Nonlinear Programming Algorithm Research Articles

Improved Feasible SQP Algorithm for Nonlinear Programs with Equality Constrained Sub-Problems

This paper is concerned with an improved feasible sequential quadratic programming (FSQP) method for nonlinear programs. As compared with the existing SQP methods which required solving the QP sub-problem with inequality constraints in single iteration, in order to obtain the feasible direction, the method of this paper is only necessary to solve an equality constrained quadratic programming sub-problems. Combined the generalized projection technique, a height-order correction direction is yielded by explicit formulas, which can avoids Maratos effect. Furthermore, under some mild assumptions, the algorithm is globally convergent and its rate of convergence is one-step superlinearly. Numerical results reported show that the algorithm in this paper is effective.

A hybrid optimization approach to design of compact self-shielded super conducting magnetic resonance imaging magnet system

A hybrid optimization approach with a combination of linear programming and nonlinear programming algorithm for designing a compact self-shielded magnetic resonance imaging (MRI) superconducting magnet system is presented. The designed coils possess advantages of low construction costs, simple coil structure and the maximum magnetic strength within coils, current margin and electromagnetic stress easy to control. Firstly, in the stage of linear programming optimization, the feasible rectangular region can be divided into two-dimensional meshes, and a current map is calculated for meeting the magnetic field constraints over the surfaces of DSV sphere and 5 gauss stray field ellipse; Secondly, the current map has many nonzero current clusters and each cluster can be discretized into a solenoid. A nonlinear programming algorithm is employed to optimize the positions of all solenoids for minimizing the total coil volume and meeting all constraints including magnetic field which is the same as linear programming stage, and maximum magnetic strength, current margin and the gap between neighborhood inner coils. A 1.5 T compact self-shielded MRI superconducting magnet system is studied, the total coil length is only 1.32 m and the peak-peak homogeneity over 50 cm DSV is 10 ppm. The design approach is flexible and efficient for designing symmetrical and asymmetrical horizontal MRI and also open bi-planar MRI system.

On the Dynamic Optimization of Biped Robot

 Abstract—This paper concentrates on three important points: the selection of the suitable direct method used for suboptimal control of the biped robot, the selection of the appropriate nonlinear programming (NLP) algorithm that searches for the global minimum rather than the local minimum, and the effect of different constraints on the energy of the biped robot. To perform the mentioned points, the advantages and disadvantages of the optimal control methods were illustrated. The inverse-dynamics based optimization is preferred because of the ability to convert the original optimal control into algebraic equations which are easy to deal with. The inverse-dynamics-based optimization was classified as spline and the finite difference based optimization. Due to the easy use of the latter, it was used for investigating seven cases with different constraints for 6-DOF biped robot during the single support phase (SSP). Hybrid genetic-sequential quadratic programming (GA-SQP) was used for simulation of the target robot with MATLAB. It can be concluded that more imposed constraints on the biped robot, more energy is needed. In general, more energy can be required in the case of (1) restriction of the swing foot to be level to the ground and (2) reducing the hip height or constraining the hip to move in constant height.

A more efficient algorithm for Convex Nonparametric Least Squares

Convex Nonparametric Least Squares (CNLSs) is a nonparametric regression method that does not require a priori specification of the functional form. The CNLS problem is solved by mathematical programming techniques; however, since the CNLS problem size grows quadratically as a function of the number of observations, standard quadratic programming (QP) and Nonlinear Programming (NLP) algorithms are inadequate for handling large samples, and the computational burdens become significant even for relatively small samples. This study proposes a generic algorithm that improves the computational performance in small samples and is able to solve problems that are currently unattainable. A Monte Carlo simulation is performed to evaluate the performance of six variants of the proposed algorithm. These experimental results indicate that the most effective variant can be identified given the sample size and the dimensionality. The computational benefits of the new algorithm are demonstrated by an empirical application that proved insurmountable for the standard QP and NLP algorithms.

An Improved Control Vector Iteration Approach for Nonlinear Dynamic Optimization (I) Problems Without Path Constraints

This study proposes an efficient indirect approach for general nonlinear dynamic optimization problems without path constraints. The approach incorporates the virtues both from indirect and direct methods: it solves the optimality conditions like the traditional indirect methods do, but uses a discretization technique inspired from direct methods. Compared with other indirect approaches, the proposed approach has two main advantages: (1) the discretized optimization problem only employs unconstrained nonlinear programming (NLP) algorithms such as BFGS (Broyden-Fletcher-Goldfarb-Shanno), rather than constrained NLP algorithms, therefore the computational efficiency is increased; (2) the relationship between the number of the discretized time intervals and the integration error of the four-step Adams predictor-corrector algorithm is established, thus the minimal number of time intervals that under desired integration tolerance can be estimated. The classic batch reactor problem is tested and compared in detail with literature reports, and the results reveal the effectiveness of the proposed approach. Dealing with path constraints requires extra techniques, and will be studied in the second paper.

