Complexity analysis for a third-order algorithm to reach second-order stationarity - Silva, David Ricardo Barreto Lima (2024)
We develop a third-order interior-point-trust-region algorithm for nonconvex and non-negative constrained optimization with convergence to a second-order stationary point. Usually, a p-th order derivative p 3 is used only to improve the complexity bounds for finding a first-order stationary point or a p-th order stationary point to within a tolerance. Namely, when only the second-order derivative is considered, the version of our algorithm is known to achieve a second-order stationary point within tolerance > 0 in at most O(^(3)) iterations, while we show that using the third-order derivative, this complexity is improved to O(^(2)). The price to pay for achieving this result is that at each iteration of the algorithm, we solve a cubic ball constrained subproblem, which is considerably harder than its quadratic counterpart.