arXiv subjects

Mitsuaki Obara

Publications and source records attributed to Mitsuaki Obara.

5 recordsLinked to original sources

arXiv2025

Local near-quadratic convergence of Riemannian interior point methods

We consider Riemannian optimization problems with inequality and equality constraints and analyze a class of Riemannian interior point methods for solving them. The algorithm of interest consists of outer and inner iterations. We show that, under standard assumptions, the algorithm achieves local superlinear convergence by solving a linear system at each outer iteration, removing the need for further computations in the inner iterations. We also provide a specific update for the barrier parameter that achieves local near-quadratic convergence of the algorithm. We apply our results to the method proposed by Obara, Okuno, and Takeda (2026) and show its local superlinear and near-quadratic convergence with an analysis of the second-order stationarity. To our knowledge, this is the first algorithm for constrained optimization on Riemannian manifolds that achieves both local convergence and global convergence to a second-order stationary point. Numerical results support the theoretical analyses of the proposed methods.

math.OC↗

arXiv2025

A primal-dual interior point trust region method for second-order stationary points of Riemannian inequality-constrained optimization problems

We consider Riemannian inequality-constrained optimization problems. Such problems inherit the benefits of Riemannian approach developed in the unconstrained setting and naturally arise from applications in control, machine learning, and other fields. We propose a Riemannian primal-dual interior point trust region method (RIPTRM) for solving them. We prove its global convergence to an approximate Karush-Kuhn-Tucker point and a weak second-order stationary point. Under the strict complementarity condition, this result reduces to global convergence to a second-order stationary point. To the best of our knowledge, this is the first algorithm that incorporates the trust region strategy for constrained optimization on Riemannian manifolds, and has the second-order convergence property for optimization problems on Riemannian manifolds with nonlinear inequality constraints. We conduct numerical experiments in which we introduce a truncated conjugate gradient method and an eigenvalue-based subsolver for RIPTRM to approximately and exactly solve the trust region subproblems, respectively. Empirical results show that RIPTRMs consistently find solutions with high accuracy. Additionally, we observe that RIPTRM with the exact search direction shows promising performance in an instance where the Hessian of the Lagrangian has a large negative eigenvalue.

math.OC↗

arXiv2023

Stable Linear System Identification with Prior Knowledge by Riemannian Sequential Quadratic Optimization

We consider an identification method for a linear continuous time-invariant autonomous system from noisy state observations. In particular, we focus on the identification to satisfy the asymptotic stability of the system with some prior knowledge. To this end, we propose to model this identification problem as a Riemannian nonlinear optimization (RNLO) problem, where the stability is ensured through a certain Riemannian manifold and the prior knowledge is expressed as nonlinear constraints defined on this manifold. To solve this RNLO, we apply the Riemannian sequential quadratic optimization (RSQO) that was proposed by Obara, Okuno, and Takeda (2022) most recently. RSQO performs quite well with theoretical guarantee to find a point satisfying the Karush-Kuhn-Tucker conditions of RNLO. In this paper, we demonstrate that the identification problem can be indeed solved by RSQO more effectively than competing algorithms.

math.OC↗

arXiv2022

Optimal design of lottery with cumulative prospect theory

Lotteries are a prevalent form of gambling between a seller and buyers. Designing a lottery requires a model of how buyers make decisions when confronted with uncertain outcomes. Cumulative prospect theory (CPT) is a descriptive model that captures people's propensity to overestimate extreme events and their different attitudes toward gains and losses. In this study, we design a lottery that maximizes the seller's profit when the buyers' decision-making adheres to the CPT framework. The main difficulty is the nonconvexity of the CPT framework, which we overcome by reformulating the problem as a three-level optimization problem and characterizing its optimal solution. Based on the analysis, we propose a linear-time algorithm that computes the optimal lottery. Furthermore, we present an efficient algorithm applicable to a broader setting with a ticket price constraint. This is the first study to employ the CPT framework in designing an optimal lottery with more than two outcomes.

cs.GT↗

arXiv2021

Sequential Quadratic Optimization for Nonlinear Optimization Problems on Riemannian Manifolds

We consider optimization problems on Riemannian manifolds with equality and inequality constraints, which we call Riemannian nonlinear optimization (RNLO) problems. Although they have numerous applications, the existing studies on them are limited especially in terms of algorithms. In this paper, we propose Riemannian sequential quadratic optimization (RSQO) that uses a line-search technique with an ell_1 penalty function as an extension of the standard SQO algorithm for constrained nonlinear optimization problems in Euclidean spaces to Riemannian manifolds. We prove its global convergence to a Karush-Kuhn-Tucker point of the RNLO problem by means of parallel transport and the exponential mapping. Furthermore, we establish its local quadratic convergence by analyzing the relationship between sequences generated by RSQO and the Riemannian Newton method. Ours is the first algorithm that has both global and local convergence properties for constrained nonlinear optimization on Riemannian manifolds. Empirical results show that RSQO finds solutions more stably and with higher accuracy compared with the existing Riemannian penalty and augmented Lagrangian methods.

math.OC↗