An Enumerative NonLinear Programming approach to direction finding with a general spatially spread electromagnetic vector sensor array

In this paper, the geometry of a general spatially spread electromagnetic vector sensor (SSEMVS) array is firstly presented. This array has a special cross product structure of the array manifold vector. The g parameters are then defined to uniquely characterize the cross product. With g parameters, the identifiability analysis of the cross product vector based direction finding algorithms of the general SSEMVS array is carried out. The analysis shows that the array has to be carefully designed to avoid ambiguity directions. Finally, a cross-product based Enumerative NonLinear Programming (ENLP) algorithm for the direction of arrival (DOA) estimation with the general SSEMVS array is proposed. The algorithm finds the optimum estimation of DOA in least square sense. Extensive simulations show that the proposed algorithm outperforms existing algorithms, and can approach the CRB effectively.

An Extended Sequential Quadratically Constrained Quadratic Programming Algorithm for Nonlinear, Semidefinite, and Second-Order Cone Programming

This paper is concerned with nonlinear, semidefinite, and second-order cone programs. A general algorithm, which includes sequential quadratic programming and sequential quadratically constrained quadratic programming methods, is presented for solving these problems. In the particular case of standard nonlinear programs, the algorithm can be interpreted as a prox-regularization of the Solodov sequential quadratically constrained quadratic programming method presented in Mathematics of Operations Research (2004). For such type of methods, the main cost of computation amounts to solve a linear cone program for which efficient solvers are available. Usually, “global convergence results” for these methods require, as for the Solodov method, the boundedness of the primal sequence generated by the algorithm. The other purpose of this paper is to establish global convergence results without boundedness assumptions on any of the iterative sequences built by the algorithm.

On the generalization of ECP and OA methods to nonsmooth convex MINLP problems

In this article, generalization of some mixed-integer nonlinear programming algorithms to cover convex nonsmooth problems is studied. In the extended cutting plane method, gradients are replaced by the subgradients of the convex function and the resulting algorithm shall be proved to converge to a global optimum. It is shown through a counterexample that this type of generalization is insufficient with certain versions of the outer approximation algorithm. However, with some modifications to the outer approximation method a special type of nonsmooth functions for which the subdifferential at any point is a convex combination of a finite number of subgradients at the point can be considered. Numerical results with extended cutting plane method are also reported.

Least squares and output error identification algorithms for continuous time systems with unknown time delay operating in open or closed loop

This paper presents two off-line output error identification algorithms for linear continuous-time systems with unknown time delay from sampled data. The proposed methods (for open and closed loop systems) use a Nonlinear Programming algorithm and need an initialization step that is also proposed from a modification of the Yang algorithm. Simulations, as illustrated by Monte-Carlo runs, show that the obtained parameters are unbiased and very accurate.

Presenting an algorithm of integer nonlinear multiple objective programming in conditions of uncertainty for balanced scorecard method (case study in Islamic Azad University, Semnan Branch)

A B S T R A C T Balanced scorecard is a performance appraisal method planned for measuring the organizational efficiency to develop their strategies. Organization's strategies in a specified period are the main inputs in this model. Furthermore, due to nature, experts' opinions play the vital role in determining the strategies. In this research, the proposed algorithm is designed by using fuzzy set covering problem and non-linear multiple objective integer programming (zero and one variables), so that it can be useful to choose the best combination of strategies for specified period of time with the least deviation in expertsopinions. The presented model is carried out in Islamic Azad University, Semnan Branch. The results indicate that the designed model can provide the best combination of strategies for entering into the balanced scorecard system.

Determination of the Complex Moduli of Viscoelastic Materials by a Free-Vibration Method

Based on the finite-element vibration equations of viscoelastic beam in Laplace domain and the corresponding analytical solution, a semi-analytical frequency equation, which contains the complex modulus of the viscoelastic material, is derived. With several simple measuring instruments, the vibration parameters of the viscoelastic beams can be measured. According to the semi-analytical frequency equation, the complex modulus of the viscoelastic material can be calculated by a nonlinear programming algorithm. As an illustrative example, the vibration test of free-free beams made of viscoelastic expoxy composite material is presented. A five-parameter complex modulus is assumed and evaluated.

Optimization Design of Geotechnical Engineering Based on Reliability and Lingo

Algorithm of most optimization calculation is favor for the most technology personnel, for its result is ideal and it doesn’t need derivative of performance function. Combined with theory of nonlinear programming and optimization algorithm of reliability, calculating program is written by means of Lingo, which optimization calculation of geotechnical engineering based on reliability, and source program is grasped for the engineers. Combined example, calculating program is given, which are realized by means of Lingo, which is nonlinear programming software. By the studying, the aim is to improve calculating efficiency for most of engineers and to push forward application of structural reliability theory in geotechnical engineering.

