Barrierparameter

Barrierparameter refers to the scalar that weights the logarithmic barrier term in barrier methods, a class of interior-point algorithms used for constrained optimization. It controls how strongly the method penalizes nearing the boundary of the feasible region. As the barrier parameter increases, the influence of the barrier term diminishes, allowing the iterates to approach the true optimum of the original problem while staying feasible in the interior.

In a typical setup, consider a constrained problem such as minimize f(x) subject to Ax ≤ b and x ≥ 0. The barrier formulation replaces the objective with a barrier-augmented one: minimize f(x) - (1/t) sum_i log(b_i − a_i^T x) or, for nonnegativity constraints, minimize f(x) - (1/t) sum_j log x_j, where t > 0 is the barrierparameter. Equivalently, one can view it as maximizing t f(x) plus a barrier term. The barrier term blows up near the boundary, keeping iterates strictly feasible (inside the interior).

Algorithmically, barriermethods traverse a central path: for each t, compute the minimizer x(t) of the barrier problem, then increase t (commonly t_{k+1} = μ t_k with μ > 1) and repeat until the duality gap or other convergence criteria meet a tolerance. The barrierparameter thus mediates the trade-off between rapid objective improvement and maintenance of interior feasibility. Different formulations use slightly different notations (t, μ, etc.), but the underlying idea is consistent across primal, dual, and primal-dual interior-point methods.

Barrierparameter is widely used in linear programming, convex optimization, and semidefinite programming, often via logarithmic barrier terms or log-det barriers, providing a smooth path to the optimum while respecting all constraints.