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 :
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 :
- .
Read more about this topic: Interior Point Method
Famous quotes containing the words interior, point and/or method:
“The exterior must be joined to the interior to obtain anything from God, that is to say, we must kneel, pray with the lips, and so on, in order that proud man, who would not submit himself to God, may be now subject to the creature.”
—Blaise Pascal (1623–1662)
“The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)
“You that do search for every purling spring
Which from the ribs of old Parnassus flows,
And every flower, not sweet perhaps, which grows
Near thereabouts into your poesy wring;
You that do dictionary’s method bring
Into your rhymes, running in rattling rows;”
—Sir Philip Sidney (1554–1586)