Sequential Quadratic Programming Technique Research Articles

Optimization on response surface models for the optimal manufacturing conditions of dairy tofu

The purpose of this research was to create response surface models through regression on experimental data and to apply the Sequential Quadratic Programming (SQP) and Genetic Algorithms (GAs) on the models to obtain optimal processing conditions for dairy tofu. The two-stage effort of obtaining a surface model using response surface methodology (RSM), and optimizing this model using GAs or SQP techniques was demonstrated to be an effective approach. Both SQP and GAs techniques were able to determine the optimal conditions for manufacturing the probiotic dairy tofu. The conditions were 1% of glucono-delta-lactone (GDL), 0% of peptides level, 3% of isomaltooligosaccharides (IMO) and 18% of milk concentrations, and they were confirmed by verification experiments. Among the SQP and two GAs employed, the SQP, modified with the multi-start capability, is the most efficient one.

Sequential Quadratic Programming for Development of a New Probiotic Dairy Tofu with Glucono‐δ‐Lactone

ABSTRACT: The purpose of this research was to evaluate the effects of various concentrations of glucono‐δ‐lactone (GDL) and skim milk powder, as well as the addition of prebiotics, on the rheology and probiotic viabilities of dairy tofu. Additionally, modern optimization techniques were applied to attempt to determine the optimal processing conditions and growth rate for the selected probiotics (Lactobacillus. acidophilus, L. casei, Bifidobacteria bifidum, and B. longum). There were 2 stages in this research to accomplish the goal. The 1st stage was to derive surface models using response surface methodology (RSM); the 2nd stage performed optimization on the models using sequential quadratic programming (SQP) techniques. The results were demonstrated to be effective. The most favorable production conditions of dairy tofu were 1% GDL, 0% peptides, 3% isomaltooligosaccharides (IMO), and 18% milk, as confirmed by subsequent verification experiments. Analysis of the sensory evaluation results revealed no significant difference between the probiotic dairy tofu and the GDL analog in terms of texture and appearance (P < 0.05). The viable numbers of probiotics were well above the recommended limit of 106 CFU/g for the probiotic dairy tofu throughout the tested storage period.

Hydrodynamic optimization of ship hull forms in shallow water

A computational method for improving hull form in shallow water with respect to wave resistance is presented. The method involves coupling ideas from two distinct research fields: numerical ship hydrodynamics and nonlinear programming techniques. The wave resistance is estimated by means of Morino’s panel method, which is extended to free surface flow and considers the influence of finite depth on the wave resistance of ships. This is linked to the optimization procedure of the sequential quadratic programming (SQP) technique, and an optimum hull form can be obtained through a series of iterations giving some design constraints. Sinkage is an important factor in shallow water, and this method considers sinkage as a hydrodynamic design constraint. The optimization procedure developed is demonstrated by selecting a Wigley (C B = 0.444) hull and the Series 60 (C B = 0.60) hull, and new hull forms are obtained at Froude number 0.316. The Froude number specified corresponds to a lower than critical speed since most of the ships operating in shallow water move below their critical speed. The numerical results of the optimization procedure indicate that the optimized hull forms yields a reduction in wave resistance.

Hybrid PSO–SQP for economic dispatch with valve-point effect

This paper presents a novel and efficient method for solving the economic dispatch problem (EDP), by integrating the particle swarm optimization (PSO) technique with the sequential quadratic programming (SQP) technique. PSO is the main optimizer and the SQP is used to fine tune for every improvement in the solution of the PSO run. PSO is a derivative free optimization technique which produces results quickly and proves itself fit for solving large-scale complex EDP without considering the nature of the incremental fuel cost function it minimizes. SQP is a nonlinear programming method which starts from a single searching point and finds a solution using the gradient information. The effectiveness of the proposed method is validated by carrying out extensive tests on three different EDP with incremental fuel cost function takes into account the valve-point loadings effects. The proposed method out-performs and provides quality solutions compared to other existing techniques for EDP considering valve-point effects are shown in general.

