Trust-Region Algorithm

Many of the methods used in Optimization Toolbox™ solvers are based on trust regions, a simple yet powerful concept in optimization.

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To understand the trust-region approach to optimization, consider the unconstrained minimization problem, minimize f(x), where the function takes vector arguments and returns scalars. Suppose that the current point is x in n-space and you want to improve by moving to a point with a lower function value. To do so, the algorithm approximates f with a simpler function q, which reasonably reflects the behavior of function f in a neighborhood N around the point x. This neighborhood is the trust region. The solver computes a trial step s by minimizing (or approximately minimizing) over N. The trust-region subproblem is

The solver updates the current point to x + s if f(x + s) < f(x); otherwise, the current point remains unchanged and the solver shrinks N (the trust region) and repeats the trial step computation.

The key questions in defining a specific trust-region approach to minimizing f(x) are how to choose and compute the approximation q (defined at the current point x), how to choose and modify the trust region N, and how accurately to solve the trust-region subproblem.

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In the standard trust-region method ([48]), the quadratic approximation q is defined by the first two terms of the Taylor approximation to F at x. The neighborhood N is usually spherical or ellipsoidal in shape. Mathematically, the trust-region subproblem is typically stated

$$\min \left\{\frac{1}{2}{s}^{T}Hs+s^{T}g\text{such that}\Vert Ds\Vert \le \Delta \right\},$$

(1)

where g is the gradient of f at the current point x, H is the Hessian matrix (the symmetric matrix of second derivatives), D is a diagonal scaling matrix, Δ is a positive scalar, and ∥ . ∥ is the 2-norm. To solve Equation 1, an algorithm (see [48]) can compute all eigenvalues of H and then apply a Newton process to the secular equation

Such an algorithm provides an accurate solution to Equation 1. However, this requires time proportional to several factorizations of H. Therefore, trust-region problems require a different approach. Several approximation and heuristic strategies, based on Equation 1, have been proposed in the literature ([42] and [50]). Optimization Toolbox solvers follow an approximation approach that restricts the trust-region subproblem to a two-dimensional subspace S ([39] and [42]). After the solver computes the subspace S, the work to solve Equation 1 is trivial because, in the subspace, the problem is only two-dimensional. The dominant work now shifts to the determination of the subspace.

The solver determines the two-dimensional subspace S with the aid of a preconditioned conjugate gradient method (described in the next section). The solver defines S as the linear space spanned by s₁ and s₂, where s₁ is in the direction of the gradient g, and s₂ is either an approximate Newton direction, that is, a solution to

or a direction of negative curvature,

The philosophy behind this choice of S is to force global convergence (via the steepest descent direction or negative curvature direction) and achieve fast local convergence (via the Newton step, when it exists).

The process of unconstrained minimization using the trust-region approach is now easy to specify:

1. Formulate the two-dimensional trust-region subproblem.

2. Solve Equation 1 to determine the trial step s.

3. If f(x + s) < f(x), then x = x + s.

4. Adjust Δ.

The solver repeats these four steps until convergence, adjusting he trust-region dimension Δ according to standard rules. In particular, the solver decreases the trust-region size if it does not accept the trial step, when f(x + s) ≥ f(x). See [46] and [49] for a discussion of this aspect.

Optimization Toolbox solvers treat important cases of f with specialized functions: nonlinear least-squares, quadratic functions, and linear least-squares. However, the underlying algorithmic ideas are the same as for the general case.

Preconditioned Conjugate Gradient Method

A popular way to solve large, symmetric, positive definite systems of linear equations Hp = –g is the method of Preconditioned Conjugate Gradients (PCG). This iterative approach requires the ability to calculate matrix-vector products of the form H·v where v is an arbitrary vector. The symmetric positive definite matrix M is a preconditioner for H. That is, M = C², where C^–1HC^–1 is a well-conditioned matrix or a matrix with clustered eigenvalues.

In a minimization context, you can assume that the Hessian matrix H is symmetric. However, H is guaranteed to be positive definite only in the neighborhood of a strong minimizer. Algorithm PCG exits when it encounters a direction of negative (or zero) curvature, that is, d^THd ≤ 0. The PCG output direction p is either a direction of negative curvature or an approximate solution to the Newton system Hp = –g. In either case, p helps to define the two-dimensional subspace used in the trust-region approach discussed in Trust-Region Methods for Nonlinear Minimization.

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