Recursive Quadratic Programming Algorithm Research Articles

A Generalized Pattern Search Algorithm Methodology for solving an Under-Determined System of Equality Constraints to achieve Power System Observability using Synchrophasors

The impact of the generalized pattern search algorithm (GPSA) on power system complete observability utilizing synchrophasors is proposed in this work. This algorithmic technique is an inherent extension of phasor measurement unit (PMU) minimization in a derivative-free framework by evaluating a linear objective function under a set of equality constraints that is smaller than the decision variables in number. A comprehensive study about the utility of such a system of equality constraints under a quadratic objective has been given in our previous paper. The one issue studied in this paper is the impact of a linear cost function to detect optimality in a shorter number of iterations, whereas the cost is minimized. The GPSA evaluates a linear cost function through the iterations needed to satisfy feasibility and optimality criteria. The other issue is how to improve the performance of convergence towards optimality using a gradient-free mathematical algorithm. The GPSA detects an optimal solution in a fewer number of iterations than those spent by a recursive quadratic programming (RQP) algorithm. Numerical studies on standard benchmark power networks show significant improvement in the maximum observability over the existing measurement redundancy generated by the RQP optimization scheme already published in our former paper.

THE MODELLING AND OPTIMIZATION OF AN ELECTRICAL VEHICLE USING JOINT ANALYSIS

Finite element analysis is an excellent method for structural analysis. For finite element analysis, accurate modelling is very important to obtain precise information. Stick modelling is a convenient way in that it is easy and simple. When the stick model is utilised, the joints are modified in the tuning process. A tuning method for the joints has been developed. Joints are modelled by designated beam elements. and the beam properties are tuned to have target values which are evaluated from experiments. An optimisation method called 'Goal Programming' is employed to impose the target values. With the tuned joints, the entire optimisation has been carried out. Using a recursive quadratic programming algorithm, the optimisation process determines the configuration of the entire structure and sizes of all the sections. For example, the structure of an electrical vehicle is modelled and analysed by the method. The stick model works very well since the structure is made of aluminium frames. Although the example is for an electrical vehicle, this method can be applied to general vehicle structures.

Structural topology optimization for the natural frequency of a designated mode

The homogenization method and the density function method are common approaches to evaluate the equivalent material properties for design cells composed of matter and void. In this research, using a new topology optimization method based on the homogenized material with a penalty factor and the chessboard prevention strategy, we obtain the optimal layout of a structure for the natural frequency of a designated mode. The volume fraction of nodes of each finite element is chosen as the design variable and a total material usage constraint is imposed. In this paper, the subspace method is used to evaluate the eigenvalue and its corresponding eigenvector of the structure for the designated mode and the recursive quadratic programming algorithm, PLBA algorithm, is used to solve the topology optimization problem.

Analysis and Optimization of Bellows With General Shape

A bellows is a component in piping systems which absorbs mechanical deformation with flexibility. Its geometry is an axially symmetric shell which consists of two toroidal shells and one annular plate or conical shell. In order to analyze the bellows, this study presents the finite element analysis using a conical frustum shell element. A finite element analysis program is developed to analyze various bellows. The formula for calculating the natural frequency of bellows is made by the simple beam theory. The formula for fatigue life is also derived by experiments. A shape optimal design problem is formulated using multiple objective optimization. The multiple objective functions are transformed to a scalar function with weighting factors. The stiffness, strength, and specified stiffness are considered as the multiple objective function. The formulation has inequality constraints imposed on the natural frequencies, the fatigue limit, and the manufacturing conditions. Geometric parameters of bellows are the design variables. The recursive quadratic programming algorithm is utilized to solve the problem.

Robust Recursive Quadratic Programming Algorithm Model with Global and Superlinear Convergence Properties

A new, robust recursive quadratic programming algorithm model based on a continuously differentiable merit function is introduced. The algorithm is globally and superlinearly convergent, uses automatic rules for choosing the penalty parameter, and can efficiently cope with the possible inconsistency of the quadratic search subproblem. The properties of the algorithm are studied under weak a priori assumptions; in particular, the superlinear convergence rate is established without requiring strict complementarity. The behavior of the algorithm is also investigated in the case where not all of the assumptions are met. The focus of the paper is on theoretical issues; nevertheless, the analysis carried out and the solutions proposed pave the way to new and more robust RQP codes than those presently available.

