Profile confidence limits all start with a goal SSE. This is a sum of squared errors (or sum of loss function) that an F test considers significantly different from the solution SSE at the given alpha level. If the loss function is specified to be a negative log-likelihood, then a Chi-square quantile is used instead of an F quantile. For each parameter’s upper confidence limit, the parameter value is increased until the SSE reaches the goal SSE. As the parameter value is moved up, all the other parameters are adjusted to be least squares estimates subject to the change in the profiled parameter. Conceptually, this is a compounded set of nested iterations. Internally there is a way to do this with one set of iterations developed by Johnston and DeLong. See SAS/STAT 9.1 vol. 3 pp. 1666-1667.
Diagram of Confidence Limits for Parameters shows the contour of the goal SSE or negative likelihood, with the least squares (or least loss) solution inside the shaded region:
Diagram of Confidence Limits for Parameters

Help created on 9/19/2017