This work is concerned with the convergence rate analysis of the Dou- glas–Rachford splitting (DRS) method for finding a zero of the sum of two maximally monotone operators. We obtain an exact rate of convergence for the DRS algorithm and demonstrate its sharpness in the setting of convex feasibility problems. Further- more, we investigate the … Read more
This paper presents a novel hybrid algorithm for minimizing the sum of a continuously differentiable loss function and a nonsmooth, possibly nonconvex, sparsity‑promoting regularizer. The proposed method adaptively switches between solving a reweighted \(\ell_1\)-regularized subproblem and performing an inexact subspace Newton step. The reweighted \(\ell_1\)-subproblem admits an efficient closed-form solution via the soft-thresholding operator, thereby … Read more
In Online Convex Optimization (OCO), when the stochastic gradient has a finite variance, many algorithms provably work and guarantee a sublinear regret. However, limited results are known if the gradient estimate has a heavy tail, i.e., the stochastic gradient only admits a finite \(\mathsf{p}\)-th central moment for some \(\mathsf{p}\in\left(1,2\right]\). Motivated by it, this work examines … Read more
In this paper, the second-order directional derivative of the optimal value function and the optimal solution function are obtained for a strongly stable parametric problem with non-unique Lagrange multipliers. Some properties of the Lagrange multipliers are proved. It is justified that the second-order directional derivative of the optimal solution function for the parametric problem can … Read more
This paper introduces a projection and contraction-type algorithm that features an extrapolation from the past, reducing the two values of the cost operator inherent in the original projection and contraction algorithm to a single value at the current iteration. Strong convergence results of the proposed algorithm are proved in Hilbert spaces. Experimental results on testing … Read more
In this work, we investigate the application of Davis–Yin splitting (DYS) to convex optimization problems and demonstrate that swapping the roles of the two nonsmooth convex functions can result in a faster convergence rate. Such a swap typically yields a different sequence of iterates, but its impact on convergence behavior has been largely understudied or … Read more
In this paper, we show that minimax rational approximations can be enhanced by introducing a controlling parameter on the denominator of the rational function. This is implemented by adding a small set of linear constraints to the underlying optimization problem. The modification integrates naturally into approximation models formulated as linear programming problems. We demonstrate our … Read more
We study a bilevel optimization framework for hyperparameter learning in variational models, with a focus on sparse regression and classification tasks. In particular, we consider a weighted elastic-net regularizer, where feature-wise regularization parameters are learned through a bilevel formulation. A key novelty of our approach is the use of a Forward-Backward (FB) reformulation of the … Read more
This paper is devoted to the Lipschitz analysis of the solution sets and optimal values for a class of parametric optimization problems involving a polyhedral feasible set mapping and a quadratic objective function with arametric linear part. Recall that a multifunction is said to be polyhedral if its graph is the union of finitely many polyhedral … Read more
In this paper, we study sparse inverse covariance matrix estimation incorporating partly smooth nonconvex regularizers. To solve the resulting regularized log-determinant problem, we develop DIIR-QUIC—a novel Damped Inexact Iteratively Reweighted algorithm based on QUadratic approximate Inverse Covariance (QUIC) method. Our approach generalizes the classic iteratively reweighted \(\ell_1\) scheme through damped fixed-point updates. A key novelty … Read more