Quadratic Programming Technique Research Articles

Nonconvex Economic Dispatch by Integrated Artificial Intelligence

This paper presents a new algorithm by integrating evolutionary programming (EP), tabu search (TS), and quadratic programming (QP) methods to solve the nonconvex economic dispatch problem (NED). A hybrid EP and TS were used for quality control and the Fletcher's quadratic programming technique was used for solving. EP and TS determine the segment of a cost curve used, which is piecewise quadratic natured. Operation constraints are modeled as linear equality or inequality equations, resulting in a typical QP problem. Fletcher's QP was chosen to enhance the performance. The fitness function is constructed from priorities without penalty terms. Numerical results show that the proposed method is more effective than other previously developed evolutionary computation algorithms.

Optimal design of composite ChamberCore structures

An optimization procedure has been developed to uniquely and efficiently determine the “best” local geometry design of a new composite ChamberCore structure. This procedure is based on minimization of the total mass of a single composite ChamberCore subject to a set of design and stress constraints. The stress constraints are obtained in closed form based on the composite box-beam model for various composite lamination designs and loading conditions. The optimization problem statement is constructed and then solved using the VMCON optimization program, which is an iterative sequential quadratic programming (SQP) technique based on Powell's algorithm. The sensitivity of the solution of the optimal geometry to the values of parameters that characterize the structural durability and the failure mechanism is discussed.

Nonconvex economic dispatch by integrated artificial intelligence

This paper presents a new algorithm by integrating evolutionary programming (EP), tabu search (TS) and quadratic programming (QP) methods to solve the nonconvex economic dispatch problem (NED). A hybrid EP and TS were used for quality control, and Fletcher's quadratic programming technique for solving. EP and TS determines the segment of a cost curve used, which is piecewise quadratic natured. Operation constraints are modeled as linear equality or inequality equations, resulting in a typical QP problem. Fletcher's QP was chosen to enhance the performance. The fitness function is constructed from priorities without penalty terms. Numerical results show that the proposed method is more effective than other previously developed evolutionary computation algorithms.

Structural damage identification using static test data and changes in frequencies

A structural damage identification algorithm using static test data and changes in natural frequencies is presented in this paper. To locate damage in the structure, the Damage Signature Matching (DSM) technique is improved through a proper definition of Measured Damage Signatures (MDS) and Predicted Damage Signatures (PDS). The effect of damage severity can be eliminated effectively as the result that the first-order approximation of changes in static deformation and natural frequencies are employed jointly in the damage signatures. The damage location, then, can be detected successfully by matching MDS with PDS. After obtaining the possible damage location, an iterative estimation scheme for solving non-linear optimization programming problems, which is based on the quadratic programming technique, is proposed to predict the damage extent. A remarkable characteristic of the present approach is that it can be directly applied in the cases of incomplete measured data. The correctness and the effectiveness of the algorithm are proved by two examples: A planar truss model with the numerically simulated data and a beam with two-fixed ends, of which the static response and the natural frequencies are obtained experimentally. The results show that the proposed algorithm is efficient for the damage identification.

Probabilistic regularization in inverse optical imaging.

The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical analysis of the noisy data is presented. Particular emphasis is placed on the question of the positivity constraint, which is incorporated into the probabilistically regularized solution by means of a quadratic programming technique. Numerical examples illustrating the main steps of the algorithm are also given.

A trust region method based on interior point techniques for nonlinear programming

An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented.

Optimisation techniques for electrical power systems. Part 1: Mathematical optimisation methods

This is the first tutorial of a three-part series. The series discusses modern optimisation techniques applicable to power system problems and, by considering relevant examples, will explain some of the concepts, issues and terminology in this field. This first part gives an introduction to mathematical programming based methods including linear programming, quadratic programming, nonlinear programming, integer programming, mixed-integer programming, dynamic programming, and Lagrangian decomposition techniques. Practical issues of applying these methods to power systems are discussed.

DESIGN OPTIMIZATION OF MARINE ENGINE-MOUNT SYSTEM

Design optimization of marine engine–mount system for vibration control is presented in this paper. The engine is modelled as a rigid body with supports connected to a rigid floor. The mounts are modelled as three-dimensional isolators with hysteresis damping. The objective is to select the stiffness coefficient and orientations of individual mount in order to minimize the vertical force transmitted from the engine to the floor to control the structure-borne noise. Constraints are imposed to keep the isolator static and dynamic deflection within the desired limits and a minimum gap between system natural frequency and engine excitation frequency for avoiding possible system resonance. The sequential quadratic programming (SQP) technique has been applied as the optimization algorithm. The typical force and moment of a 4-stroke engine with one cylinder is analyzed and input in the optimization system. The results of optimization are compared with that of a conventional engineering design with the isolators working under their maximum allowable deflection. This comparison shows that, as compared with force transmission of the conventional isolation system, the value of optimized system is only half for multi-frequencies excitation and one-fourth for single frequency excitation. The mechanism of vibration isolation involved in the optimization is investigated with the frequency response of the system. Sensitivity study of the system with the variation of the design parameters is also carried out and proves that the minimum obtained is the global value in the range of design parameters.

