arXiv subjects

Van-Bong Nguyen

Publications and source records attributed to Van-Bong Nguyen.

4 recordsLinked to original sources

arXiv2025

A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem

We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with smallest radius $r$ such that $B(a,r)$ contains the intersection $\cap_{i=1}^mB(a_i,r_i)$ of $m$ given balls $B(a_i,r_i).$ We show that $G(\Bbb R^n)$ is convex if and only if ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}\le n-1.$ Otherwise, ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}=n$ and $G(\Bbb R^n)$ is not convex. In this case we propose a new set $G(\Bbb R^n)^\bullet$ which allows to show that if $m=n$ then $G(\Bbb R^n)^\bullet$ is convex even $G(\Bbb R^n)$ is not. Importantly, the separable property of $G(\Bbb R^n)^\bullet$ then implies the separable property for $G(\Bbb R^n).$ As a result, a new progress on solving the SEB problem is obtained.

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arXiv2023

Positive semidefinite interval of matrix pencil and its applications for the generalized trust region subproblems

We are concerned with finding the set $I_{\succeq}(A,B)$ of real values $μ$ such that the matrix pencil $A+μB$ is positive semidefinite. If $A, B$ are not simultaneously diagonalizable via congruence (SDC), $I_{\succeq}(A,B)$ either is empty or has only one value $μ.$ When $A, B$ are SDC, $I_{\succeq}(A,B),$ if not empty, can be a singleton or an interval. Especially, if $I_{\succeq}(A,B)$ is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval $I_{\succ}(A,B).$ If $A, B$ are both singular, then even $I_{\succeq}(A,B)$ is an interval, its interior may not be $I_{\succ}(A,B),$ but $A, B$ are then decomposed to block diagonals of submatrices $A_1, B_1$ with $B_1$ nonsingular such that $I_{\succeq}(A,B)=I_{\succeq}(A_1,B_1).$ Applying $I_{\succeq}(A,B),$ the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.

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arXiv2023

Simultaneous diagonalization via congruence of $m$ real symmetric matrices and its implications in optimization

Let $\{C_1, C_2, \ldots, C_m\},~m\ge2$ be a collection of $n\times n$ real symmetric matrices. The objective of the paper is to offer an algorithm that finds a common congruence matrix $R$ such that $R^TC_iR$ is real diagonal for every $C_i;$ or reports none of such kind. The problem, referred to as the simultaneously diagonalization via congruence (SDC in short), seems to be of pure linear algebra at first glance. However, for quadratically constrained quadratic programming (QCQP), if the quadratic forms are SDC, their joint range set is a closed convex polyhedral cone, which opens the possibility to extend the classical $\mathcal{S}$-lemma for more than two symmetric matrices. In addition, under the SDC assumption of quadratic forms, QCQP can be recast in separable forms which is usually easier to tackle. It is thus important to have a standard procedure for determining whether or not the SDC property holds for the underlined quadratic optimization problem. Our result solves a long standing problem posed by Hiriart-Urruty in 2007.

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arXiv2014

An SDP Approach For Solving Quadratic Fractional Programming Problems

This paper considers a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. As is well-known, the fractional objective function can be replaced by a parametric family of quadratic functions, which makes (P) highly related to, but more difficult than a single quadratic programming problem subject to a similar constraint set. The task is to find the optimal parameter $λ^*$ and then look for the optimal solution if $λ^*$ is attained. Contrasted with the classical Dinkelbach method that iterates over the parameter, we propose a suitable constraint qualification under which a new version of the S-lemma with an equality can be proved so as to compute $λ^*$ directly via an exact SDP relaxation. When the constraint set of (P) is degenerated to become an one-sided inequality, the same SDP approach can be applied to solve (P) {\it without any condition}. We observe that the difference between a two-sided problem and an one-sided problem lies in the fact that the S-lemma with an equality does not have a natural Slater point to hold, which makes the former essentially more difficult than the latter. This work does not, either, assume the existence of a positive-definite linear combination of the quadratic terms (also known as the dual Slater condition, or a positive-definite matrix pencil), our result thus provides a novel extension to the so-called "hard case" of the generalized trust region subproblem subject to the upper and the lower level set of a quadratic function.

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