Interior Point Method - Primal-dual Interior Point Method For Nonlinear Optimization

Primal-dual Interior Point Method For Nonlinear Optimization

The primal-dual method's idea is easy to demonstrate for constrained nonlinear optimization. For simplicity consider the all-inequality version of a nonlinear optimization problem:

minimize subject to .

The logarithmic barrier function associated with (1) is

Here is a small positive scalar, sometimes called the "barrier parameter". As converges to zero the minimum of should converge to a solution of (1).

The barrier function gradient is

where is the gradient of the original function and is the gradient of .

In addition to the original ("primal") variable we introduce a Lagrange multiplier inspired dual variable (sometimes called "slack variable")

(4) is sometimes called the "perturbed complementarity" condition, for its resemblance to "complementary slackness" in KKT conditions.

We try to find those which turn gradient of barrier function to zero.

Applying (4) to (3) we get equation for gradient:

where the matrix is the constraint Jacobian.

The intuition behind (5) is that the gradient of should lie in the subspace spanned by the constraints' gradients. The "perturbed complementarity" with small (4) can be understood as the condition that the solution should either lie near the boundary or that the projection of the gradient on the constraint component normal should be almost zero.

Applying Newton's method to (4) and (5) we get an equation for update :

\begin{pmatrix} W & -A^T \\ \Lambda A & C \end{pmatrix}\begin{pmatrix} p_x \\ p_\lambda \end{pmatrix}=\begin{pmatrix} -g + A^T \lambda \\ \mu 1 - C \lambda \end{pmatrix}

where is the Hessian matrix of and is a diagonal matrix of .

Because of (1), (4) the condition

should be enforced at each step. This can be done by choosing appropriate :

.

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