Investigation of a System Identification Methodology in the Context of the ASCE Benchmark Problem

This article briefly presents the theory for a system identification and damage detection algorithm for linear systems, and discusses the effectiveness of such a methodology in the context of a benchmark problem that was proposed by the ASCE Task Group in Health Monitoring. The proposed approach has two well-defined phases: (1) identification of a state space model using the Observer/Kalman filter identification algorithm, the eigensystem realization algorithm, and a nonlinear optimization approach based on sequential quadratic programming techniques, and (2) identification of the second-order dynamic model parameters from the realized state space model. Structural changes (damage) are characterized by investigating the changes in the second-order parameters of the reference and damaged models. An extensive numerical analysis, along with the underlying theory, is presented in order to assess the advantages and disadvantages of the proposed identification methodology.

Optimum design of structures with stress and displacement constraints using the force method

A new structural analysis and optimization algorithm is developed to determine the minimum weight of structures with the truss and beam-type members under displacement and stress constraints. The algorithm combines the mathematical programming based on the sequential quadratic programming (SQP) technique and the finite element technique based on the integrated force method. The equilibrium matrix is generated automatically through the finite element analysis while the compatibility matrix is obtained directly using the displacement–deformation relations and the single value decomposition (SVD) technique. By combining the equilibrium and compatibility matrices with the force–displacement relations, the equations of equilibrium with the element forces as variables are obtained. The proposed method is extremely efficient to analyze and optimize the truss and beam structures under stress and displacement constraints. The computational effort required by the force method is found to be significantly lower than that of the displacement method. The effect of the geometric nonlinearity in the structural optimization problems under the stress and displacement constraints were also investigated and it is illustrated that the geometric nonlinearity is not an important issue in these types of problems and hence, it does not affect the final optimum solution significantly. Four examples illustrate the procedure and allow the results to be compared with those reported in the literatures.

A sequential quadratic programming with a dual parametrization approach to nonlinear semi-infinite programming

In the sequel of the work reported in Liu et al. (1999), in which a method based on a dual parametrization is used to solve linear-quadratic semi-infinite programming (SIP) problems, a sequential quadratic programming technique is proposed to solve nonlinear SIP problems. A merit function to measure progress toward the solution and a procedure to compute the penalty parameter are also proposed.

Nonlinear System Identification Tools Applied to the Modelling of Submarine Dynamics

This paper presents initial results on the use of nonlinear systems identification algorithms to estimate the hydrodynamic coefficients in fully coupled, nonlinear submarine motion equations. Parameter identification algorithms rely on the sequential quadratic programming technique. The paper briefly describes the form of the non-linear equations used in the submarine simulation, describes the system identification algorithms developed and then illustrates their application to three test cases of interest. These are based on horizontal motion only and aim to identify the model coefficients in the cases when these are completely unknown, partially known, and partially known but with an error on the known values respectively.

On the Identification of the Parameters of a Viscoelastic Material Law for Polymeric Foams

AbstractIn this paper, the gradient‐based sequential quadratic programming (SQP) technique is used to determine the parameters of a finite viscoelastic material law for the description of polymeric foams. Due to the cellular structure of foams, the resulting non‐linear behaviour is reproduced by an Ogden‐type law in the framework of the Theory of Porous Media (TPM) [1]. The conformity between experimental data and numerical simulation is improved by the minimization of a least squares functional under consideration of equality and inequality constraints. The result is shown by a finite element based back‐analysis of unconfined compression tests.

Optimization of the Growth Rate of Probiotics in Fermented Milk Using Genetic Algorithms and Sequential Quadratic Programming Techniques

Prebiotics (peptides, N-acetyglucoamine, fructo-oligosaccharides, isomalto-oligosaccharides and galacto- oligosaccharides) were added to skim milk in order to improve the growth rate of contained Lactobacillus acidophilus, Lactobacillus casei, Bifidobacterium longum and Bifidobacterium bifidum. The purpose of this research was to study the potential synergy between probiotics and prebiotics when present in milk, and to apply modern optimization techniques to obtain optimal design and performance for the growth rate of the probiotics using a response surface-modeling technique. To carry out response surface modeling, the regression method was performed on experimental results to build mathematical models. The models were then formulated as an objective function in an optimization problem that was consequently optimized using a genetic algorithm and sequential quadratic programming approach to obtain the maximum growth rate of the probiotics. The results showed that the quadratic models appeared to have the most accurate response surface fit. Both SQP and GA were able to identify the optimal combination of prebiotics to stimulate the growth of probiotics in milk. Comparing both methods, SQP appeared to be more efficient than GA at such a task. (Asian-Aust. J. Anim. Sci. 2003. Vol 16, No. 6 : 894-902)

