Programme Gaspard MONGE pour l’Optimisationet la Recherche Operationnelle (PGMO)
”Nothing in the world takes place without optimization, and there is nodoubt that all aspects of the world that have a rational basis can be
explained by optimization methods”
L.Euler (1744)
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 20
Programme Gaspard MONGE pour l’Optimisationet la Recherche Operationnelle (PGMO)
”Nothing in the world takes place without optimization, and there is nodoubt that all aspects of the world that have a rational basis can be
explained by optimization methods”
L.Euler (1744)
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 20
Some anniversaries ...
Henri Poincare
... died in 1912.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 19
Some anniversaries ...
Henri Poincare
... died in 1912.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 19
Alan Turing Turing Memorial
... born in 1912.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 18
Leonid Kantorovich
... born in 1912.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 17
Paul Sabatier, Nobel prize winner in 1912
Paul Sabatier Fermat
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 16
1962-1963: Birth of modern convex analysis and optimization.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 15
A variational approach of the rank function.
Jean-Baptiste HIRIART-URRUTYand
Hai Yen LE
Institut de Mathematiques de ToulouseUniversite Paul Sabatier
Septembre 2012
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 11
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 10
The rank in linear algebra or matricial calculus
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 9
The rank in linear algebra or matricial calculus
Let p := min(m, n), let
rank : Mm,n(R) −→ {0, 1, . . . , p}A 7−→ rank A.
The properties of the rank function are well-known in the context of linearalgebra or matricial calculus. Any book on these subjects presents them;there even are compilations of them. We recall here the most basic ones:
1 rank A = rank AT ; rank A = rank (AAT ) = rank (ATA).
2 If the product AB can be done,
rank (AB) ≤ min(rank A, rank B) (Sylvester inequality).
3 rank A = 0 if and only if A = 0; rank (cA) = rank A for c 6= 0.
4 |rank A− rank B| ≤ rank (A + B) ≤ rank A + rank B.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 8
The rank in linear algebra or matricial calculus
Let p := min(m, n), let
rank : Mm,n(R) −→ {0, 1, . . . , p}A 7−→ rank A.
The properties of the rank function are well-known in the context of linearalgebra or matricial calculus. Any book on these subjects presents them;there even are compilations of them. We recall here the most basic ones:
1 rank A = rank AT ; rank A = rank (AAT ) = rank (ATA).
2 If the product AB can be done,
rank (AB) ≤ min(rank A, rank B) (Sylvester inequality).
3 rank A = 0 if and only if A = 0; rank (cA) = rank A for c 6= 0.
4 |rank A− rank B| ≤ rank (A + B) ≤ rank A + rank B.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 8
The rank in linear algebra or matricial calculus
Theorem
1 Let A ∈Mn(R) be non-null diagonalizable, with all real eigenvalues.Then
rank A ≥ (tr A)2
tr (A2),
where tr B denotes the trace of the matrix B.
2 Let A ∈ Sn(R) be non-null and positive definite. Then
tr A
λmax(A)≤ rank A ≤ tr A
λmin(A),
where λmax(A) (resp. λmin(A)) stands for the maximal (resp.minimal) eigenvalue of A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 7
The rank in linear algebra or matricial calculus
Theorem
1 Let A ∈Mn(R) be non-null diagonalizable, with all real eigenvalues.Then
rank A ≥ (tr A)2
tr (A2),
where tr B denotes the trace of the matrix B.
2 Let A ∈ Sn(R) be non-null and positive definite. Then
tr A
λmax(A)≤ rank A ≤ tr A
λmin(A),
where λmax(A) (resp. λmin(A)) stands for the maximal (resp.minimal) eigenvalue of A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 7
The rank from the topological and semi-algebraic viewpoint
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 6
The rank from the topological and semi-algebraic viewpoint
The only (useful) topological property of the rank function is that it islower-semicontinuous: if Aν → A in Mm,n(R), then
lim infν→+∞
rank Aν ≥ rank A.
For k ∈ {0, 1, . . . , p}, consider now the following two subsets of Mm,n(R):
Sk := {A ∈Mm,n(R)| rank A ≤ k},
Σk := {A ∈Mm,n(R)| rank A = k}.
Theorem
1 Σp is an open dense subset of Sp =Mm,n(R).
2 If k < p, the interior of Σk is empty and its closure is Sk .
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 5
The rank from the topological and semi-algebraic viewpoint
The only (useful) topological property of the rank function is that it islower-semicontinuous: if Aν → A in Mm,n(R), then
lim infν→+∞
rank Aν ≥ rank A.
For k ∈ {0, 1, . . . , p}, consider now the following two subsets of Mm,n(R):
Sk := {A ∈Mm,n(R)| rank A ≤ k},
Σk := {A ∈Mm,n(R)| rank A = k}.
Theorem
1 Σp is an open dense subset of Sp =Mm,n(R).
2 If k < p, the interior of Σk is empty and its closure is Sk .
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 5
The rank from the topological and semi-algebraic viewpoint
The only (useful) topological property of the rank function is that it islower-semicontinuous: if Aν → A in Mm,n(R), then
lim infν→+∞
rank Aν ≥ rank A.
