China
Africa
Explore chapters and articles related to this topic
Nonlinear Optimization
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
SAS Institute, Inc. provides a general nonlinear optimization package that runs on various platforms. SAS offers several techniques including Newton–Raphson, Quasi-Newton, conjugate gradient, Nelder-Mead simplex, hybrid Quasi-Newton, and Gauss–Newton methods, which comprise special routines for quadratic optimization problems. The Quasi-Newton methods use the gradient to update an approximation to the inverse of the Hessian and is applicable where the objective function has continuous first and second derivatives in the feasible region. SAS OPTMODEL provides a general nonlinear optimization problem solver. SAS/OR handles nonconvex nonlinear optimization problems that may have many locally optimal solutions that are not globally optimal. SAS/OR applies multiple global and local search algorithms in parallel to solve difficult optimization problems such as those having discontinuous or non-differentiable functions, to identify global optima.
Dialectics of Nature: Inspiration for Computing
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
This approach is known as quasi-Newton methods, also known as variable metric methods. Eventually, quasi-Newton methods construct an approximation of the Hessian matrix H or the inverse of the Hessian matrix H−1 in an iterative way. The approximations H^ will converge to H−1 near the solution point. H^k≈(xk+1−xk)(gk+1−gk)
Nonlinear Optimization Methods
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
The conjugate direction methods and quasi-Newton’s methods are intermediate between the steepest descent and Newton’s method. The conjugate direction methods are motivated by the need to accelerate the typically slow convergence of the steepest descent methods. Conjugate direction methods, as can be seen in Table 3.1, define the search direction by utilizing the gradient vector of the objective function of the current iteration and the information on the gradient and search direction of the previous iteration. The motivation of quasi-Newton methods is to avoid inverting the Hessian matrix as required by Newton’s method. These methods use approximations to the inverse Hessian with a different form of approximation for the different quasi-Newton methods. Detailed descriptions and theoretical development can be found in textbooks such as Luen-berger (1984), Fletcher (1980), Dennis and Schnable (1983), Gill et al. (1981), and Edgar and Himmelblau (1988).
Distributed Newton and Quasi-Newton methods for formation control of autonomous vehicles
Published in Ships and Offshore Structures, 2020
Mansour Karkoub, Huiwei Wang, Tzu-Sung Wu
Various optimisation methods for solving (5) in a distributed manner are available, such as the distributed gradient descent (DGD) algorithm (Nedić and Ozdaglar, 2009) and the distributed dual average (DDA) algorithm (Duchi et al., 2012). For the objective function (5), these methods converge linearly to a solution neighbourhood, but ignore its inherent quadratic characteristics. It is motivated sufficiently to investigate a second-order distributed optimisation method, such as Newton's method, for addressing this problem. Newton's method has two advantages, one is the existence of the domain of attraction, and the other is known as local quadratic convergence. A disadvantage in Newton's method, however, is that the Hessian (and/or its inverse) will be determined at each iteration, which involves the evaluation of scalar functions at least. In any case, this is a computationally intensive operation. In order to reduce the overall computational effort of the Hessian, some techniques for replacing the Hessian by its approximation have appeared in the literature, generally referred to as quasi-Newton methods. This section will investigate solutions for the AUVs system based on distributed Newton and quasi-Newton methods.
A family of quasi-Newton methods for unconstrained optimization problems
Published in Optimization, 2018
Newton's method is an iterative method, which requires the Hessian matrix at every iteration for finding extrema. In quasi-Newton methods, the Hessian matrix does not need to be computed, so it can be used when the Hessian is unavailable or difficult to calculate. Several recent computational studies have shown that the quasi-Newton methods are effective methods for solving minimization problems [16]. These methods use only first derivatives to make an approximation to at each step instead of performing the computational work of evaluating and inverting .
Analyzing the role of the Inf-Sup condition for parameter identification in saddle point problems with application in elasticity imaging
Published in Optimization, 2020
Baasansuren Jadamba, Akhtar A. Khan, Michael Richards, Miguel Sama, Christiane Tammer
For discretization, we use the finite element library FreeFem++ [42]. We solve the optimization problem by using the IPOPT optimization library integrated with FreeFem++. We recall that IPOPT is a software library for large scale nonlinear constrained optimization, which implements a primal-dual interior-point method (see [43]). We approximate the Hessian by a BFGS update quasi-Newton method. We recall that IPOPT permits box constraints, and as the lower bound, we set and as the upper In all numerical experiments, we take as the regularization space.