Gradient-based optimization of a viscoelastic Ogden type model for cellular polymers

The parameter identification is the interface between a theoretical material model and its application in numerical computations. Only by an accurate identification of the theoretically introduced material parameters, an applicable simulation of the material is achieved. An increasing standard of the parameter identification is set by the requirements of complex material models used in computer-aided engineering. A common identification strategy is a gradient-based optimization of a least-squares functional, e. g. the sequential quadratic programming (SQP) technique. In this paper, the SQP method is used to optimize material models of cellular polymers. In particular, the optimization is shown for a viscoelastic polyurethane (PU) foam. Due to the high-grade nonlinear material behaviour, the foam is modelled by a finite viscoelastic Ogden type law in the framework of the Theory of Porous Media (TPM).

Evaluation of modal transformation matrices for overhead transmission lines and underground cables by optimization method

The paper develops a numerical procedure based on constrained optimization for evaluating the modal transformation matrices of overhead transmission lines and underground cables. In the method, the eigenvalue-eigenvector equation is transformed into an objective function which is then minimized for finding eigenvalues and eigenvectors. The method developed avoids completely the problem of eigenvalue switching which causes the discontinuities of eigenvectors encountered in the QR-method. Initial values for starting the optimization at each frequency are the eigenvalues and eigenvectors evaluated at the preceding frequency. This technique leads to the set of smooth eigenvectors that define the modal transformation matrices. The form of the objective function derived in the paper for minimization together with the sequential quadratic programming technique guarantees that convergence to valid eigenvalues and eigenvectors is achieved. Using the method, transformation matrices for a wide range of frequency for representative double-circuit line and underground cable are evaluated.

An Interior Point Constrained Trust Region Method for a Special Class of Nonlinear Semidefinite Programming Problems

In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming (SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e.g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.

Evaluation of Modal Transformation Matrices for Overhead Transmission Lines and Underground Cables by Optimization Methods

This article develops a numerical procedure based on constrained optimization for evaluating the modal transformation matrices of overhead transmission lines and underground cables. In the method, the eigenvalue-eigenvector equation is transformed into an objective function, which is then minimized for finding eigenvalues and eigenvectors. The developed method completely avoids the problem of eigenvalue switching, which causes the discontinuities of eigenvectors encountered in the QR method. Initial values for starting the optimization at each frequency are the eigenvalues and eigenvectors evaluated at the preceding frequency. This technique leads to the set of smooth eigenvectors that define the modal transformation matrices. The form of the objective function derived in the paper for minimization, together with the sequential quadratic programming technique, guarantees that convergence to valid eigenvalues and eigenvectors is achieved. Using the method, transformation matrices for a wide range of frequency for representative double-circuit line and underground cable are evaluated.

Optimal weight design of rotor systems with oil-film bearings subjected to frequency constraints

The purpose of this study is to investigate optimal weight design of rotors. The analysis results of this work for dynamic analysis of rotors are compared to those available in the literature. An excellent agreement is observed, which validates the formulation and implementation of the model developed. Further analysis is conducted for rotors with oil-film bearings. The stability of the system is studied by examining the real parts of the eigenvalues and the logarithmic decrement. The sequential quadratic programming technique is used for the minimum weight design. The effect of minimum weight design on the distribution of rotor shape, weight saving, and system stability is examined. It is demonstrated that significant reduction of rotor weight can be achieved. Weight reduction of more than 80% is possible for a simple rotor analyzed, whereas for a realistic complex rotor with oil-lubricated bearings, an original rotor weight of 5.4886 kg can be reduced to 1.7267 kg after being optimized.

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.

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.

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.

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.

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.