arXiv subjects

Hou-Duo Qi

Publications and source records attributed to Hou-Duo Qi.

9 recordsLinked to original sources

arXiv2026

A Robust EDM Optimization Approach for 3D Single-Source Localization with Angle and Range Measurements

Accurate source localization in Multi-Platform Radar Networks (MPRNs) benefits from exploiting both range and angle measurements under robust estimation. In this paper, we propose a robust Euclidean distance matrix (EDM) optimization model that simultaneously integrates range measurements, angle information, and the least absolute deviation ($\ell_1$-norm) criterion for the case of 3D single-source localization (3DSSL). A key theoretical contribution of this work is the rigorous reformulation of {existing} 3D angle measurements into simple box constraints on the Euclidean distances. Unlike previous approximations, we achieve this by reducing each of the 3D angle measurements to a two-dimensional nonlinear optimization problem, whose global minimum and maximum solutions can be characterized and utilized to get the lower and upper bounds of the distances from the unknown source to the sensors. To solve the resulting rank-constrained EDM problem, we develop an efficient algorithm based on the majorization penalty method. Extensive numerical experiments confirm that the new EDM model significantly outperforms leading solvers in terms of localization accuracy and computational efficiency, particularly in low Signal-to-Noise Ratio (SNR) scenarios.

eess.SP↗

arXiv2026

On subspace-constrained preconditioning for randomized iterative methods

In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular, we design a QR-like factorization that transforms the original linear system into an equivalent block-orthogonal form, thus avoiding the full-rank assumptions adopted in existing work. Moreover, this reformulation reduces the problem to solving a smaller linear system with a favorable singular value distribution, provided an appropriate initial point is employed. The proposed framework can be implemented implicitly within the iteration and does not require explicitly constructing either a preconditioner matrix or a preconditioned linear system, which eliminates the prohibitive cost of forming a fully preconditioned system. Furthermore, we construct orthogonalized search directions from stochastic gradients and develop accelerated variants of the framework. We prove that the proposed algorithmic framework converges linearly in expectation. Numerical experiments demonstrate the benefits of the proposed preconditioning strategy.

math.NA↗

arXiv2025

Randomized iterative methods for generalized absolute value equations: Solvability and error bounds

Randomized iterative methods, such as the Kaczmarz method and its variants, have gained growing attention due to their simplicity and efficiency in solving large-scale linear systems. Meanwhile, absolute value equations (AVE) have attracted increasing interest due to their connection with the linear complementarity problem. In this paper, we investigate the application of randomized iterative methods to generalized AVE (GAVE). Our approach differs from most existing works in that we tackle GAVE with non-square coefficient matrices. We establish more comprehensive sufficient and necessary conditions for characterizing the solvability of GAVE and propose precise error bound conditions. Furthermore, we introduce a flexible and efficient randomized iterative algorithmic framework for solving GAVE, which employs randomized sketching matrices drawn from user-specified distributions. This framework is capable of encompassing many well-known methods, including the Picard iteration method and the randomized Kaczmarz method. Leveraging our findings on solvability and error bounds, we establish both almost sure convergence and linear convergence rates for this versatile algorithmic framework. Finally, we present numerical examples to illustrate the advantages of the new algorithms.

math.NA↗

arXiv2024

Analytic analysis of the worst-case complexity of the gradient method with exact line search and the Polyak stepsize

We give a novel analytic analysis of the worst-case complexity of the gradient method with exact line search and the Polyak stepsize, respectively, which previously could only be established by computer-assisted proof. Our analysis is based on studying the linear convergence of a family of gradient methods, whose stepsizes include the one determined by exact line search and the Polyak stepsize as special instances. The asymptotic behavior of the considered family is also investigated which shows that the gradient method with the Polyak stepsize will zigzag in a two-dimensional subspace spanned by the two eigenvectors corresponding to the largest and smallest eigenvalues of the Hessian.

math.OC↗

arXiv2023

Sparse SVM with Hard-Margin Loss: a Newton-Augmented Lagrangian Method in Reduced Dimensions

The hard margin loss function has been at the core of the support vector machine (SVM) research from the very beginning due to its generalization capability.On the other hand, the cardinality constraint has been widely used for feature selection, leading to sparse solutions. This paper studies the sparse SVM with the hard-margin loss (SSVM-HM) that integrates the virtues of both worlds. However, SSVM-HM is one of the most challenging models to solve. In this paper, we cast the problem as a composite optimization with the cardinality constraint. We characterize its local minimizers in terms of {\rm P}-stationarity that well captures the combinatorial structure of the problem. We then propose an inexact proximal augmented Lagrangian method (iPAL). The different parts of the inexactness measurements from the {\rm P}-stationarity are controlled at different scales in a way that the generated sequence converges both globally and at a linear rate. This matches the best convergence theory for composite optimization. To make iPAL practically efficient, we propose a gradient-Newton method in a subspace for the iPAL subproblem. This is accomplished by detecting active samples and features with the help of the proximal operator of the hard margin loss and the projection of cardinality constraint. Extensive numerical results on both simulated and real datasets demonstrate that the proposed method is fast, produces sparse solution of high accuracy, and can lead to effective reduction on active samples and features when compared with several leading solvers.

