We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are only achieved by non-autonomous methods via open-loop damping (e.g., Nesterov’s algorithm), we show that our system is the first one featuring a closed-loop damping while exhibiting a rate arbitrarily close to the optimal one. … Read more
In this paper, we consider a general low-rank matrix optimization problem which is modeled by a general Schatten p-quasi-norm (${\rm 0<p<1}$) regularized matrix optimization. For this nonconvex nonsmooth and non-Lipschitz matrix optimization problem, based on the matrix p-thresholding operator, we first propose a fixed point continuation algorithm with extrapolation (FPCAe) for solving it. Secondly, we … Read more
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In this paper, we consider a partial sparse and partial group sparse optimization problem, where the loss function is a continuously differentiable function (possibly nonconvex), and the penalty term consists of two parts associated with sparsity and group sparsity. The first part is the $\ell_0$ norm of ${\bf x}$, the second part is the $\ell_{2,0}$ … Read more
Recently Grimmer [1] showed for smooth convex optimization by utilizing longer steps periodically, gradient descent’s state-of-the-art O(1/T) convergence guarantees can be improved by constant factors, conjecturing an accelerated rate strictly faster than O(1/T) could be possible. Here we prove such a big-O gain, establishing gradient descent’s first accelerated convergence rate in this setting. Namely, we … Read more
Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such … Read more
The limited memory steepest descent method (Fletcher, 2012) for unconstrained optimization problems stores a few past gradients to compute multiple stepsizes at once. We review this method and propose new variants. For strictly convex quadratic objective functions, we study the numerical behavior of different techniques to compute new stepsizes. In particular, we introduce a method … Read more
Derivative-free algorithms seek the minimum of a given function based only on function values queried at appropriate points. Although these methods are widely used in practice, their performance is known to worsen as the problem dimension increases. Recent advances in developing randomized derivative-free techniques have tackled this issue by working in low-dimensional subspaces that are … Read more
The main computational cost per iteration of adaptive cubic regularization methods for solving large-scale nonconvex problems is the computation of the step \(s_k\), which requires an approximate minimizer of the cubic model. We propose a new approach in which this minimizer is sought in a low dimensional subspace that, in contrast to classical approaches, is … Read more