Identification of geometric shapes and material properties of inclusions in two-dimensional finite bodies by boundary parameterization

A system identification scheme based on the boundary element method (BEM) is proposed to determine geometric shape and elastic material properties of an inclusion in a finite body. The proposed algorithm is based on the minimization of least squared errors between measured displacement and calculated displacement by a boundary element model. A regularization function that consists of the geometric term and material property term is added to the error function to overcome ill-posedness of inverse problems. To deal with the shape variation of an inclusion during the optimization process, the boundary parameterization technique is applied to the BEM. The recursive quadratic programming (RQP) technique with line search is used for optimization.

Shape Optimization of RC Flexural Members

A natural velocity field method for shape optimization of reinforced concrete (RC) flexural members has been demonstrated. The possibility of shape optimization by modifying the shape of an initially rectangular section, in addition to variation of breadth and depth along the length, has been explored. Necessary shape changes have been computed using the sequential quadratic programming (SQP) technique. Genetic algorithm has been used to optimize the diameter and number of main reinforcement bars. A limit-state design approach has been adopted for the nonprismatic RC sections. Such relevant issues as formulation of optimization problem, finite-element modeling, and solution procedure have been described. These design examples—a simply supported beam, a cantilever beam, and a two-span continuous beam, all under uniformly distributed loads—have been optimized. The results show a significant savings (40–56%) in material and cost and also result in aesthetically pleasing structures. This procedure will lead to considerable cost saving, particularly in cases of mass-produced precast members and a heavy cast-in-place member such as a bridge girder.

Compressive buckling of laminates with an embedded delamination

In this paper, a buckling analysis of laminates with an embedded delamination has been conducted by employing a finite-element method based on the Mindlin plate theory. An effective solution method has been put forward to deal with the contact problem in the buckling mode. In this method, an iterative updating process incorporating first-order sensitivity analysis and quadratic programming technique has been proposed for computing fictitious forces in the contact area, which can be used to calculate stiffness parameters of some artificial springs. Finally, the penetration between two delaminated layers in the buckling mode can be effectively prevented by inserting these artificial springs into the overlapped area in the buckling mode. Numerical examples indicated that this method is very efficient for solving the contact problem in buckling analysis. Its accuracy and convergence speed are very high. The numerical results demonstrate that the contact analysis is very important for buckling analysis in some cases. The effects of various sizes, shapes and positions of the delamination and the fiber angle of the sublaminate on the buckling load have also been investigated.

Optimal Induction Motor Design by the Use of NLP-Approximation Approach and SQP-Method

In computational terms, design optimization for any electrical machine constitutes a mixedinteger programming (MIP) task because, along with continuously variable design parameters, which are allowed to assume any value within a specified range, there exist parameters that may only assume discrete values; finding an optimal solution for such a combination of design variables represents a computational challenge. If, on the other hand, for the purpose of searching for the optima, discrete variables could be treated as continuously variable quantities, the task would be considerably simpler. Then a range of modern optimization methods based on gradient search techniques could be employed in determining the search direction. This approach would be tantamount to having converted the MIP problem into a nonlinear programming (NLP) problem. This paper describes the application of a sequential quadratic programming (SQP) technique in design optimization for an induction motor. It demonstrates how a MIP-problem can be successfully approached using an NLP-approximation to simplify the task of finding optima. The design algorithm, implemented on a desktop computer, allows globally optimized designs to be found with relative ease, unlikely to be achievable with conventional design methods. Aspects of algorithm implementation are discussed, including the formulation of the NLP-approximation, convergence speed, and the nature of convergence.