Development of finite element analysis program and simplified formulas of bellows and shape optimization

Bellows is a component in piping systems which absorbs mechanical deformation with flexibility. Its geometry is an axial symmetric shell which consists of two toroidal shells and one annular plate or conical shell. In order to analyze bellows, this study presents the finite element analysis using a conical frustum shell element. A finite element analysis is developed to analyze various bellows. The validity of the developed program is verified by the experimental results for axial and lateral stiffness. The formula for calculating the natural frequency of bellows is made by the simple beam theory. The formula for fatigue life is also derived by experiments. The shape optimal design problem is formulated using multiple objective optimization. The multiple objective functions are transformed to a scalar function by weighting factors. The stiffness, strength and specified stiffness are considered as the multiple objective function. The formulation has inequality constraints imposed on the fatigue limit, the natural frequencies, and the manufacturing conditions. Geometric parameters of bellows are the design variables. The recursive quadratic programming algorithm is selected to solve the problem. The results are compared to existing bellows, and the characteristics of bellows is investigated through optimal design process. The optimized shape of bellows is expected to give quite a good guideline to practical design.

Optimal control of vibration suppression in flexible systems via dislocated sensor/actuator positioning

In this study, we present a new controller design method that not only effectively suppresses vibration in flexible systems, but also has the ability to save control energy. The proposed method allows integrated determination of sensor/actuator location and feedback gain by minimizing the sum of the integral flexible system energy and the integral control energy. Also, the cost function is characterized by an effective representation of control systems, and is determined via an efficient solution of the Lyapunov equation. The optimization problem is solved by a recursive quadratic programming algorithm. The feasibility of applying this method to a simple and flexible structure confirms the direct relationship between our optimization criterion and effectiveness in vibration suppression.

Mechanical behavior of U-shaped bellows and shape optimal design using multiple objective optimization method

A bellows is frequently applied in piping systems for absorbing mechanical movement. Its geometry is an axial symmetric shell, which is composed of two toroidal shells and one annular plate. The mechanical behavior of U-shaped bellows under axial force and internal pressure is estimated by changing the dimensions of the geometric parameters. The changing ranges of the geometric dimensions is so selected as to invest the results with practical environments in many fields. The minimization of strength and spring rate is considered simultaneously as a multiple objective function. The weighting objective method is implemented, in which a vector function is transformed to a scalar function. The structure is analyzed by the energy method for toroidal sections. Optimization is carried out by the Recursive Quadratic Programming algorithm.

Collocated Sensor/Actuator Positioning and Feedback Design in the Control of Flexible Structure System

This paper presents a new control design method for the control of flexible systems that not only guarantees closed-loop asymptotic stability but also effectively suppresses vibration. This method allows integrated determination of actuator/sensor locations and feedback gain via minimization of an energy criterion, which is chosen as the integrated total energy stored in the system. The energy criterion is determined via an efficient solution of the Lyapunov equation and minimized with a quasi-Newton or recursive quadratic programming algorithm. The prerequisite for this optimal design method is that the controlled system be asymptotically stable. This study shows that when the controller structure is a collocated direct velocity feedback design with positive definite feedback gain, the number and placement of actuators/sensors are the only factors needed to determine necessary and sufficient conditions for ensuring closed-loop asymptotic stability. The application of this method to a simple flexible structure confirms the direct relationship between our optimization criterion and effectiveness in vibration suppression.

A recursive quadratic programming algorithm that uses a new nondifferentiable penalty functions

In this paper, a recursive quadratic programming algorithm is proposed and studied. The line search functions used are Han’s nondifferentiable penalty functions with a second order penalty term. In order to avoid maratos effect, Fukushima’s mixed direction is used as the direction of line search. Finally, we prove the global convergence and the local second order convergence of the algorithm.

Effective Active Damping Design for Suppression of Vibration in Flexible Systems via Dislocated Sensor/Actuator Positioning

In this paper we present a new controller design method that effectively suppresses vibration in flexible systems. The method allows integrated determination of sensor/actuator locations and feedback gain via minimization of an energy criterion which represents the integrated total energy stored in the system. This energy criterion is characterized by an effective representation of vibration, determined via an efficient solution of the Lyapunov equation and minimized with a quasi-Newton or recursive quadratic programming algorithm. The application of this method to a simple flexible structure confirms the direct relationship between our optimization criterion and effectiveness of vibration suppression.

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.

Recursive quadratic programming algorithm that uses an exact augmented Lagrangian function

An algorithm for nonlinear programming problems with equality constraints is presented which is globally and superlinearly convergent. The algorithm employs a recursive quadratic programming scheme to obtain a search direction and uses a differentiable exact augmented Lagrangian as line search function to determine the steplength along this direction. It incorporates an automatic adjustment rule for the selection of the penalty parameter and avoids the need to evaluate second-order derivatives of the problem functions. Some numerical results are reported.

