Trust Region algorithm

A. R. Conn, N. I. M. Gould, and Ph. L. Toint, SIAM J. Numer. Anal., 25, 433 (1988). Global Convergence of a Class of Trust Region Algorithms for Optimization with Simple Bounds. 

The strategy used to choose the trust region radius d at each iteration is very important in trust region algorithms. In this regard, it useful to monitor the following ratio ... 

NOTE In the Fit options, with the Trust-Region algorithm selected, try the three robust cases (Off, EAR, and Bisquare) and see which case gives the best fit. 

After having established a well-validated predictive model, the inverse problem can be approached finding optimum process parameters to adjust a PSD with certain mean size. Key process parameters are time-temperature profiles which lead to distinct PSDs. Using a trust-region algorithm and the existing population balance model, suitable time-temperature profiles were obtained after 99 iterations for the narrow PSD and 115 iterations... 

The PSLP algorithm is a steepest descent procedure applied to the exact Lx penalty function (see Section 8.4). It uses a trust region strategy (see Section 6.3.2) to guar-... 

If predk = 0, then no changes Ax within the rectangular trust region (8.58) can reduce PI below the value P1(0, x ). Then x is called a stationary point of the nonsmooth function P, that is, the condition predk = 0 is analogous to the condition V/(x ) = 0 for smooth functions. If predk = 0, the PSLP algorithm stops. Otherwise predk > 0, so we can compute the ratio of actual to predicted reduction. 

Step 4 rejects the new point and decreases the step bounds if ratiok < 0. This step can only be repeated a finite number of times because, as the step bounds approach zero, the ratio approaches 1.0. Step 6 decreases the size of the trust region if the ratio is too small, and increases it if the ratio is close to 1.0. Zhang et al. (1986) proved that a similar SLP algorithm converges to a stationary point of P from any initial point. 

The quality of line search in these nonlinear CG algorithms is crucial. (Typically, line searches are used rather than the trust region methods.) Adjustments must be made not only to preserve the mutual conjugacy of the search directions—a property critical for finite termination of the method—but also to ensure that each generated direction is one of descent. A technique known as... 

G. Korte, Eds., pp. 256-287, Springer-Verlag, New York, 1983. Recent Developments in Algorithms and Software for Trust Region Methods. 

D. M. Gay, ACM Trans. Math. Software, 9, 503 (1983). Algorithm 611. Subroutines for Unconstrained Minimization Using a Model/Trust-Region Approach. 

To solve the equations of problem P ), the optimization algorithms used are Levenberg-Marquardt and Trust-Region procedures. These methods enable computation of the solution by using the Jacobian matrix and the Hessian matrix (or its approximation) related to the objective function E(Y) [57]. 

The SAO algorithm proposed in [6] was used as a basis of the algorithm proposed here (Figure 3). The key features of this algorithm are related to the way by which the base metamodel is adapted and assessed, the trust region updating procedure and the... 

The default algorithm is trust-region-reflective . It is not the aim of this book to comment on the solution methods but for the interested reader, the details can be found in specialized literature [3,4]. 

If we relax the condition that the algorithms should be globally convergent, then quasi-Newton methods may be quite useful and cost-effective, as demonstrated by Culot et al [49]. These authors use a trust-region based quasi-Newton method with an exact initial Hessian and converge in a modest number of iterations (ten to twenty) provided the initial guess is a reasonable one. 

The basic structure of an iterative local optimization algorithm is one of greedy descent . It is based on one of the following two algorithmic frameworks line-search or trust-region methods. Both are found throughout the literature and in software packages and are essential components of effective... 

Algorithm [A2] Basic Descent Using A Trust Region Subsearch... 

LANCELOT Philippe Toint pht raath.fundp.ac.be Various Newton methods for constrained and unconstrained nonlinear optimization, specializing in laige-scale problems and including a trust-region Newton method and an algorithm for nonlinear least squares that exploits partial separability... 

This kind of algorithms firstly constructs a model function (trust region) mk whose behavior near the current solution Xk is similar to that of the actual objective function/. Then, these methods choose the step and the direction to find the approximate optimizer of the model in this trust region. 

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