Buckling analysis of delaminated laminates with consideration of contact in buckling mode

This paper concerns the buckling load analysis of laminates with a pre-existing delamination, using the finite element method based on the Mindlin plate theory. To deal with the contact problem in the buckling mode, an effective algorithm is presented. In this method, an iterative updating process based on the first-order sensitivity analysis and the quadratic programming technique is proposed to compute the fictitious forces in contacting areas at first. These fictitious forces are then transferred into the stiffness parameters of some artificial springs. The original stiffness matrix of system can be modified, using these artificial springs. Finally, the penetration between two delaminated layers in the buckling mode can be prevented effectively. Numerical examples show that this method is very efficient to solve the contact problem in eigenvalue analysis from the viewpoint of its accuracy, stability and convergence speed. The effects of contact and delamination size on the buckling load analysis are also investigated. Copyright © 1999 John Wiley & Sons, Ltd.

Dynamical models of NGC 3115

We present new dynamical models of the S0 galaxy N3115, making use of the available published photometry and kinematics as well as two-dimensional TIGER spectrography. The models are based on a detailed model of the luminosity distribution built using an MGE fit on HST/WFPC2 and ground-based photometric data. We first examined the kinematics in the central 40 arcsec in the light of two-integral f(E,J) models. Jeans equations were used to constrain the mass-to-light ratio, and the central dark mass, the existence of which was suggested by previous studies. The even part of the distribution function was then retrieved via the Hunter & Qian formalism. We thus confirmed that the velocity and dispersion profiles in the central region could be well-fitted with a two-integral model, given the presence of a central dark mass of ∼109 M⊙. However, no two-integral model could fit the h3 profile around a radius of about 25 arcsec where the outer disc dominates the surface brightness distribution. Three-integral analytical models were therefore built using a quadratic programming technique. These models showed that three-integral components do indeed provide a reasonable fit to the kinematics, including the higher Gauss--Hermite moments. Again, models without a central dark mass failed to reproduce the observed kinematics in the central arcsec. This clearly supports the presence of a nuclear black hole of at least 6.5×108 M⊙ in the centre of NGC 3115. These models were finally used to estimate the importance of the dark matter in the outer part of NGC 3115, suggested by the flat stellar rotation curve observed by Capaccioli et al. This study finally points out the difficulty of integrating independently published data in a coherent and consistent way, thus demonstrating the importance of taking into account the details of the instrumental setup and the reduction processes.

On-line optimization of free radical bulk polymerization reactors in the presence of equipment failure

An on-line optimizing control scheme has been developed for bulk polymerization of free radical systems. The effects of random errors, as well as one kind of a major disturbance (heating system failure), have been studied. A model-based, inferential state estimation scheme was incorporated to estimate, on-line, the parameters of the model (and thereby, the monomer conversion and molecular weight of the polymer) using experimental data on temperature and viscosity. A sequential quadratic programming technique was used for this purpose. A major disturbance, such as heating system failure, leads to a deteriorated final product unless an on-line optimal temperature trajectory (history) is recomputed and implemented on the reactor. Genetic algorithm was used for this purpose. It has been found that, if the “sensing” of the major temperature deviation from the optimal value and rectification of the heating system is achieved well in advance of the onset of the Trommsdroff effect, use of a reoptimized temperature history is sufficient to produce the desired product without significantly altering reaction time. However, if such a disturbance occurs late, a single-shot intermediate addition of an optimal amount of initiator needs to be used in addition to changing the temperature history to produce polymers having the desired properties in the minimum reaction time. Other types of failures can similarly be handled using the methodology developed. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 71: 2101–2120, 1999

Large-Scale Direct Optimal Control Applied to a Re-Entry Problem

We present the numerical solution of an atmospheric re-entry problem for the Space Shuttle. We discretize the control and state with identical grid and use a large-scale successive quadratic programming technique. With the help of sliding horizon and successive ree nement of the discretization, we can solve on a workstation a problem with 1600 grid intervals, an unusually large e gure for this kind of real-world optimal control problem. HE technique called direct optimal control (DOC) consistsof discretizing an optimal control problem and then solving the resulting nonlinearprogramming problem. Itisoftenopposed to the techniques based on Pontryagin’ s principle, in which the control is expressed as a function of the state and costate, reducing the optimality system (in the simplest case ) to a two-point boundary-value problem, which can be solved by a multiple shooting algorithm. 1 The advantages of each method have been discussed thoroughly by many authors, among them Pesch 2 and Betts. 3 It is recognized that multiple shooting is most effective when the starting point (for the state and costate )is good. In terms of complexity,this algorithm is optimal in the sense that the computational effort is (in the case of an integration scheme of order 1 ) proportional to the number of points used when integrating the differential system. In addition, the integration can be done using a device for controlling the precision. The drawbacks are that the method may have dife culties in converging if the starting point is poor, which may occur often as it is not easy to give good initial values for the costate. In addition, any structural change in the constraints implies a modie cation of the system of equations to be solved. The advantage ofa prioridiscretizing an optimal control problem is that it is a general method, not so sensitive to an initial guess for the costate, which allows one to use the software already available for solving nonlinear programming problems. In the past, this kind oftechnique has oftenbeen combinedwith a low-dimension parameterization of the control. 4 In that case, the nonlinear programming problem has a small number of variables and a large number of constraints: the distributed control and state constraints. Effective algorithms exist for dealing with this kind of structure, the so-called active set methods. 5 However, parameterizing the control destroys the local structure of the optimal control problems. It is dife cult to evaluate how far the solution of the parameterized problem is from the solution of the original problem. Anotherpossibility istodiscretizethe controlusingthe samegrid intervals as for the state. The aim of this paper is to explore such a possibility. The disadvantage we have to face is the dife culty of solving the resulting large-scale nonlinear programming problem. In particular, it seems dife cult to obtain the same computational complexity as for multiple shooting. Rather, we may hope to obtain a less precise estimate of the optimal control, but it will be easier to obtain due to the generality of the method. Some results along this line were obtained by Betts and Huffman. 6;7 In this paper we study the application of a large-scale DOC algorithmtothe problemofatmosphericre-entryoftheSpaceShuttle.In

