arXiv subjects

Makoto Yamashita

Publications and source records attributed to Makoto Yamashita.

At least 37 records · Page 2Linked to original sources

arXiv2024

Dual Spectral Projected Gradient Method for Generalized Log-det Semidefinite Programming

Log-det semidefinite programming (SDP) problems are optimization problems that often arise from Gaussian graphic models. A log-det SDP problem with an l1-norm term has been examined in many methods, and the dual spectral projected gradient (DSPG) method by Nakagaki et al.~in 2020 is designed to efficiently solve the dual problem of the log-det SDP by combining a non-monotone line-search projected gradient method with the step adjustment for positive definiteness. This paper extends the DSPG method for solving a generalized log-det SDP problem involving additional terms to cover more structures in Gaussian graphical models in a unified style. We establish the convergence of the proposed method to the optimal value. We conduct numerical experiments to illustrate the efficiency of the proposed method.

math.OC↗

arXiv2024

A new dual spectral projected gradient method for log-determinant semidefinite programming with hidden clustering structures

In this paper, we propose a new efficient method for a sparse Gaussian graphical model with hidden clustering structures by extending a dual spectral projected gradient (DSPG) method proposed by Nakagaki et al.~(2020). We establish the global convergence of the proposed method to an optimal solution, and we show that the projection onto the feasible region can be solved with a low computational complexity by the use of the pool-adjacent-violators algorithm. Numerical experiments on synthesis data and real data demonstrate the efficiency of the proposed method. The proposed method takes 0.91 seconds to achieve a similar solution to the direct application of the DSPG method which takes 4361 seconds.

math.OC↗

arXiv2024

T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization

We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into the standard SDP relaxation with block-diagonal structures. The convergence of the T-SDP relaxation to the optimal value of a given constrained POP is established under moderate assumptions as the relaxation level increases. Additionally, the feasibility and optimality of the T-SDP relaxation are discussed. Numerical results illustrate that the proposed T-SDP relaxation enhances numerical efficiency.

math.OC↗

arXiv2023

Exact Matrix Completion via High-Rank Matrices in Sum-of-Squares Relaxations

We study exact matrix completion from partially available data with hidden connectivity patterns. Exact matrix completion was shown to be possible recently by Cosse and Demanet in 2021 with Lasserre's relaxation using the trace of the variable matrix as the objective function with given data structured in a chain format. In this study, we introduce a structure for the objective function so that the resulting sum-of-squares (SOS) relaxation, the dual of Lasserre's SDP relaxation, produces a rank-($N$-1) solution, where $N$ denotes the size of variable matrix in the SOS relaxation. Specifically, the arrowhead structure is employed for the coefficient matrix of the objective function. We show that a matrix can be exactly completed through the SOS relaxation when the connectivity of given data is not explicitly displayed or follows a chain format. The theoretical exactness is proved using the rank of the Gram matrix for the SOS relaxation. We also present numerical algorithms designed to find the coefficient matrix in the SOS relaxation. Numerical experiments illustrate the validity of the proposed algorithm.

math.OC↗

arXiv2023

An infeasible interior-point arc-search method with Nesterov's restarting strategy for linear programming problems

An arc-search interior-point method is a type of interior-point methods that approximates the central path by an ellipsoidal arc, and it can often reduce the number of iterations. In this work, to further reduce the number of iterations and computation time for solving linear programming problems, we propose two arc-search interior-point methods using Nesterov's restarting strategy that is well-known method to accelerate the gradient method with a momentum term. The first one generates a sequence of iterations in the neighborhood, and we prove that the convergence of the generated sequence to an optimal solution and the computation complexity is polynomial time. The second one incorporates the concept of the Mehrotra-type interior-point method to improve numerical performance. The numerical experiments demonstrate that the second one reduced the number of iterations and computational time. In particular, the average number of iterations was reduced compared to existing interior-point methods due to the momentum term.

math.OC↗

arXiv2023

Deformation of algebras associated with group cocycles

We study deformation of algebras with coaction symmetry of reduced algebra of discrete groups, where the deformation parameter is given continuous family of group $2$-cocycles. When the group satisfies the Baum-Connes conjecture with coefficients, we obtain isomorphism of K-groups of the deformed algebras. This extends both the $θ$-deformation of Rieffel on $\mathbb T^n$-actions, and a recent result of Echterhoff, Lück, Phillips, and Walters on the K-groups on the twisted group algebras.

math.OA↗

arXiv2023

Crossed Product Equivalence of Quantum Automorphism Groups

We compare the algebras of the quantum automorphism group of finite-dimensional C$^\ast$-algebra $B$, which includes the quantum permutation group $S_N^+$, where $N = \dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $Γ$ lead to isomorphic $\ast$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups.