For k ∈ {0, 1, . . . , p}, consider now the following two subsets of Mm,n(R):
Sk := {A ∈Mm,n(R)| rank A ≤ k},
Σk := {A ∈Mm,n(R)| rank A = k}.
Theorem
1 Σp is an open dense subset of Sp =Mm,n(R).
2 If k < p, the interior of Σk is empty and its closure is Sk .
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 5
The rank from the topological and semi-algebraic viewpoint
Jean-Baptiste Hiriart-Urruty
Bases, outils et principes pour l'analyse variationnelle
Jean-Baptiste Hiriart-Urruty
Bases, tools and principles for variational analysis
Mathematics
L’étude mathématique des problèmes d’optimisation, ou de ceux dits variationnels de manière générale (c’est-à-dire, « toute situation où il y a quelque chose à minimiser sous des contraintes »), requiert en préalable qu’on en maîtrise les bases, les outils fondamentaux et quelques principes. Le présent ouvrage est un cours répondant en partie à cette demande, il est principalement destiné à des étudiants de Master en formation, et restreint à l’essentiel. Sont abordés successivement : La semicontinuité inférieure, les topologies faibles, les résultats fondamentaux d’ existence en optimisation ; Les conditions d’optimalité approchée ; Des développements sur la projection sur un convexe fermé, notamment sur un cône convexe fermé ; L’analyse convexe dans son rôle opératoire ; Quelques schémas de dualisation dans des problèmes d’optimisation non convexe structurés ; Une introduction aux sous-différentiels généralisés de fonctions non différentiables.
The mathematical study of optimization problems, or of the so-called variational ones (that is to say, “every situation where there is something to be minimized under constraints”), requires first having a solid understanding of bases, fundamental tools and principles. The present book consists of lecture notes that partly meet these requirements; it is mainly intended for students at the Master level, and focuses on the essentials. The following points are addressed: Lower semi-continuity, weak topologies, and fundamental existence results in optimization; Conditions for approximate optimality; Developments in the projection on a closed convex set, more specifically on a closed convex cone: Convex analysis in its operational role; Some duality schemes for specially structured non-convex optimization problems; and An introduction to the generalized sub-differentials of non-smooth functions.
ISBN 978-3-642-30734-8
Hiriart-U
rruty
Mathématiques et Applications 70
Bases, outils et principes pour l'analyse variationnelle
Jean-Baptiste Hiriart-Urruty
M&A70
Bases, outils et principes pour l'analyse variationnelle
Mathématiques et Applications 70
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 4
Best approximation in terms of rank
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 3
Best approximation in terms of rank
The best approximation problem that we consider in this section is asfollows: Given A ∈Mm,n(R) of rank r ,
(Ak)
{Minimize ‖A−M‖M ∈ Sk
The problem of course depends on the choice of ‖.‖; it is solved when ‖.‖is either the Frobenius-Schur norm or the spectral norm.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 2
Best approximation in terms of rank
Theorem (Eckart and Young)
Let A ∈Mm,n(R) and A = UΣAV be a SVD of A. Choose ‖.‖ as either‖.‖F or ‖.‖sp. Then
Ak := UDkV ,
(where Dk is obtained from D by keeping σ1, . . . , σk and putting 0 in theplace of σk+1, . . . , σr ) is a solution of the best approximation problem(Ak). For the Frobenius-Schur norm case, Ak is the unique solution in(Ak) when σk > σk+1.The optimal values in (Ak) are as following:
minM∈Sk
‖A−M‖F =
√√√√ r∑i=k+1
σ2i ;
minM∈Sk
‖A−M‖sp = σk+1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 1
Global minimization of the rank
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 0
Global minimization of the rank
Global minimization
(P)
{Minimize rank(A)
A ∈ C
Example for C. Aij given for (i , j) ∈ K ⊂ {1, . . . ,m} × {1, . . . , n}.
Theorem (”Esclarecimiento de Valparaiso”)
Every admissible point in (P) is a local minimizer.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -1
Global minimization of the rank
Global minimization
(P)
{Minimize rank(A)
A ∈ C
Example for C. Aij given for (i , j) ∈ K ⊂ {1, . . . ,m} × {1, . . . , n}.
Theorem (”Esclarecimiento de Valparaiso”)
Every admissible point in (P) is a local minimizer.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -1
Relaxed form of the rank function
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -2
Relaxed form of the rank function
For R > 0, we define
A ∈Mm,n(R) 7−→ rankR(A) :=
{rank of A if ‖A‖sp ≤ R;
+∞ otherwise.
Theorem (Fazel, 2002)
We have:
A ∈Mm,n(R) 7−→ co (rankR)(A) :=
{1R ‖A‖∗ if ‖A‖sp ≤ R;+∞ otherwise.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -3
Relaxed form of the rank function
For R > 0, we define
A ∈Mm,n(R) 7−→ rankR(A) :=
{rank of A if ‖A‖sp ≤ R;
+∞ otherwise.
