arXiv subjects

Woosuk L. Jung

Publications and source records attributed to Woosuk L. Jung.

6 recordsLinked to original sources

arXiv2026

Projection, Degeneracy, and Singularity Degree for Spectrahedra

Facial reduction, FR, is a regularization technique for convex programs where the strict feasibility constraint qualification, CQ, fails.Though this CQ holds generically, failure is pervasive in applications such as semidefinite relaxations of hard discrete optimization problems. In this paper we relate FR to the analysis of the convergence behaviour of a semismooth Newton root finding method for the projection onto a spectrahedron, i.e., onto the intersection of a linear manifold and the semidefinite cone. We examine the effect of failure of strict feasibility on the projection problem. In the process, we derive an elegant formula for the projection onto a face of the semidefinite cone obtained via regularization and discuss pathologies that arise in the absence of strict feasibility. We show further that the ill-conditioning of the Jacobian of the Newton method near optimality characterizes the degeneracy of the nearest point in the spectrahedron. We apply the results, both theoretically and empirically, to the problem of finding nearest points to the sets of: (i) correlation matrices or the elliptope; and (ii) semidefinite relaxations of permutation matrices or the vontope, i.e., the feasible sets for the semidefinite relaxations of the max-cut and quadratic assignment problems, respectively.

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arXiv2026

Optimal Diagonal Preconditioning Beyond Worst-Case Conditioning: Theory and Practice of Omega Scaling

We study optimal diagonal preconditioning using the classical worst-case $κ$-condition number and the averaging-based $ω$-condition number. For the $κ$-optimal preconditioning problem, we derive an affine-based pseudoconvex reformulation with three key advantages: all stationary points are global minima, subgradients are inexpensive to compute, and the optimization variable is an $n$-dimensional vector rather than an $n\times n$ matrix as in semidefinite programming (SDP) approaches. We develop a simple and highly efficient subgradient method, with convergence guarantees, for solving this pseudoconvex formulation that is substantially more scalable and accurate than existing SDP-based methods. For the $ω$-condition number, we provide explicit characterizations of optimal diagonal and block diagonal preconditioners. In particular, we show that several classical preconditioners, including Jacobi and row/column normalization, are $ω$-optimal, and that matrix balancing schemes monotonically reduce $ω$ and converge to stationary points of the two-sided problem. To the best of our knowledge, this is the first unified and explicit characterization of optimality conditions for both $κ$ and $ω$-based preconditioning. Our numerical experiments further reveal a striking phenomenon: although $κ$-optimal preconditioners achieve stronger reductions in the worst-case condition number, $ω$-optimal preconditioners are substantially cheaper to compute and yield better performance for iterative methods such as preconditioned conjugate gradient (PCG) and least squares method (LSQR). Moreover, applying $ω$-optimal scaling to linear systems that are already $κ$-optimally preconditioned leads to further improvements in PCG iterations.

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arXiv2025

Exact Solutions for the NP-hard Wasserstein Barycenter Problem using a Doubly Nonnegative Relaxation and a Splitting Method

The so-called \emph{simplified} Wasserstein barycenter problem, also known as the cheapest hub problem, consists in selecting one point from each of $k$ given sets, each set consisting of $n$ points, with the aim of minimizing the sum of distances to the barycenter of the $k$ chosen points. This problem is known to be NP-hard. We compute the Wasserstein barycenter by exploiting the Euclidean distance matrix structure to obtain a facially reduced doubly nonnegative, DNN, relaxation. The facial reduction provides a natural splitting for applying the symmetric alternating directions method of multipliers (sADMM) to the DNN relaxation. The sADMM method exploits structure in the subproblems to find strong upper and lower bounds. In addition, we extend the problem to allow varying $n_j$ points for the $j$-th set. The purpose of this paper is twofold. First we want to illustrate the strength of this DNN relaxation with the natural splitting approach mentioned above. Our numerical tests then illustrate the surprising success on random problems, as we generally, efficiently, find the provable exact solution of this NP-hard problem. Comparisons with current commercial software illustrate this surprising efficiency. However, we demonstrate and prove that there is a duality gap for problems with \emph{enough} multiple optimal solutions, and that this arises from problems with highly symmetrized structure.

