange multiplier is that it conveys the sensitivity information when resources are required to be proportionally input. To compute the weighted minimum norm Lagrange multiplier, we propose two algorithms. One is the penalty function method with numeric stability, and the other is the accelerated gradient method with fewer arithmetic operations and a convergence rate of \begin{document}$ O(\frac{1}{k^2}) $\end{document} . Furthermore, we propose a two-phase procedure to compute a particular subset of shadow prices that belongs to the set of bounded Lagrange multipliers. This subset is particularly attractive since all its elements are computable shadow prices. We report the numerical results for randomly generated problems.

Stochastic Gradient-Based Optimal Signal Control With Energy Consumption Bounds

This paper develops a stochastic gradient-based optimization model for traffic signal control with bounds on network-level vehicular energy consumption. The signal control problem is formulated as a mixed-integer linear mathematical program, which incorporates inequality constraints to limit the total energy consumption in the network. The developed stochastic gradient approximation algorithm provides a near-optimal solution to the non-convex optimization problem. To account for the energy consumption constraints, a penalty function method leveraging the pseudo gradient estimation technique is developed. Empirical results from a signalized arterial street show that it is possible to achieve optimized signal settings at the desired energy consumption bound without compromising delay. Further, we report the sensitivity of the energy bounds to the mobility metrics-system delay. Our novel gradient-approximation-based solution technique offers a functional and feasible way to accommodate non-convex energy consumption bounds within a signal control optimization model to achieve maximal mobility with minimal energy consumption.

An exact minimax penalty function approach to solve multitime variational problems

This paper aims to examine an appropriateness of the exact minimax penalty function method applied to solve the partial differential inequation (PDI) and partial differential equation (PDE) constrained multitime variational problems. The criteria for equivalence between the optimal solutions of a multitime variational problem with PDI and PDE constraints and its associated unconstrained penalized multitime variational problem is studied in this work. We also present some examples to validate the results derived in the paper.