Analysis of Surface Tractions in Complex Fretting Fatigue Cycles Using Quadratic Programming

Real fretting fatigue cycles often involve complex loading where the fretting force may vary with a different phase or frequency to the other loads on the components. Analysis of the contact tractions in such cases can be difficult. Here we present a quadratic programming technique which enables the shear tractions to be determined in a relatively straightforward manner. A number of sample load histories are investigated and it is shown that the shear traction cycle rapidly shakes down to a steady state. Further, the results are relatively insensitive to the manner in which loading is commenced.

A BOUNDARY ELEMENT MODEL OF PLASTICITY‐INDUCED FATIGUE CRACK CLOSURE

This paper describes a plane stress boundary element model of plasticity‐induced fatigue crack closure. A simple Dugdale‐type strip yield zone is used and quadratic programming techniques are employed to establish crack shape, stress and plastic deformation. The technique is extremely effective and the model can be readily implemented on a personal computer. Predictions of crack closure behaviour are produced for cracks growing under constant amplitude loading, and also following an overload or overload/underload cycle. These results are compared with an empirical R‐ratio correction due to Walker and with experimental measurements taken from the literature. The model is found to give good predictions of crack behaviour under constant amplitude loading. Predictions for crack closure levels following an overload cycle give qualitative agreement with experimental results; the differences observed may well be due to the different definition of crack closure in the experiments.

Real-time economic dispatch with line flow and emission constraints using quadratic programming

The presence of multiple constraints due to network line flow limits and emission allowances in the economic dispatch of modern power systems makes the conventional Lambda-Delta iterative approach no longer effective. This paper proposes a practical strategy based on quadratic programming (QP) techniques to solve the real-time economic dispatch problem. It formulates the problem with a quadratic objective function based on the unit's cost curves in quadratic or piecewise-quadratic forms. The operation constraints are modeled as linear equality/inequality equations, resulting in a typical QP problem. Goal programming techniques are also incorporated in the formulation which guarantees the best available solution even under infeasible conditions. In addition, the proposed strategy formulates the problem in the second phase dispatch in real-time by including a set of emergency control variables to provide effective control strategies for properly relieving constraint violations if they exist. The effectiveness of the proposed strategy is demonstrated by an example power dispatch problem.

Optimisation of biochemical reactors—an analysis of different approximations of fed-batch operation

In this work we analyse the fed-batch operation of a biochemical reactor. In the first part of the work we consider a repeated fed-batch operation. The reaction is assumed to follow Haldane kinetics, i.e. it is characterised by substrate inhibition. The feed rate of the substrate is chosen as the control variable. The entire duration of the operation is divided into different subintervals. We optimise the system performance by approximating the feed flow rate using (i) discrete pulses, (ii) a constant flow rate over different sub-intervals. In the first strategy the equations lend themselves to an analytical solution. For the second case we use a shooting method coupled with a sequential quadratic programming technique to obtain the constant flow rates in the different sub-intervals. This has been analysed for two scenarios: (i) equal duration of sub-intervals and (ii) unequal duration of sub-intervals. The objective here is to maximise the biomass production over one cycle. The effect of converting a non-linear constraint to a linear constraint by reformulating the problem on the numerical convergence has also been investigated. In the second part of this work we have extended our method to investigate a non-repeated fed-batch reactor operation. Here the objective is to maximise the product formed.