arXiv subjects

Hòa T. Bùi

Publications and source records attributed to Hòa T. Bùi.

3 recordsLinked to original sources

arXiv2026

Variational Analysis in Spectral Decomposition Systems

This work is concerned with the variational analysis of functions defined on Euclidean spaces whose values depend solely on certain invariants (``spectrum'') of their arguments, a class we term ``spectral functions.'' Building on our previous work \cite{PartI} on the convex analysis of such functions, we work in the abstract framework of spectral decomposition systems, which covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices, and allows the derivation of new results in more general settings such as normal decomposition systems. The main results of this work provide constructive formulae for computing the regular, limiting, and Clarke subdifferentials of a spectral function in terms of the corresponding objects of the associated invariant function. Finally, we obtain a generalization of Lidski\uı's theorem on the spectrum of additive perturbations of Hermitian matrices to arbitrary spectral decomposition systems.

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arXiv2026

Convex Analysis in Spectral Decomposition Systems

This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We propose a notion of a spectral decomposition system which brings together a wide array of settings underlying important applications such as Fourier-phase-invariant functions, mixed-norm regularization, and functions of eigenvalues or (signed) singular values of matrices. We are particularly motivated by algorithmic requirements for evaluating convex analytical objects. Thus, a central contribution is a novel reduced minimization principle that enables the constructive reduction of minimization problems involving spectral functions to those of the simpler associated invariant functions. This result is then leveraged to explicitly evaluate the conjugates, subgradients, and set-valued Bregman proximity operators of spectral functions.

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arXiv2025

Cutting plane methods with gradient-based heuristics

Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their computational performance can be highly variable. Recent research has shown that constructing cutting planes at the projection of infeasible points onto the feasible set can significantly improve the performance of cutting plane approaches. Motivated by this, we examine whether constructing cuts at feasible points closer to the optimal solution set could further enhance the effectiveness of cutting plane methods. We propose a hybrid method that combines the global convergence guarantees of cutting plane methods with the local exploration capabilities of first-order optimization techniques. Specifically, we use projected gradient methods as a heuristic to identify promising regions of the solution space and generate tighter, more informative cuts. We focus on binary optimization problems with convex differentiable objective functions, where projection operations can be efficiently computed via mixed-integer linear programming. By constructing cuts at points closer to the optimal solution set and eliminating non-optimal regions, the algorithm achieves better approximation of the feasible region and faster convergence. Numerical experiments confirm that our approach improves both the quality of the solution and computational efficiency across different solver configurations. This framework provides a flexible foundation for further extensions to more general discrete domains and offers a promising heuristic to the toolkit for nonlinear discrete optimization.

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