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arXiv2025

On the local and global minimizers of the smooth stress function in Euclidean Distance Matrix problems

We consider the nonconvex minimization problem, with quartic objective function, that arises in the exact recovery of a configuration matrix $P\in \R^{nd}$ of $n$ points when a Euclidean distance matrix, \EDMp, is given with embedding dimension $d$. It is an open question in the literature whether there are conditions such that the minimization problem admits a local nonglobal minimizer, \lngmp. We prove that all second-order stationary points are global minimizers whenever $n \leq d + 1$. {And, for $d=1$ and $n\geq 7>d+1$, we present an example where we can analytically exhibit a local nonglobal minimizer. For more general cases,} we numerically find a second-order stationary point and then prove that there indeed exists a nearby \lngm for the quartic nonconvex minimization problem. Thus, we answer the previously open question about their existence in the affirmative. Our approach to finding the \lngm is novel in that we first exploit the translation and rotation invariance to remove the singularities of the Hessian, and reduce the size of the problem from $nd$ variables in $P$ to $(n-1)d - d(d-1)/2$ variables. This allows for stabilizing Newton's method, and for finding examples that satisfy the strict second order sufficient optimality conditions. The motivation for being able to find global minima is to obtain \emph{exact recovery} of the configuration matrix, even in the cases where the data is noisy and/or incomplete, without resorting to approximating solutions from convex (semidefinite programming) relaxations. In the process of our work we present new insights into when \lngmp s of the smooth stress function do and do not exist.

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arXiv2024

The $ω$-Condition Number: Applications to Optimal Preconditioning and Low Rank Generalized Jacobian Updating

Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g.,in quasi-Newton methods. Motivated by the latter, we study a nonclassic matrix condition number, the $ω$-condition number, $ω$ for short. $ω$ is the ratio of the arithmetic and geometric means of the singular values, rather than largest and smallest. Moreover, unlike the latter classical $κ$ condition number, $ω$ is not invariant under inversion, an important point that allows one to recall that it is the conditioning of the inverse that is important. Our study is in the context of optimal conditioning for: (i) low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) iterative methods for linear systems; (iia) clustering of eigenvalues; (iib) convergence rates; and (iic) estimating the actual condition of a linear system. We emphasize that the simple functions in $ω$ allow one to exploit optimality conditions and derive explicit formulae for $ω$-optimal preconditioners of special structure. Connections to partial Cholesky type sparse preconditioners are made that modify the iterates of Cholesky decomposition by including the entire diagonal at each iteration. Our results confirm the efficacy of using the $ω$-condition number compared to the classical $κ$-condition number.

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arXiv2024

Single Element Error Correction/ in a Euclidean Distance Matrix

We consider the \emph{exact} error correction of a noisy Euclidean distance matrix, EDM, where the elements are the squared distances between $n$ points in $R^d$. For our problem we are given two facts: (i) the embedding dimension, $d$, (ii) \emph{exactly one} distance in the data is corrupted by \emph{nonzero noise}. But we do \underline{not} know the magnitude nor position of the noise. Thus there is a combinatorial element to the problem. We present three solution techniques. These use three divide and conquer strategies in combination with three versions of facial reduction that use: exposing vectors, facial vectors, and Gale transforms. This sheds light on the connections between the various forms of facial reduction related to Gale transforms. Our highly successful empirics confirm the success of these approaches as we can solve huge problems of the order of $100,000$ nodes in approximately one minute to machine precision. \\Our algorithm depends on identifying whether a principal submatrix of the \EDM contains the corrupted element. We provide a theorem for doing this that is related to the existing results for identifying \emph{yielding} elements, i.e.,~we provide a characterization for guaranteeing the perturbed EDM remains an EDM with embedding dimension $d$. The characterization is particularly simple in the $d=2$ case. \\In addition, we characterize when the intuitive approach of the nearest EDM problem, solves our problem. In fact, we show that this happens if, and only if, the original distance element is $0$, degenerate, and the perturbation is negative.

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