Optimization algorithms aim to minimize a cost function by generating successive iterates of increasing relevance. Central theoretical questions include: do the iterates converge? If so, do they converge to points of interest---ideally global minimizers? And at what rate? Answering these questions typically requires structural assumptions on the problem, and the classical convex framework provides partial answers.

However, many practical applications give rise to nonconvex problems, where algorithms can exhibit complex dynamics. Indeed, for structural reasons, nonconvex problems frequently admit nonisolated minimizers---that is, a continuum of minimizers---and lack convexity even locally. At a global scale, they feature saddle points and spurious local minimizers, which can slow convergence or trap iterates.

Yet, nonconvex methods are widely used and often perform strikingly well in practice. Their theoretical understanding remains limited compared to convex counterparts, though significant progress has been made in recent decades. This thesis contributes to this ongoing effort by developing new analytical tools for understanding certain classes of nonconvex problems. We focus on differentiable functions of continuous variables.

The first part of the thesis focuses on local convergence beyond the standard assumption of local strong convexity. We identify alternative favorable conditions and derive superlinear convergence to nonisolated minimizers for several algorithms, including adaptive regularized Newton with cubics and trust-region methods.

The second part investigates benign nonconvex landscapes---settings in which all second-order critical points are global minimizers. These landscapes are particularly well-behaved, as they enable saddle-point avoidance and convergence to global minimizers. We show that several problems arising in applications enjoy benign landscapes, sometimes after a nonconvex relaxation---that is, by increasing the dimension of the search space.