Barrier function

In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region of an optimization problem. Such functions are used to replace inequality constraints by a penalizing term in the objective function that is easier to handle.

The two most common types of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated by their connection with primal-dual interior point methods.

Consider the following constrained optimization problem:

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

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.

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.

For logarithmic barrier functions, ${\displaystyle g(x,b)}$ is defined as ${\displaystyle -\log(b-x)}$ when ${\displaystyle x<b}$ and ${\displaystyle \infty }$ otherwise (in 1 dimension. See below for a definition in higher dimensions). This essentially relies on the fact that ${\displaystyle \log(t)}$ tends to negative infinity as ${\displaystyle t}$ tends to 0.

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