math.OC↗

arXiv2023

iNALM: An inexact Newton Augmented Lagrangian Method for Zero-One Composite Optimization

Zero-One Composite Optimization (0/1-COP) is a prototype of nonsmooth, nonconvex optimization problems and it has attracted much attention recently. The augmented Lagrangian Method (ALM) has stood out as a leading methodology for such problems. The main purpose of this paper is to extend the classical theory of ALM from smooth problems to 0/1-COP. We propose, for the first time, second-order optimality conditions for 0/1-COP. In particular, under a second-order sufficient condition (SOSC), we prove the R-linear convergence rate of the proposed ALM. In order to identify the subspace used in SOSC, we employ the proximal operator of the 0/1-loss function, leading to an active-set identification technique. Built around this identification process, we design practical stopping criteria for any algorithm to be used for the subproblem of ALM. We justify that Newton's method is an ideal candidate for the subproblem and it enjoys both global and local quadratic convergence. Those considerations result in an inexact Newton ALM (iNALM). The method of iNALM is unique in the sense that it is active-set based, it is inexact (hence more practical), and SOSC plays an important role in its R-linear convergence analysis. The numerical results on both simulated and real datasets show the fast running speed and high accuracy of iNALM when compared with several leading solvers.

math.OC↗

arXiv2021

A Lagrange-Newton Algorithm for Sparse Nonlinear Programming

The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning, pattern recognition, finance and management, etc. However, the computational challenge posed by SNP has not yet been well resolved due to the nonconvex and discontinuous $\ell_0$-norm involved. In this paper, we resolve this numerical challenge by developing a fast Newton-type algorithm. As a theoretical cornerstone, we establish a first-order optimality condition for SNP based on the concept of strong $β$-Lagrangian stationarity via the Lagrangian function, and reformulate it as a system of nonlinear equations called the Lagrangian equations. The nonsingularity of the corresponding Jacobian is discussed, based on which the Lagrange-Newton algorithm (LNA) is then proposed. Under mild conditions, we establish the locally quadratic convergence and the iterative complexity estimation of LNA. To further demonstrate the efficiency and superiority of our proposed algorithm, we apply LNA to solve two specific application problems arising from compressed sensing and sparse high-order portfolio selection, in which significant benefits accrue from the restricted Newton step in LNA.

math.OC↗

arXiv2020

Global and Quadratic Convergence of Newton Hard-Thresholding Pursuit

Algorithms based on the hard thresholding principle have been well studied with sounding theoretical guarantees in the compressed sensing and more general sparsity-constrained optimization. It is widely observed in existing empirical studies that when a restricted Newton step was used (as the debiasing step), the hard-thresholding algorithms tend to meet halting conditions in a significantly low number of iterations and are very efficient. Hence, the thus obtained Newton hard-thresholding algorithms call for stronger theoretical guarantees than for their simple hard-thresholding counterparts. This paper provides a theoretical justification for the use of the restricted Newton step. We build our theory and algorithm, Newton Hard-Thresholding Pursuit (NHTP), for the sparsity-constrained optimization. Our main result shows that NHTP is quadratically convergent under the standard assumption of restricted strong convexity and smoothness. We also establish its global convergence to a stationary point under a weaker assumption. In the special case of the compressive sensing, NHTP effectively reduces to some of the existing hard-thresholding algorithms with a Newton step. Consequently, our fast convergence result justifies why those algorithms perform better than without the Newton step. The efficiency of NHTP was demonstrated on both synthetic and real data in compressed sensing and sparse logistic regression.

math.OC↗

arXiv2014

Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP) to reconstruct such faithful representations. While those SDP models are capable of producing high quality configuration numerically, they suffer two major drawbacks. One is that there exist no theoretically guaranteed bounds on the quality of the configuration. The other is that they are slow in computation when the data points are beyond moderate size. In this paper, we propose a convex optimization model of Euclidean distance matrices. We establish a non-asymptotic error bound for the random graph model with sub-Gaussian noise, and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low-rank matrix up to a logarithmic factor. Our results partially explain why MVU and MVE often work well. Moreover, we develop a fast inexact accelerated proximal gradient method. Numerical experiments show that the model can produce configurations of high quality on large data points that the SDP approach would struggle to cope with.

stat.ML↗