Feasible direction method for bilevel programming problem

In this article, we investigate the application of feasible direction method for an optimistic non-linear bilevel programming problem. The convex lower level problem of an optimistic non-linear bilevel programming problem is replaced by relaxed KKT conditions. The feasible direction method developed by Topkis and Veinott [D.M. Topkis and A.F.jun. Veinott, On the convergence of some feasible direction algorithms for nonlinear programming, SIAM J. Control Optim. 5(1967), pp. 268–279] is applied to the auxiliary problem to get a Bouligand stationary point for an optimistic bilevel programming problem.

Handling infeasibility in a large-scale nonlinear optimization algorithm

Practical Nonlinear Programming algorithms may converge to infeasible points. It is sensible to detect this situation as quickly as possible, in order to have time to change initial approximations and parameters, with the aim of obtaining convergence to acceptable solutions in further runs. In this paper, a recently introduced Augmented Lagrangian algorithm is modified in such a way that the probability of quick detection of asymptotic infeasibility is enhanced. The modified algorithm preserves the property of convergence to stationary points of the sum of squares of infeasibilities without harming the convergence to KKT points in feasible cases.

A superlinearly convergent numerical algorithm for nonlinear programming

In this paper, a new algorithm is proposed to solve the nonlinear constrained optimization problems. Unlike sequential quadratic programming (SQP) type methods, this algorithm does not involve solutions of quadratic programs. It is merely necessary to solve systems of linear equations with small scale to obtain a direction. The scheme is based on an idea of ε-effective active set strategies. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions.

Gauss–Newton method

AbstractThe Gauss–Newton method for nonlinear regression is described, along with its close relationship with the Fisher scoring method. Its many extensions, sometimes in the form of Fisher scoring, are briefly reviewed, including those for coping with large‐residual problems in nonlinear regression, for optimizing a likelihood or other performance criterion, and for dealing with parameters with constraints.WIREs Comput Stat2012 doi: 10.1002/wics.1202This article is categorized under:Algorithms and Computational Methods > Quadratic and Nonlinear ProgrammingAlgorithms and Computational Methods > Numerical Methods

Optimization of Impulsive Trajectories from a Circular Orbit to an Excess Velocity Vector

Feasible transfer trajectories are constructed to serve as initial guesses for determining constrained optimal impulsive escape trajectories, from a circular orbit to a target hyperbolic excess velocity vector. The proximity of these feasible solutions to their corresponding optima are quantified. The objective of this work is to improve the feasible solutions, thereby enabling general anytime escape trajectories to be targeted that result in substantially reduced fuel expenditure. The procedure is currently restricted to one- and three-impulse escape trajectories from a circular parking orbit. Two separate three-impulse methods are presented, each specific to whether the time of flight is free or fixed. Numerical results, obtained using nonlinear programming algorithms, are presented to validate the robustness of the method. Analysis and results are nondimensional, and therefore are applicable to departures from a circular orbit about any celestial body.

Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive

In this paper, a polymorphic uncertain nonlinear programming (PUNP) model is constructed to formulate the problem of maximizing the V-belt's fatigue life according to the practical engineering design conditions. The model is converted into an equivalent interval programming only involved with interval parameters for any given degree of membership and confidence level. Then, a deterministic equivalent formulation (DEF) for the original model is obtained based on the concept of possibility degree for the order of two interval numbers. An algorithm, called sampling based algorithm, is developed to find a robust optimal design scheme for maximizing the fatigue life of the V-belt. Case study is employed to demonstrate the validity and the practicability of the constructed model and the algorithm.

A Sequential Linear Constraint Programming Algorithm for NLP

A new method for nonlinear programming (NLP) using sequential linear constraint programming (SLCP) is described. Linear constraint programming (LCP) subproblems are solved by a new code using a recently developed spectral gradient method for minimization. The method requires only first derivatives and avoids having to store and update approximate Hessian or reduced Hessian matrices. Globalization is provided by a trust region filter scheme. Open source production quality software is available. Results on a large selection of CUTEr test problems are presented and discussed and show that the method is reliable and reasonably efficient.

A More Efficient Algorithm for Convex Nonparametric Least Squares

Convex Nonparametric Least Squares (CNLS) is a nonparametric regression method that does not require a priori specification of the functional form. The CNLS problem is solved by mathematical programming techniques; however, since the CNLS problem size grows quadratically as a function of the number of observations, standard Quadratic Programming (QP) and Nonlinear Programming (NLP) algorithms are inadequate for handling large samples, and the computational burdens become significant even for relatively small samples. This study proposes a generic algorithm that improves the computational performance in small samples and is able to solve problems that are currently unattainable. A Monte Carlo simulation is performed to evaluate the performance of six variants of the proposed algorithm. These experimental results indicate that a particular variant is most efficient given the sample size and the dimensionality. The computational benefits of the new algorithm are demonstrated by an empirical application that proved insurmountable for the standard QP and NLP algorithms.