Optimal design of sandwich beams

Minimum weight design of sandwich beams is considered in this paper. A small deflection theory is presented for determining the stresses and deformations in variable thickness elastic sandwich beams that are symmetric about their axes. The face sheets are treated as membranes; the core is assumed to carry only transverse shear. The contribution of the face sheet membrane forces (by virtue of their slopes) to the transverse shear is accounted for in this analysis. A deflection constraint as well as stress constraints are considered. The formulated optimal design problem is solved by the optimization system IDESIGN3 which is based on the Recursive Quadratic Programming Algorithm.

Optimization With Discrete Variables Via Recursive Quadratic Programming: Part 2—Algorithm and Results

A discrete recursive quadratic programming algorithm is developed for a class of mixed discrete constrained nonlinear programming (MDCNP) problems. The symmetric rank one (SR1) Hessian update formula is used to generate second order information. Also, strategies, such as the watchdog technique (WT), the monotonicity analysis technique (MA), the contour analysis technique (CA), and the restoration of feasibility have been considered. Heuristic aspects of handling discrete variables are treated via the concepts and convergence discussions of Part I. This paper summarizes the details of the algorithm and its implementation. Test results for 25 different problems are presented to allow evaluation of the approach and provide a basis for performance comparison. The results show that the suggested method is a promising one, efficient and robust for the MDCNP problem.

The positioning of sensors and actuators in vibration control of flexible system.

A method for the determination of sensors and actuators positioning and feedback gains for the active vibration control of flexible structures is presented. This method is based on the minimization of the minimum function of a quadratic cost functional in the standard optimal control. The optimal criterion is determined via Riccati equations, and it is minimized with a recursive quadratic programming algorithm. The application of this method to a cantilever beam yields several dislocated sensor and actuator locations, which are locally optimal. An extension to the case when the model filters are used is also examined. In this case, the location of the actuator is found to be more sensitive than that of the sensors. This method is clear for the control efficiency, and has the freedom of varying the weighting matrix.

Recursive quadratic programming algorithm for color matching

AbstractA recursive quadratic programming (RQP) algorithm for color matching is described. RQP does not work on the assumption that the difference of spectral reflectance between sample and match can be approximated by a first‐order expansion. Color matching is instead obtained only by agreement of a trichromatic specification. A numerical example showing a possible field of application of RQP is presented.

Dynamic response optimization using an active set RQP algorithm

AbstractA recently developed recursive quadratic programming (RQP) algorithm is applied to dynamic response design problems. The algorithm incorporates updated Hessians of the Lagrange function and uses an active set strategy in which only a subset of the constraints is included in the direction finding QP subproblem. Hessian updating with dynamic response constraints presents some difficulties. The primary difficulty is that the total number of constraints and the location of time points where they have to be imposed can change from iteration to iteration. This can cause inconsistencies in Hessian updating if proper numerical procedures are not used. A numerical procedure to handle the situation is developed, implemented and evaluated. Automatic restarting procedures are necessary for proper convergence of the algorithm. The new algorithm is robust as well as more efficient than the purely linear algorithm. The active set strategy plays an important role for application to dynamic response problems. The RQP algorithms that do not use such a strategy are not applicable to this class of problems.

An Investigation of Pshenichnyi’s Recursive Quadratic Programming Method for Engineering Optimization

This paper discusses Pshenichnyi’s recursive quadratic programming algorithm for use in engineering optimization problems. An evaluation of the original algorithm is offered and several modifications are presented. The modifications include; addition of a variable metric update of the Hessian, an improved active set criterion, direct inclusion of the variable bounds, a divergence control mechanism, and updating schemes for the algorithm parameters. Implementations of the original algorithm and the modified algorithm were tested against the Sandgren test set of 23 engineering optimization problems. The results indicate that the modified algorithm was able to solve 20 of the 23 test problems while the original algorithm solved only 11. The modified algorithm was more efficient than the original on all the test problems.

The Optimal Design of Symmetric Laminated Beams Considering Damping

The task of designing symmetric laminated beams with optimal stiffness and damping is considered by incorporating the damping model of Adams, et al. into an optimization algorithm. The design variables are the fiber orientation in each ply, the stacking se quence, and also the thickness of each ply in the laminate. A recursive quadratic program ming algorithm is employed for the optimization process and numerical results are presented for several test problems including the constraint condition of invariable flexural rigidity.