Convergence of trust-region algorithms in metric spaces

Abstract: Trust-region algorithms can be applied to very abstract optimization problems because they do not require a specific direction of descent or gradient. This has lead to recent interest in them, in particular in the area of integer optimal control problems, where the infinite-dimensional problem formulations do not assume vector space structure. We analyze a trust-region algorithm in the abstract setting of a metric space, a setting in which integer optimal control problems with total variation regularization can be formulated. Our analysis avoids a reset of the trust-region radius upon acceptance of the iterates when proving convergence to stationary points. This reset has been present in previous analyses of trust-region algorithms for integer optimal control problems. Our computational benchmark shows that the runtime can be considerably improved when avoiding this reset, which is now theoretically justified.