Optimization.TrustRegion Method

Trust region algorithm to find bounded minimum of vector function.

Additional parameters LB and UB define x lower and/or upper bounds.

Trust region algorithm for finding minimum of vector function.

What it computes. Solves the nonlinear least-squares problem
min_x 1/2 || F(x)||_2^2 for the residual vector $F(x)=y-f(x)$ using Intel MKL's derivative trust-region solver (dtrnlsp; the bounded overload uses dtrnlspbc with the box LB <= x <= UB). At each step a trial step is accepted only if it stays inside the current trust region, whose radius is grown or shrunk from the agreement between the model and the true decrease. The Jacobian A(x)=d F/d x is supplied by JacProc or, when that is nil, approximated numerically.

Domain. X holds the n starting variables; Y
receives the m residual components (its length sets m). This is a double-precision routine; drive it with mvDouble vectors. EPSArray are six non-negative stopping thresholds; Rs ≥ 0 sets the initial trust radius (0 selects the default 100).

Defined behaviour. Returns the iteration count; X holds
the minimizer and Y the residual there. StopReason is Dew.Math.TOptStopReason.OptResConverged when || F|| falls below EPSArray[1], or one of Dew.Math.TOptStopReason.OptResSmallGrad,

Dew.Math.TOptStopReason.optResSmallJacobian, Dew.Math.TOptStopReason.OptResSmallStep,
Dew.Math.TOptStopReason.OptResMaxIter per the MKL stop code. Building MtxVec

Core (no MKL) raises an exception; this routine is not available under .NET.

The Trust Region (TR) algorithms are relatively new iterative algorithms for solving nonlinear optimization problems. They are widely used in power engineering, finance, applied mathematics, physics, computer science, economics, sociology, biology, medicine, mechanical engineering, chemistry, and other areas. TR methods have global convergence and local super convergence, which differenciates them from line search methods and Newton methods. TR methods have better convergence when compared with widely-used Newton-type methods.

The main idea behind TR algorithm is calculating a trial step and checking if the next values of x belong to the trust region. Calculation of the trial step is strongly associated with the approximation model.

For more on TR algorithm, check the following links:

• http://iridia.ulb.ac.be/~fvandenb/optimization/optimizationIntro.html
• http://en.wikipedia.org/wiki/Trust_region

Note:
This function is currently not supported under .NET. The objective function calls are threaded. The objective function should not be allocating any memory on the heap because calling threads are created outside of Delphi/BCB code.