barriermethod

Barriermethod, commonly referred to in optimization as the barrier method, is an approach for solving constrained problems by embedding a barrier term in the objective function to keep iterates inside the feasible region. The typical setting is to minimize f0(x) subject to x belonging to a feasible set D defined by inequality constraints, such as ai(x) ≤ 0. A standard barrier is φ(x) = -∑ log(-ai(x)) for interior-feasible problems or φ(x) = -∑ log xi for bound constraints. The barrier-augmented problem becomes minimize f0(x) + μ φ(x), where μ > 0 is the barrier parameter. For each fixed μ, the subproblem is solved (often with Newton-type methods) to obtain x(μ). As μ decreases toward zero, the solution moves toward the boundary of D but remains strictly interior; the limit, if it exists, is an optimal solution of the original constrained problem.

Barriermethod is a class of interior-point methods and is closely related to the central-path concept, which traces a path of interior points as μ varies. Variants include primal barrier methods, dual barrier methods, and logarithmic barrier methods, with implementations spanning linear programming, convex quadratic programming, and nonlinear programming. Convergence generally requires smoothness and appropriate convexity or structure; performance depends on problem size, sparsity, and how μ is updated.

Applications of barriermethods appear across operations research, engineering optimization, economics, and machine learning, where large-scale constrained problems are common. While modern interior-point techniques have evolved, barrier methods remain foundational both theoretically and in practical algorithms.