What is trust region reflective algorithm?

‘trust-region-reflective’ requires you to provide a gradient, and allows only bounds or linear equality constraints, but not both. Within these limitations, the algorithm handles both large sparse problems and small dense problems efficiently. It is a large-scale algorithm; see Large-Scale vs. Medium-Scale Algorithms.

What is the importance of Cauchy point in trust region methods?

Cauchy point is the minimizer of the sub-problem along the steepest descent direction within the trust-region as shown in the figure below.

What is a Cauchy point?

The Cauchy point is the point lying on the gradient which minimises the quadratic model subject to the step being within the trust region. By iteratively finding the Cauchy point the local minimum can be found. The convergence of the technique is inefficient, being similar to that of the steepest descent algorithm.

What is SQP Matlab?

The sqp algorithm combines the objective and constraint functions into a merit function. The algorithm attempts to minimize the merit function subject to relaxed constraints. This modified problem can lead to a feasible solution.

What is trust region radius?

The trust region is defined as the ball about xk such that ‖ x − x k ‖ 2 = ‖ s ‖ ≤ δ , where δ is called the trust region radius (Trust region methods can handle the case Hk = ∇2f(xk), even if the Hessian is not positive definite, but here we assume that the model Hessian Hk is symmetric and positive definite.).

How do you approximate Hessian?

One method for approximating the Hessian matrix is to use difference approximations. Difference approximation methods exploit the fact that each column of the Hessian can be approximated by taking the difference between two instances of the gradient vector evaluated at two nearby points.

What is the key feature of the trust-region-dogleg algorithm?

The key feature of the trust-region-dogleg algorithm is the use of the Powell dogleg procedure for computing the step d, which minimizes Equation 3. For a detailed description, see Powell [34].

How to solve trust-region sub-problem?

(Note that hessian or approximate hessian will be evaluated in dogleg method) The most widely used method for solving a trust-region sub-problem is by using the idea of conjugated gradient (CG) method for minimizing a quadratic function since CG guarantees convergence within a finite number of iterations for a quadratic programming.

What is TRM (trust region method)?

Trust-region method (TRM) is one of the most important numerical optimization methods in solving nonlinear programming (NLP) problems. It works in a way that first define a region around the current best solution, in which a certain model (usually a quadratic model) can to some extent approximate the original objective function.

What is the trust-region of a function?

In most cases, the trust-region is defined as a spherical area of radius in which the trust-region subproblem lies. If we are using the quadratic model to approximate the original objective function, then our optimization problem is essentially reduced to solving a sequence of trust-region subporblems