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.^[1] 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.

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 re-formulated as

minimize f(x) + c(x),

where c(x) = ∞ 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.^[2]

Logarithmic barrier function

For logarithmic barrier functions, is defined as when and otherwise (in 1 dimension. See below for a definition in higher dimensions). This essentially relies on the fact that tends to negative infinity as tends to 0.

This introduces a gradient to the function being optimized which favors less extreme values of (in this case values lower than ), 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 which should be limited to be strictly lower than , add .

Formal definition

Minimize subject to

Assume strictly feasible:

Define logarithmic barrier

See also

• Penalty method

References

External links

• Lecture 14: Barrier method from Professor Lieven Vandenberghe of UCLA

Wikimedia Commons has media related to Newton Method.

Optimization: Algorithms, methods, and heuristics
Unconstrained nonlinear: Methods calling … … functions Golden section search Interpolation methods Line search Nelder–Mead method Successive parabolic interpolation … and gradients Convergence Trust region Wolfe conditions Quasi–Newton BFGS and L-BFGS DFP Symmetric rank-one (SR1) Other methods Gauss–Newton Gradient Levenberg–Marquardt Conjugate gradient Truncated Newton … and Hessians Newton's method
Constrained nonlinear General Barrier methods Penalty methods Differentiable Augmented Lagrangian methods Sequential quadratic programming Successive linear programming
Convex optimization Convex minimization Cutting-plane method Reduced gradient (Frank–Wolfe) Subgradient method Linear and quadratic Interior point Affine scaling Ellipsoid algorithm of Khachiyan Projective algorithm of Karmarkar Basis-Exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm of Lemke
Combinatorial Paradigms Approximation algorithm Dynamic programming Greedy algorithm Integer programming Branch & bound or cut Graph algorithms Minimum spanning tree Bellman–Ford Borůvka Dijkstra Floyd–Warshall Johnson Kruskal Network flows Dinic Edmonds–Karp Ford–Fulkerson Push–relabel maximum flow
Metaheuristics Evolutionary algorithm Hill climbing Local search Simulated annealing Tabu search
Categories Algorithms and methods Heuristics Software