Searcharxiv⌕ Search

arXiv subjects

Hoa T. Bui

Publications and source records attributed to Hoa T. Bui.

13 recordsLinked to original sources
arXiv2024

Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

In this work, we consider a class of convex optimization problems in a real Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those established for the linear programming setting in Nurminski (2015) by considering problems that: (i) may have multiple solutions, (ii) do not satisfy strict complementary conditions, and (iii) possess non-linear convex constraints. As a by-product of our analysis, we provide a quantitative estimate on the required distance between the infeasible point and the feasible set in order for its projection to be a solution of the problem. Our analysis relies on a "sharpness" property of the constraint set; a new property we introduce here.

math.OC↗
arXiv2023

The linkedness of cubical polytopes: beyond the cube

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is \textit{$k$-linked} if its graph is $k$-linked. In a previous paper \cite{BuiPinUgo20a} we proved that every cubical $d$-polytope is $\floor{d/2}$-linked. Here we strengthen this result by establishing the $\floor{(d+1)/2}$-linkedness of cubical $d$-polytopes, for every $d\ne 3$. A graph $G$ is {\it strongly $k$-linked} if it has at least $2k+1$ vertices and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We say that a polytope is (strongly) \textit{$k$-linked} if its graph is (strongly) $k$-linked. In this paper, we also prove that every cubical $d$-polytope is strongly $\floor{d/2}$-linked, for every $d\ne 3$. These results are best possible for this class of polytopes.

math.CO↗
arXiv2023

Cutting Plane Algorithms are Exact for Euclidean Max-Sum Problems

This paper studies binary quadratic programs in which the objective is defined by a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems includes the capacitated, generalised and bi-level diversity problems as special cases. We introduce two exact cutting plane algorithms to solve this class of optimisation problems. The new algorithms remove the need for a concave reformulation, which is known to significantly slow down convergence. We establish exactness of the new algorithms by examining the concavity of the quadratic objective in a given direction, a concept we refer to as directional concavity. Numerical results show that the algorithms outperform other exact methods for benchmark diversity problems (capacitated, generalised and bi-level), and can easily solve problems of up to three thousand variables.

math.OC↗
arXiv2022

An exact cutting plane method for solving p-dispersion-sum problems

This paper aims to answer an open question recently posed in the literature, that is to find a fast exact method for solving the p-dispersion-sum problem (PDSP), a nonconcave quadratic binary maximization problem. We show that, since the Euclidean distance matrix defining the quadratic term in (PDSP) is always conditionally negative definite, the cutting plane method is exact for (PDSP) even in the absence of concavity. As such, the cutting plane method, which is primarily designed for concave maximisation problems, converges to the optimal solution of the (PDSP). The numerical results show that the method outperforms other exact methods for solving (PDSP), and can solve to optimality large instances of up to two thousand variables.

math.OC↗
arXiv2022

Cutting plane algorithms for nonlinear binary optimization

Current state-of-the-art methods for solving discrete optimization problems are usually restricted to convex settings. In this paper, we propose a general approach based on cutting planes for solving nonlinear, possibly nonconvex, binary optimization problems. We provide a rigorous convergence analysis that quantifies the number of iterations required under different conditions. This is different to most other work in discrete optimization where only finite convergence is proved. Moreover, using tools from variational analysis, we provide necessary and sufficient dual optimality conditions.

math.OC↗
arXiv2021

Necessary Conditions for Non-Intersection of Collections of Sets

This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections.

math.OC↗
arXiv2021

The linkedness of cubical polytopes: The cube

The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is \textit{$k$-linked} if its graph is $k$-linked. We establish that the $d$-dimensional cube is $\lfloor(d+1)/2\rfloor$-linked, for every $d\ne 3$; this is the maximum possible linkedness of a $d$-polytope. This result implies that, for every $d\ge 1$, a cubical $d$-polytope is $\lfloor{d/2}\rfloor$-linked, which answers a question of Wotzlaw \cite{Ron09}. Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph $G$ is {\it strongly $k$-linked} if it has at least $2k+1$ vertices and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We show that cubical 4-polytopes are strongly $2$-linked and that, for each $d\ge 1$, $d$-dimensional cubes are strongly $\lfloor{d/2}\rfloor$-linked.

math.CO↗
arXiv2021

A Note on the Finite Convergence of Alternating Projections

We establish sufficient conditions for finite convergence of the alternating projections method for two non-intersecting and potentially nonconvex sets. Our results are based on a generalization of the concept of intrinsic transversality, which until now has been restricted to sets with nonempty intersection. In the special case of a polyhedron and closed half space, our sufficient conditions define the minimum distance between the two sets that is required for alternating projections to converge in a single iteration.

math.OC↗
arXiv2020

Zero Duality Gap in View of Abstract Convexity

Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of nonconvex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) $\varepsilon-$subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach--Alaoglu--Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of nonconvex, non-differentiable problems.

math.FA↗
arXiv2019

Connectivity of cubical polytopes

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any $d\ge 3$, the graph of a cubical $d$-polytope with minimum degree $δ$ is $\min\{δ,2d-2\}$-connected. Second, we show, for any $d\ge 4$, that every minimum separator of cardinality at most $2d-3$ in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself.

math.CO↗
arXiv2018

Extremality, Stationarity and Generalized Separation of Collections of Sets

The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analyzed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining 'extremal' statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.

math.OC↗
arXiv2018

About Extensions of the Extremal Principle

In this article, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions and expand the applicability of the generalised separability results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity and Lipschitz-like properties of set-valued mappings.

math.OC↗