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The Unconstrained Nonlinear Programming Solver |
Trust-Region Algorithm |
One of the techniques of NLPU (TECH=CGTR) implements a trust-region algorithm. These types of algorithms determine the search direction by solving the following optimization problem:
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Note that the objective function of the preceding problem is a quadratic approximation of the objective function of the original problem. Also the constraint defines a region around which the quadratic function is trusted to be an accurate approximation of
. The shape of that region can be thought of as a sphere with radius
. The size of the trust-region radius
is selected based on how well the quadratic function
approximates the original objective function
. The ratio
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defines a measure of how well approximates
within the trust region. If
is close to
, then
is a good approximation of
and the radius
is allowed to increase. By letting
increase, the algorithm can take larger steps in subsequent iterations, permitting the algorithm to converge at a faster rate to the optimal solution of the original problem.
On the other hand, if is not close to
, the quadratic model is considered to be a poor approximation to
for the given trust-region radius. Because the direction
is generated with respect to the quadratic model, it is important that a sufficient level of accuracy is maintained between model and objective. In this case, the accuracy of the quadratic model is improved by reducing the trust-region radius
. A more in-depth discussion about trust-region methods can be found in Nocedal and Wright (1999).
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