This short note considers and resolves the apparent contradiction between known worst-case complexity results for first and second-order methods for solving unconstrained smooth nonconvex optimization problems and a recent note by Jarre (2011) implying a very large lower bound on the number of iterations required to reach the solution’s neighbourhood for a specific problem with … Read more
The complexity of finding epsilon-approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(epsilon^{-2}) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order shorts-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, … Read more
This paper examines worst-case evaluation bounds for finding weak minimizers in unconstrained optimization. For the cubic regularization algorithm, Nesterov and Polyak (2006) and Cartis, Gould and Toint (2010) show that at most O(epsilon^{-3}) iterations may have to be performed for finding an iterate which is within epsilon of satisfying second-order optimality conditions. We first show … Read more