Stochastic Trust Region Methods with Trust Region Radius Depending on Probabilistic Models

We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models. Especially, we propose a specific algorithm, termed STRME, in which the trust-region radius depends linearly on the latest model gradient. The complexity of STRME method in non-convex, convex and strongly convex settings has all been analyzed, which matches the existing algorithms based on probabilistic properties. In addition, several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods

A New Two-dimensional Model-based Subspace Method for Large-scale Unconstrained Derivative-free Optimization: 2D-MoSub

This paper proposes the method 2D-MoSub (2-dimensional model-based subspace method), which is a novel derivative-free optimization (DFO) method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems. 2D-MoSub combines 2-dimensional quadratic interpolation models and trust-region techniques to iteratively update the points and explore the 2-dimensional subspace. 2D-MoSub's framework includes initialization, constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. Experimental results demonstrate the effectiveness and efficiency of 2D-MoSub in solving a variety of optimization problems.Comment: 22 page

Symplectic Discretization Approach for Developing New Proximal Point Algorithms

Proximal point algorithms have found numerous applications in the field of convex optimization, and their accelerated forms have also been proposed. However, the most commonly used accelerated proximal point algorithm was first introduced in 1967, and recent studies on accelerating proximal point algorithms are relatively scarce. In this paper, we propose high-resolution ODEs for the proximal point operators for both closed proper convex functions and maximally monotone operators, and present a Lyapunov function framework to demonstrate that the trajectories of our high-resolution ODEs exhibit accelerated behavior. Subsequently, by symplectically discretizing our high-resolution ODEs, we obtain new proximal point algorithms known as symplectic proximal point algorithms. By decomposing the continuous-time Lyapunov function into its elementary components, we demonstrate that symplectic proximal point algorithms possess $O(1/k^2)$ convergence rates

Derivative-Free Optimization with Transformed Objective Functions (DFOTO) and the Algorithm Based on the Least Frobenius Norm Updating Quadratic Model

Derivative-free optimization problems are optimization problems where derivative information is unavailable. The least Frobenius norm updating quadratic interpolation model function is one of the essential under-determined model functions for model-based derivative-free trust-region methods. This article proposes derivative-free optimization with transformed objective functions and gives a trust-region method with the least Frobenius norm model. The model updating formula is based on Powell's formula. The method shares the same framework with those for problems without transformations, and its query scheme is given. We propose the definitions related to optimality-preserving transformations to understand the interpolation model in our method. We prove the existence of model optimality-preserving transformations beyond translation transformation. The necessary and sufficient condition for such transformations is given. The affine transformation with a positive multiplication coefficient is not model optimality-preserving. We also analyze the corresponding least Frobenius norm updating model and its interpolation error when the objective function is affinely transformed. Convergence property of a provable algorithmic framework containing our model is given. Numerical results of solving test problems and a real-world problem with the implementation NEWUOA-Trans show that our method can successfully solve most problems with objective optimality-preserving transformations, even though such transformations will change the optimality of the model function. To our best knowledge, this is the first work providing the model-based derivative-free algorithm and analysis for transformed problems with the function evaluation oracle (not the function-value comparison oracle). This article also proposes the "moving-target" optimization problem.Comment: 42 pages, Derivative-Free Optimization with Transformed Objective Functions (DFOTO) and the Algorithm Based on the Least Frobenius Norm Updating Quadratic Mode

Riemannian preconditioned algorithms for tensor completion via tensor ring decomposition

We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets -- movie ratings, hyperspectral images, and high-dimensional functions -- suggest that the proposed algorithms are more efficient and have better reconstruction ability than other candidates.Comment: 25 pages, 7 figures, 5 table

Low-rank optimization on Tucker tensor varieties

In the realm of tensor optimization, low-rank tensor decomposition, particularly Tucker decomposition, stands as a pivotal technique for reducing the number of parameters and for saving storage. We embark on an exploration of Tucker tensor varieties -- the set of tensors with bounded Tucker rank -- in which the geometry is notably more intricate than the well-explored geometry of matrix varieties. We give an explicit parametrization of the tangent cone of Tucker tensor varieties and leverage its geometry to develop provable gradient-related line-search methods for optimization on Tucker tensor varieties. The search directions are computed from approximate projections of antigradient onto the tangent cone, which circumvents the calculation of intractable metric projections. To the best of our knowledge, this is the first work concerning geometry and optimization on Tucker tensor varieties. In practice, low-rank tensor optimization suffers from the difficulty of choosing a reliable rank parameter. To this end, we incorporate the established geometry and propose a Tucker rank-adaptive method that is capable of identifying an appropriate rank during iterations while the convergence is also guaranteed. Numerical experiments on tensor completion with synthetic and real-world datasets reveal that the proposed methods are in favor of recovering performance over other state-of-the-art methods. Moreover, the rank-adaptive method performs the best across various rank parameter selections and is indeed able to find an appropriate rank.Comment: 46 pages, 14 figures, 1 tabl