The Reduced Space Shooting Method for Calculating the Peak Periodic Solutions of Nonlinear Systems

2018 ◽  
Vol 13 (6) ◽  
Author(s):  
Haitao Liao ◽  
Wenwang Wu
Keyword(s):  
Nonlinear Systems ◽  
Optimization Problem ◽  
Shooting Method ◽  
Hybrid Approach ◽  
Equality Constraints ◽  
Bound Constraints ◽  
Sqp Method ◽  
Decomposition Scheme ◽  
Reduced Space ◽  
Nonlinear Equality

A hybrid approach which combines the reduced sequential quadratic programing (SQP) method with the shooting method is proposed to search the worst resonance response of nonlinear systems. The shooting method is first employed to construct the nonlinear equality constraints for the constrained optimization problem. Then, the complex optimization problem is simplified and solved numerically by the reduced SQP method. By virtue of the coordinate basis decomposition scheme which exploits the gradients of nonlinear equality constraints, the nonlinear equality constraints are eliminated, resulting in a simple optimization problem subject to bound constraints. Moreover, the second-order correction (SOC) technique is adopted to overcome Maratos effect. The novelty of the approach described lies in the capability to efficiently handle nonlinear equality constraints. The effectiveness of the proposed algorithm is demonstrated by two benchmark examples seen in the literature.

A Frequency Domain Method for Calculating the Failure Probability of Nonlinear Systems With Random Uncertainty

2018 ◽  
Vol 140 (4) ◽  
Author(s):  
Haitao Liao ◽  
Wenwang Wu
Keyword(s):  
Vibration Frequency ◽  
Failure Probability ◽  
Harmonic Balance ◽  
Optimization Problem ◽  
Equality Constraints ◽  
Vibration Response ◽  
Frequency Domain Method ◽  
Balance Equations ◽  
Random Uncertainty ◽  
Nonlinear Equality

A hybrid approach is proposed to evaluate the probability of unacceptable performance with respect to uncertain parameters. The evaluation of structural reliability and the solution of maximum vibration response are performed simultaneously. A constrained optimization problem is deduced for which several techniques have been developed to obtain the reliability index. The nonlinear equality constraints of the optimization problem are constructed based on the harmonic balance equations, the optimality condition of the maximum vibration response with respect to the vibration frequency and the limit state failure function. With the nonlinear equality constraints imposed on the harmonic balance equations and the derivative of the maximum vibration response with respect to the vibration frequency, the inner loop for solving the maximum vibration response is avoided. The sensitivity gradients are derived by virtue of the adjoint method. The original optimization formulation is then solved by means of the sequential quadratic programming method (SQP) method. Finally, the developed approach has been verified by comparison with reference values from Monte Carlo simulation (MCS). Numerical results reveal that the proposed method is capable of predicting the failure probability of nonlinear structures with random uncertainty.

scholarly journals Nonlinear Steady-State Optimization of Large-Scale Gas Transmission Networks

Energies ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 2832
Author(s):  
Andrzej J. Osiadacz ◽  
Małgorzata Kwestarz
Keyword(s):  
Steady State ◽  
Large Scale ◽  
Optimization Problem ◽  
Operating Cost ◽  
Network Efficiency ◽  
Major Objective ◽  
Transmission Networks ◽  
Sqp Method ◽  
Gas Network ◽  
Compressor Power

The major optimization problem of the gas transmission system is to determine how to operate the compressors in a network to deliver a given flow within the pressure bounds while using minimum compressor power (minimum fuel consumption or maximum network efficiency). Minimization of fuel usage is a major objective to control gas transmission costs. This is one of the problems that has received most of the attention from both practitioners and researchers because of its economic impact. The article describes the algorithm of steady-state optimization of a high-pressure gas network of any structure that minimizes the operating cost of compressors. The developed algorithm uses the “sequential quadratic programming (SQP)” method. The tests carried out on the real network segment confirmed the correctness of the developed algorithm and, at the same time, proved its computational efficiency. Computational results obtained with the SQP method demonstrate the viability of this approach.

Solving Mono- and Multi-Objective Problems Using Hybrid Evolutionary Algorithms and Nelder-Mead Method

2021 ◽  
Vol 12 (4) ◽  
pp. 98-116
Author(s):  
Noureddine Boukhari ◽  
Fatima Debbat ◽  
Nicolas Monmarché ◽  
Mohamed Slimane
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Optimization Problems ◽  
State Of The Art ◽  
Hybrid Approach ◽  
Optimization Methods ◽  
Global Optimum ◽  
Stochastic Methods ◽  
Local Optima ◽  
Portfolio Optimization Problem

Evolution strategies (ES) are a family of strong stochastic methods for global optimization and have proved their capability in avoiding local optima more than other optimization methods. Many researchers have investigated different versions of the original evolution strategy with good results in a variety of optimization problems. However, the convergence rate of the algorithm to the global optimum stays asymptotic. In order to accelerate the convergence rate, a hybrid approach is proposed using the nonlinear simplex method (Nelder-Mead) and an adaptive scheme to control the local search application, and the authors demonstrate that such combination yields significantly better convergence. The new proposed method has been tested on 15 complex benchmark functions and applied to the bi-objective portfolio optimization problem and compared with other state-of-the-art techniques. Experimental results show that the performance is improved by this hybridization in terms of solution eminence and strong convergence.

scholarly journals Mixed E-duality for E-differentiable Vector Optimization Problems Under (Generalized) V-E-invexity

2021 ◽  
Vol 2 (3) ◽  
Author(s):  
Najeeb Abdulaleem
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Equality Constraints ◽  
Vector Optimization Problems ◽  
Duality Theorems ◽  
Differentiable Vector

AbstractIn this paper, a class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. The so-called vector mixed E-dual problem is defined for the considered E-differentiable vector optimization problem with both inequality and equality constraints. Then, several mixed E-duality theorems are established under (generalized) V-E-invexity hypotheses.

scholarly journals Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

2021 ◽  
pp. 66-70
Author(s):  
S. Homeniuk ◽  
S. Grebenyuk ◽  
D. Gristchak
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Analytical Solution ◽  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Numerical Solutions ◽  
Hybrid Approach ◽  
Time Dependent ◽  
Forced Oscillations

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters. The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters. The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions. Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions. Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

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