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We are using the Matlab optimization toolbox function fmincon() to solve a constrained minimization with only equality constraints. We wish to find out which particular constrained optimization method the solver uses to solve our problem.

We read from the documentation (http://www.mathworks.com/help/optim/ug/constrained-nonlinear-optimization-algorithms.html) that by default the solver uses the 'interior-point' option. Since we don't have inequality constraints, my understanding is that the interior-point feature of the algorithm is irrelevant. Reading further under the section 'fmincon Interior Point Algorithm' in the above documentation site, and stripping away the irrelevant details (e.g. the concepts barrier function and slack variables play no role here), it appears that the solver is using ideas from both of the penalty method and the Lagrange multiplier method, combined with a quasi-Newton method (BFGS to be specific).

Note 1: The documentation mentions both 'KKT' and 'merit function', which suggests that the algorithm uses idea from both the penalty method and the Lagrange multiplier method. And this is the beginning of our confusion.]

Note 2: We provide the solver with only gradients, but not Hessians, for the objective and constraint functions. So we believe that the solver is ultimately using BFGS. (BFGS is indeed mentioned in the documentation.)

In the nonlinear optimization literature, there is a well-known method, called the augmented Lagrangian method that cleverly combines the idea of the penalty method and the Lagrange multiplier method. Does fmincon() implements that? (The term 'augmented Lagrangian' is nowhere to be seen in the documentation.)

Thanks in advance for any advice.

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