Optimization Toolbox

Box Constraints

The box constrained problem is of the form

(3-7)

where l is a vector of lower bounds, and u is a vector of upper bounds. Some (or all) of the components of may be equal to and some (or all) of the components of may be equal to The method generates a sequence of strictly feasible points. Two techniques are used to maintain feasibility while achieving robust convergence behavior. First, a scaled modified Newton step replaces the unconstrained Newton step (to define the two-dimensional subspace ). Second, reflections are used to increase the step-size.

The scaled modified Newton step arises from examining the Kuhn-Tucker necessary conditions for Eq. 3-7

(3-8)

where

and the vector is defined below, for each

• If and then
• If and then
• If and then
• If and then

The nonlinear system Eq. 3-8 is not differentiable everywhere; nondifferentiability occurs when Hence we avoid such points by maintaining strict feasibility, i.e., restricting

The scaled modified Newton step for Eq. 3-8 is defined as the solution to the linear system

(3-9)

where

(3-10)

and

(3-11)

Here plays the role of the Jacobian of Each diagonal component of the diagonal matrix equals 0, -1 or 1. If all the components of l and u are finite, At a point where may not be differentiable. We define at such a point. Nondifferentiability of this type is not a cause for concern because, for such a component, it is not significant which value takes. Further will still be discontinuous at this point, but the function is continuous.

Second, reflections are used to increase the step-size. A (single) reflection step is defined as follows. Given a step that intersects a bound constraint, consider the first bound constraint crossed by p; assume it is the ith bound constraint (either the ith upper or ith lower bound). Then the reflection step except in the ith component where

Linear Equality Constraints

Nonlinear Least Squares