Optimization Trajectory
Constrained optimization problems are solved efficiently by algorithms that traverse the feasible region through its interior rather than along its boundary. Modern implementations of interior-point methods find wide application in optimal control, sensor network localization, and real-time model predictive control. By avoiding the combinatorial complexity of active-set methods, these algorithms achieve polynomial-time complexity for a broad class of convex problems.
Mathematical Formulation
The algorithm proceeds by adding a barrier function to the objective, which becomes arbitrarily large as the boundary of the feasible set is approached. A logarithmic barrier is typically used, with a positive scaling parameter that is sequentially decreased toward zero as the iteration progresses. At each step, Newton method is applied to solve the resulting system of nonlinear equations, which yields a search direction that points toward the optimal solution.
The step length along this direction is restricted to ensure that the next iterate remains strictly inside the feasible region. This sequential reduction of the barrier parameter generates a sequence of points known as the central path, which eventually terminates at the true constrained minimum.
Numerical Stability
In high-accuracy sensor calibration systems, the numerical precision of the optimization solver determines the limit of the achievable calibration quality. The linear systems solved at each iteration of the method become increasingly ill-conditioned as the barrier parameter approaches zero. Solvers must employ specialized factorization techniques to maintain accuracy in these final steps.
Algorithmic Limitation
The computational cost is dominated by the factorization of large, sparse matrices at each iteration. This constraint makes the method less suitable for extremely high-frequency control loops on resource-constrained embedded hardware. In such hardware environments, alternative solvers with lower per-iteration costs are often preferred.
