barrier function#Logarithmic barrier function

Consider the following constrained optimization problem:

:minimize {{math|f(x)}}

:subject to {{math|x ≤ b}}

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

:minimize {{math|f(x) + c(x)}},

:where {{math|c(x) {{=}} ∞}} if {{math|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 {{mvar|c}}, and therefore the objective function {{math|f(x) + c(x)}}, is discontinuous, preventing the use of calculus to solve it.

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

:minimize {{math|f(x) + μ g(x)}}

where {{math|μ > 0}} is a free parameter. This problem is not equivalent to the original, but as {{mvar|μ}} approaches zero, it becomes an ever-better approximation.{{cite book |first=Robert J. |last=Vanderbei |authorlink=Robert J. Vanderbei |title=Linear Programming: Foundations and Extensions |year=2001 |publisher=Kluwer |pages=277–279}}

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

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 $-\backslash log(b\_i-x\_i)$.

Minimize $\backslash mathbf\; c^Tx$ subject to $\backslash mathbf\; a\_i^T\; x\; \backslash le\; b\_i,\; i\; =\; 1,\backslash ldots,m$

Assume strictly feasible:

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

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

+\infty & \text{otherwise}

• Penalty method
• Augmented Lagrangian method

{{reflist}}

• [http://www.seas.ucla.edu/~vandenbe/ee236a/lectures/barrier.pdf Lecture 14: Barrier method] from Professor Lieven Vandenberghe of UCLA