Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives

Flores-Bazán, Fabián; Vera, Cristián Garay

doi:10.1007/s10898-015-0267-6

Cited by 22 publications

(22 citation statements)

References 15 publications

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“…The desired result now follows from (19) and (21). Remark 4: As was mentioned in Remark 3, calculating the nullspace constant is NP-hard in general.…”

Section: B Polynomial Time Guarantees For Sparse Recoverymentioning

confidence: 90%

Sparse Recovery over Graph Incidence Matrices

Zhao¹,

Kaba²,

Vidal

et al. 2018

2018 IEEE Conference on Decision and Control (CDC)

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Classical results in sparse recovery guarantee the exact reconstruction of s-sparse signals under assumptions on the dictionary that are either too strong or NP-hard to check. Moreover, such results may be pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We derive necessary and sufficient conditions for sparse recovery, which depend on properties of the cycles of the graph that can be checked in polynomial time. We also derive support-dependent conditions for sparse recovery that depend only on the intersection of the cycles of the graph with the support of the signal. Finally, we exploit sparsity properties on the measurements and the structure of incidence matrices to propose a specialized sub-graph-based recovery algorithm that outperforms the standard 1-minimization approach.

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“…The desired result now follows from (19) and (21). Remark 4: As was mentioned in Remark 3, calculating the nullspace constant is NP-hard in general.…”

Section: B Polynomial Time Guarantees For Sparse Recoverymentioning

confidence: 90%

Sparse Recovery over Graph Incidence Matrices

Zhao¹,

Kaba²,

Vidal

et al. 2018

2018 IEEE Conference on Decision and Control (CDC)

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“…Both inequalities could be strict even for quasiconvex functions, as was proved in [5,Example 5.6]. Finally, it is important to point out that f ∞ q (u) > 0 for all u = 0 does not imply that f is coercive as the function f (x) = |x| 1+|x| shows.…”

Section: Introduction 2 Preliminaries and Basic Definitionsmentioning

confidence: 90%

“…(2.5) The problem to find an adequate definition of an asymptotic function has been studied in the last years, since the usual asymptotic function is not well suited for the description of the behavior of a nonconvex function at infinity. Several attempts to deal with the quasiconvex case has been made in [1,5,6,10] while applications to optimization can be found in [6,9].…”

Section: Introduction 2 Preliminaries and Basic Definitionsmentioning

confidence: 99%

“…The following two asymptotic functions to deal with quasiconvexity were introduced in [5]. Recall that, given a proper function f :…”

Section: Introduction 2 Preliminaries and Basic Definitionsmentioning

confidence: 99%

“…If f is lsc and quasiconvex, by [5,Theorem 4.7] we have 8) and by [5,Proposition 5.3] f ∞ (u; λ) > 0, ∀ u = 0 ⇐⇒ S λ (f ) = ∅ and compact, (2.9)…”

Section: Introduction 2 Preliminaries and Basic Definitionsmentioning

confidence: 99%

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A Quasiconvex Asymptotic Function with Applications in Optimization

Hadjisavvas

Lara

Martínez-Legaz

2018

J Optim Theory Appl

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Burgers’ Equation Revisited: Extension of Mono-Dimensional Case on a Network

Mucha

Puchalska

2022

J. Math. Fluid Mech.

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The paper deals with the analysis of Burgers’ equation on acyclic metric graphs. The main goal is to establish the existence of weak solutions in the TV—class of regularity. A key point is transmission conditions in vertices obeying the Kirchhoff law. First, we consider positive solutions at arbitrary acyclic networks and highlight two kinds of vertices, describing two mechanisms of flow splitting at the vertex. Next we design rules at vertices for solutions of arbitrary sign for any subgraph of hexagonal grid, which leads to a construction of general solutions with TV—regularity for this class of networks. Introduced transmission conditions are motivated by the change of the energy estimation.

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Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives

Cited by 22 publications

References 15 publications

Sparse Recovery over Graph Incidence Matrices

Sparse Recovery over Graph Incidence Matrices

A Quasiconvex Asymptotic Function with Applications in Optimization

Burgers’ Equation Revisited: Extension of Mono-Dimensional Case on a Network

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