Barrier function

Motivation

Consider the following constrained optimization problem:

:minimize f(x) :subject to x ≤ b

where b is some constant. If one wishes to remove the inequality constraint, the problem can be reformulated as

:minimize f(x) + c(x), :where if x b, and zero otherwise.

This problem is equivalent to the first. It gets rid of the inequality, but introduces the issue that the penalty function c, and therefore the objective function f(x) + c(x), is discontinuous, preventing the use of calculus to solve it.

A barrier function, now, is a continuous approximation g to c that tends to infinity as x approaches b from above. Using such a function, a new optimization problem is formulated, viz.

:minimize f(x) + μ g(x)

where μ 0 is a free parameter. This problem is not equivalent to the original, but as μ approaches zero, it becomes an ever-better approximation.

Logarithmic barrier function

For logarithmic barrier functions, g(x,b) is defined as -\log(b-x) when x and \infty otherwise (in one dimension; see below for a definition in higher dimensions). This essentially relies on the fact that \log t tends to negative infinity as t tends to 0.

This introduces a gradient to the function being optimized which favors less extreme values of x (in this case values lower than b), while having relatively low impact on the function away from these extremes.

Logarithmic barrier functions may be favored over less computationally expensive inverse barrier functions depending on the function being optimized.

Higher dimensions

Extending to higher dimensions is simple, provided each dimension is independent. For each variable x_i which should be limited to be strictly lower than b_i, add -\log(b_i-x_i).

Formal definition

Minimize \mathbf c^Tx subject to \mathbf a_i^T x \le b_i, i = 1,\ldots,m

Assume strictly feasible: {\mathbf x|A x

Define logarithmic barrier g(x) = \begin{cases}

\sum_{i=1}^m -\log(b_i - a_i^Tx) & \text{for } Ax +\infty & \text{otherwise} \end{cases}

References

References

1. Nesterov, Yurii. (2018). "Lectures on Convex Optimization". Springer.
2. Nocedal, Jorge. (2006). "Numerical Optimization". Springer.
3. Vanderbei, Robert J.. (2001). "Linear Programming: Foundations and Extensions". Kluwer.