Superlinear and Quadratic Convergence of Riemannian Interior Point Methods

We extend the classical primal-dual interior point algorithms from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point (RIP) method, is for solving Riemannian constrained optimization problems. Under the standard assumptions in the Riemannian setting, we establish locally superlinear, quadratic convergence for the Newton version of RIP and locally linear, superlinear convergence for the quasi-Newton version. These are generalizations of the classical local convergence theory of primal-dual interior point algorithms for nonlinear programming proposed by El-Bakry et al. and Yamashita et al. in 1996.Comment: 34page

A New Extension of Chubanov's Method to Symmetric Cones

We propose a new variant of Chubanov's method for solving the feasibility problem over the symmetric cone by extending Roos's method (2018) of solving the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (i) equivalent to that of Roos's original method (2018) and superior to that of Louren\c{c}o et al.'s method (2019) when the symmetric cone is the nonnegative orthant, (ii) superior to that of Louren\c{c}o et al.'s method (2019) when the symmetric cone is a Cartesian product of second-order cones, (iii) equivalent to that of Louren\c{c}o et al.'s method (2019) when the symmetric cone is the simple positive semidefinite cone, and (iv) superior to that of Pena and Soheili's method (2017) for any simple symmetric cones under the feasibility assumption of the problem imposed in Pena and Soheili's method (2017). We also conduct numerical experiments that compare the performance of our method with existing methods by generating instances in three types: strongly (but ill-conditioned) feasible instances, weakly feasible instances, and infeasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.Comment: 44 pages; Department of Policy and Planning Sciences Discussion Paper Series No. 1378, University of Tsukub

Centering ADMM for the Semidefinite Relaxation of the QAP

We propose a new method for solving the semidefinite (SD) relaxation of the quadratic assignment problem (QAP), called Centering ADMM. Centering ADMM is an alternating direction method of multipliers (ADMM) combining the centering steps used in the interior-point method. The first stage of Centering ADMM updates the iterate so that it approaches the central path by incorporating a barrier function term into the objective function, as in the interior-point method. If the current iterate is sufficiently close to the central path with a sufficiently small value of the barrier parameter, the method switches to the standard version of ADMM. We show that Centering ADMM (not employing a dynamic update of the penalty parameter) has global convergence properties. To observe the effect of the centering steps, we conducted numerical experiments with SD relaxation problems of instances in QAPLIB. The results demonstrate that the centering steps are quite efficient for some classes of instances

Post-Processing with Projection and Rescaling Algorithms for Semidefinite Programming

We propose the algorithm that solves the symmetric cone programs (SCPs) by iteratively calling the projection and rescaling methods the algorithms for solving exceptional cases of SCP. Although our algorithm can solve SCPs by itself, we propose it intending to use it as a post-processing step for interior point methods since it can solve the problems more efficiently by using an approximate optimal (interior feasible) solution. We also conduct numerical experiments to see the numerical performance of the proposed algorithm when used as a post-processing step of the solvers implementing interior point methods, using several instances where the symmetric cone is given by a direct product of positive semidefinite cones. Numerical results show that our algorithm can obtain approximate optimal solutions more accurately than the solvers. When at least one of the primal and dual problems did not have an interior feasible solution, the performance of our algorithm was slightly reduced in terms of optimality. However, our algorithm stably returned more accurate solutions than the solvers when the primal and dual problems had interior feasible solutions.Comment: 78 page

A complexity bound of a predictor-corrector smoothing method using CHKS-functions for monotone LCP

We propose a new smoothing method using CHKS-functions for solving linear complementarity problems. While the algorithm in [6] uses a quite large neighborhood, our algorithm generates a sequence in a relatively narrow neighborhood and employs predictor and corrector steps at each iteration. A complexity bound for the method is also provided under the assumption that the problem is monotone and has a feasibleinterior point. As a result, the bound can be improved compared to the one in [6].Includes bibliographical references (p. 16-17

LP-based Tractable Subcones of the Semidefinite Plus Nonnegative Cone

Tractable Subcones and LP-based Algorithms for Testing Copositivit