math.OA↗

arXiv2022

Exact SDP relaxations for quadratic programs with bipartite graph structures

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. Our results on the QCQPs generalize the results on QCQP with sign-definite bipartite graph structures, QCQPs with forest structures, and QCQPs with nonpositive off-diagonal data elements. Second, we propose a conversion method from QCQPs with no particular structure to the ones with bipartite graph structures. As a result, we demonstrate that a wider class of QCQPs can be exactly solved by the SDP relaxation. Numerical instances are presented for illustration.

math.OC↗

arXiv2021

Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal

We apply our previous work on the relation between groupoid homology and K-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets, and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of unstable equivalence relation.

math.KT↗

arXiv2021

Lieb type convexity for positive operator monotone decreasing functions

We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions.

math.FA↗

arXiv2021

Categorically Morita equivalent compact quantum groups

We give a dynamical characterization of categorical Morita equivalence between compact quantum groups. More precisely, by a Tannaka-Krein type duality, a unital C*-algebra endowed with commuting actions of two compact quantum groups corresponds to a bimodule category over their representation categories. We show that this bimodule category is invertible if and only if the actions are free, with finite dimensional fixed point algebras, which are in duality as Frobenius algebras in an appropriate sense. This extends the well-known characterization of monoidal equivalence in terms of bi-Hopf-Galois objects.

math.OA↗

arXiv2021

Strengthened convexity of positive operator monotone decreasing functions

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function.

math.FA↗

arXiv2021

Homology and K-theory of dynamical systems. I. torsion-free ample groupoids

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.

math.KT↗

arXiv2020

An Infeasible Interior-point Arc-search Algorithm for Nonlinear Constrained Optimization

In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search, since they compute a next iterate on a straight line determined by a search direction which approximates the central path.The proposed arc-search interior-point algorithm uses an arc for the approximation.We discuss convergence properties of the proposed algorithm.We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using less iterations but longer time than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed.

math.OC↗

arXiv2020

Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with $n$ variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than $n-1$ and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation.

math.OC↗

arXiv2019

Exploiting Aggregate Sparsity in Second Order Cone Relaxations for Quadratic Constrained Quadratic Programming Problems

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. (2001) and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP relaxation of QCQPs. Specifically, we prove that exploiting the aggregate sparsity reduces the number of second-order cones in the SOCP relaxation, and that we can simplify the matrix completion procedure by Fukuda et al. in both primal and dual of the SOCP relaxation problem without losing the max-determinant property. For numerical experiments, QCQPs from the lattice graph and pooling problem are tested as their SOCP relaxations provide the same optimal value as the SDP relaxations. We demonstrate that exploiting the aggregate sparsity improves the computational efficiency of the SOCP relaxation for the same objective value as the SDP relaxation, thus much larger problems can be handled by the proposed SOCP relaxation than the SDP relaxation.

math.OC↗

arXiv2019

Sparsity Exploitation of Accelerated Modulus-Based Gauss-Seidel Method for Interactive Rigid Body Simulations

Large-scale linear complementarity problems (LCPs) are repeatedly solved in interactive rigid-body simulations. The projected Gauss-Seidel method is often employed for LCPs, since it has advantages in computation time, numerical robustness, and memory use. Zheng and Yin (2013) proposed modulus-based matrix splitting iteration methods and showed their effectiveness for large problems, but a simple application of their approach to large-scale LCPs in interactive rigid-body simulations is not effective since such a simple application demands large matrix multiplications. In this paper, we propose a novel method derived from accelerated modulus-based matrix splitting iteration methods that fits LCPs arising in interactive rigid-body simulations. To improve the computation time, we exploit sparsity structures related to the generalized velocity vector of rigid bodies. We discuss the convergence of the proposed method for an important case that the coefficient matrix of a given LCP is a positive definite matrix. Numerical experiments show that the proposed method is more efficient than the simple application of the accelerated modulus-based Gauss-Seidel method and that the accuracy in each step of the proposed method is superior to that of the projected Gauss-Seidel method.

math.OC↗

arXiv2019

Experimental Determination of Bose-Hubbard Energies

We present the first experimental measurement of the ensemble averages of both the kinetic and interaction energies of the three-dimensional Bose--Hubbard model at finite temperature and various optical lattice depths across weakly to strongly interacting regimes, for an almost unit filling factor. The kinetic energy is obtained through Fourier transformation of a time-of-flight signal, and the interaction energy is measured using a newly developed atom-number-projection spectroscopy technique, by exploiting an ultra-narrow optical transition of two-electron atoms. The obtained experimental results can be used as benchmarks for state-of-the-art numerical methods of quantum many-body theory. As an illustrative example, we compare the measured energies with numerical calculations involving the Gutzwiller and cluster-Gutzwiller approximations, assuming realistic trap potentials and particle numbers at nonzero entropy (finite temperature); we obtain good agreement without fitting parameters. We also discuss the possible application of this method to temperature estimations for atoms in optical lattices using the thermodynamic relation. This study offers a unique advantage of cold atom system for `quantum simulators', because, to the best of our knowledge, it is the first experimental determination of both the kinetic and interaction energies of quantum many-body system.

cond-mat.quant-gas↗