The alternating-direction-method-of-multipliers-based (ADMM-based) interior point method, or ABIP method, is a hybrid algorithm that effectively combines interior point method (IPM) and first-order methods to achieve a performance boost in large-scale linear optimization. Different from traditional IPM that relies on computationally intensive Newton steps, the ABIP method applies ADMM to approximately solve the barrier penalized problem. However, similar to other first-order methods, this technique remains sensitive to condition number and inverse precision. In this paper, we provide an enhanced ABIP method with multiple improvements. First, we develop an ABIP method to solve the general linear conic optimization and establish the associated iteration complexity. Second, inspired by some existing methods, we develop different implementation strategies for the ABIP method, which substantially improve its performance in linear optimization. Finally, we conduct extensive numerical experiments in both synthetic and real-world data sets to demonstrate the empirical advantage of our developments. In particular, the enhanced ABIP method achieves a 5.8× reduction in the geometric mean of run time on 105 selected linear optimization instances from Netlib, and it exhibits advantages in certain structured problems, such as support vector machine and PageRank. However, the enhanced ABIP method still falls behind commercial solvers in many benchmarks, especially when high accuracy is desired. We posit that it can serve as a complementary tool alongside well-established solvers. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: This research was supported by the National Natural Science Foundation of China [Grants 72394360, 72394364, 72394365, 72225009, 72171141, and 72150001] and by the Program for Innovative Research Team of Shanghai University of Finance and Economics. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0017 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0017 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Optimization problems with access to only zeroth-order information of the objective function on Riemannian manifolds arise in various applications, spanning from statistical learning to robot learning. While various zeroth-order algorithms have been proposed in Euclidean space, they are not inherently designed to handle the challenging constraints imposed by Riemannian manifolds. The proper adaptation of zeroth-order techniques to Riemannian manifolds remained unknown until the pioneering work of (Li et al., 2023a). However, zeroth-order algorithms are widely observed to converge slowly and be unstable in practice. To alleviate these issues, we propose a Riemannian accelerated zeroth-order algorithm with improved robustness. Regarding efficiency, our accelerated algorithm has the function query complexity of $\mathcal{O}(\epsilon^{-7/4}d)$ for finding an $\epsilon$-approximate first-order stationary point. By introducing a small perturbation, it exhibits a function query complexity of $\tilde{\mathcal{O}}(\epsilon^{-7/4}d)$ for seeking a second-order stationary point with a high probability, matching state-of-the-art result in Euclidean space. Moreover, we further establish the almost sure convergence in the asymptotic sense through the Stable Manifold Theorem. Regarding robustness, our algorithm requires larger smoothing parameters in the order of $\tilde{\mathcal{O}}(\epsilon^{7/8}d^{-1/2})$, improving the existing result by a factor of $\tilde{\mathcal{O}}(\epsilon^{3/4})$.

In this paper, we consider an unconstrained optimization model where the objective is a sum of a large number of possibly nonconvex functions, though overall the objective is assumed to be smooth and convex. Our bid to solving such model uses the framework of cubic regularization of Newton's method. As well known, the crux in cubic regularization is its utilization of the Hessian information, which may be computationally expensive for large-scale problems. To tackle this, we resort to approximating the Hessian matrix via sub-sampling. In particular, we propose to compute an approximated Hessian matrix by either uniformly or non-uniformly sub-sampling the components of the objective. Based upon such sampling strategy, we develop accelerated adaptive cubic regularization approaches and provide theoretical guarantees on global iteration complexity of $\O(\epsilon^{-1/3})$ with high probability, which matches that of the original accelerated cubic regularization methods Jiang et al. (2020) using the full Hessian information. Interestingly, we also show that in the worst case scenario our algorithm still achieves an $O(\epsilon^{-5/6}\log(\epsilon^{-1}))$ iteration complexity bound. The proof techniques are new to our knowledge and can be of independent interets. Experimental results on the regularized logistic regression problems demonstrate a clear effect of acceleration on several real data sets.

This paper is concerned with finding an optimal algorithm for minimizing a composite convex objective function. The basic setting is that the objective is the sum of two convex functions: the first function is smooth with up to the d-th order derivative information available, and the second function is possibly non-smooth, but its proximal tensor mappings can be computed approximately in an efficient manner. The problem is to find – in that setting – the best possible (optimal) iteration complexity for convex optimization. Along that line, for the smooth case (without the second non-smooth part in the objective), Nesterov (1983) proposed an optimal algorithm for the first-order methods (d=1) with iteration complexity O( 1 / k^2 ). A high-order tensor algorithm with iteration complexity of O( 1 / k^{d+1} ) was proposed by Baes (2009) and Nesterov (2018). In this paper, we propose a new high-order tensor algorithm for the general composite case, with the iteration complexity of O( 1 / k^{(3d+1)/2} ), which matches the lower bound for the d-th order methods as established in Nesterov (2018) and Shamir et al. (2018), and hence is optimal. Our approach is based on the Accelerated Hybrid Proximal Extragradient (A-HPE) framework proposed in Monteiro and Svaiter (2013), where a bisection procedure is installed for each A-HPE iteration. At each bisection step a proximal tensor subproblem is approximately solved, and the total number of bisection steps per A-HPE iteration is bounded by a logarithmic factor in the precision required.

In this merged paper, we consider the problem of minimizing a convex function with Lipschitz-continuous $p$-th order derivatives. Given an oracle which when queried at a point returns the first $p$-derivatives of the function at that point we provide some methods which compute an $\e$ approximate minimizer in $O\left(\e^{-\frac{2}{3p+1}} \right)$ iterations. These methods match known lower bounds up to polylogarithmic factors for constant $p$.