Powell asked whether the smoothness assumptions underlying classical superlinear convergence force a fixed power law between adjacent iterates of exact-line-search variable-metric methods. We answer this question negatively for BFGS: within the smooth strongly convex setting, the smallest possible adjacent-iterate Q-order is one, and this boundary is attained by a single nonterminating run. In every finite dimension at least two, and for any prescribed radius and Hessian tolerance, we construct an infinitely differentiable, globally strongly convex objective that equals the standard quadratic outside the corresponding ball and whose Hessian remains within the prescribed tolerance of the identity in operator norm. The objective has its unique minimizer at the origin and identity Hessian there. Exact-line-search BFGS, initialized with the identity matrix and started inside that ball, converges Q-superlinearly, yet no fixed power greater than one controls all sufficiently late adjacent errors.
We study the exact Euclidean ball-proximal point method for proper, closed, convex functions, where each iteration minimizes the objective over a ball centered at the current point. Retaining the objective gap in the decrease of squared distance yields sharper bounds on objective values, stationarity, and the symmetric...
We study convex optimization over compact convex sets for twice differentiable objective functions with positive definite Hessians, assuming access to the feasible set through a linear minimization oracle. Existing second-order Frank--Wolfe methods either provide only a local linear rate over general convex sets or ach...
A global iteration complexity bound is established for the smallest gradient norm among the first $k$ iterates for the smallest gradient norm among the first $k$ iterates when the initial sublevel set is bounded.
We combine CaCuTe with sequential subspace optimization (SESOP) for smooth convex minimization under Lipschitz continuity of the Hessian. The resulting method, CaCuSESOP, achieves the global rate $O(k^{-2})$ with a fixed number of Hessian--vector products per iteration, while removing key limitations of classical SESOP...
It is proved that if $\mathcal G_N(\mathcal A)$ denotes the bound on the squared gradient norm after $N$ iterations for a method $\mathcal A, it is proved that $\limsup_{N\to\infty}N \mathcal G_N(\mathcal A) \ge 1/2$ for any method $\mathcal A$.
Greedy box splitting reads gradient signs at the centre and retains one smaller box. Its summable travel budget makes an incorrect exclusion irreversible. We determine when this geometry is safe. Halving is minimax optimal in the sign-vector oracle model on a coordinatewise sign-consistent class that need not be convex...
A. Anikin, A. Gornov, T. Zarodnyuk et al.· 0 citations
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