Theorem (Fazel, 2002)
We have:
A ∈Mm,n(R) 7−→ co (rankR)(A) :=
{1R ‖A‖∗ if ‖A‖sp ≤ R;+∞ otherwise.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -3
Relaxed form of the rank function
Proof 1(Fazel)
Calculate the Legendre-Fenchel conjugate φ∗ of φ := rankR .
Then, compute the biconjugate φ∗∗ of φ.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -4
Relaxed form of the rank function
Proof 1(Fazel)
Calculate the Legendre-Fenchel conjugate φ∗ of φ := rankR .
Then, compute the biconjugate φ∗∗ of φ.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -4
Relaxed form of the rank function
Proof 2
Here, one begins by convexifying the restricted counting function (on Rp),and then one applies fine results by A.Lewis on how to get properties onrankR from properties on cR .
Theorem
We have
x ∈ Rp 7−→ co(cR)(x) :=
{1R ‖x‖1 if ‖x‖∞ ≤ R;+∞ otherwise.
(a) The counting function (b) The l1 norm
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -5
Relaxed form of the rank function
Proof 2
Here, one begins by convexifying the restricted counting function (on Rp),and then one applies fine results by A.Lewis on how to get properties onrankR from properties on cR .
Theorem
We have
x ∈ Rp 7−→ co(cR)(x) :=
{1R ‖x‖1 if ‖x‖∞ ≤ R;+∞ otherwise.
(c) The counting function (d) The l1 norm
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -5
Relaxed form of the rank function
Proof 3
Here we look at the problem with ”geometrical glasses”: instead ofconvexifying functions directly, we firstly convexify (appropriate) sets ofmatrices and, then, get convex hulls (or quasiconvex hulls) of functions asby-products.For k ∈ {0, 1, . . . , p} and R ≥ 0, let
SRk := Sk ∩ {A ∈Mm,n(R)| ‖A‖sp ≤ R}
= {A ∈Mm,n(R)| rank A ≤ k and ‖A‖sp ≤ R}.
Theorem
We have
co SRk = {A ∈Mm,n(R)| ‖A‖sp ≤ R and ‖A‖∗ ≤ Rk}.
Ref. J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of bounded rank.
Applications to quasiconvexification and convexification of the rank function,
Optimization Letters 6(5): 841-849 (2012).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -6
Relaxed form of the rank function
Proof 3
Here we look at the problem with ”geometrical glasses”: instead ofconvexifying functions directly, we firstly convexify (appropriate) sets ofmatrices and, then, get convex hulls (or quasiconvex hulls) of functions asby-products.For k ∈ {0, 1, . . . , p} and R ≥ 0, let
SRk := Sk ∩ {A ∈Mm,n(R)| ‖A‖sp ≤ R}
= {A ∈Mm,n(R)| rank A ≤ k and ‖A‖sp ≤ R}.
Theorem
We have
co SRk = {A ∈Mm,n(R)| ‖A‖sp ≤ R and ‖A‖∗ ≤ Rk}.
Ref. J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of bounded rank.
Applications to quasiconvexification and convexification of the rank function,
Optimization Letters 6(5): 841-849 (2012).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -6
Relaxed form of the rank function
Proof 3
Here we look at the problem with ”geometrical glasses”: instead ofconvexifying functions directly, we firstly convexify (appropriate) sets ofmatrices and, then, get convex hulls (or quasiconvex hulls) of functions asby-products.For k ∈ {0, 1, . . . , p} and R ≥ 0, let
SRk := Sk ∩ {A ∈Mm,n(R)| ‖A‖sp ≤ R}
= {A ∈Mm,n(R)| rank A ≤ k and ‖A‖sp ≤ R}.
Theorem
We have
co SRk = {A ∈Mm,n(R)| ‖A‖sp ≤ R and ‖A‖∗ ≤ Rk}.
Ref. J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of bounded rank.
Applications to quasiconvexification and convexification of the rank function,
Optimization Letters 6(5): 841-849 (2012).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -6
Relaxed form of the rank function
Proof 3
Here we look at the problem with ”geometrical glasses”: instead ofconvexifying functions directly, we firstly convexify (appropriate) sets ofmatrices and, then, get convex hulls (or quasiconvex hulls) of functions asby-products.For k ∈ {0, 1, . . . , p} and R ≥ 0, let
SRk := Sk ∩ {A ∈Mm,n(R)| ‖A‖sp ≤ R}
= {A ∈Mm,n(R)| rank A ≤ k and ‖A‖sp ≤ R}.
Theorem
We have
co SRk = {A ∈Mm,n(R)| ‖A‖sp ≤ R and ‖A‖∗ ≤ Rk}.
Ref. J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of bounded rank.
Applications to quasiconvexification and convexification of the rank function,
Optimization Letters 6(5): 841-849 (2012).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -6
Relaxed form of the rank function
Proof 3
Theorem
We have:
A ∈Mm,n(R) 7−→ (rankR)q(A) =
{d 1R ‖A‖∗e if ‖A‖sp ≤ R,+∞ otherwise,
where dae stands for the smallest integer which is larger than a.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -7
Surrogates of the rank function
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -8
Surrogates of the rank function
A proposal of a continuous surrogate of the rank is as follows:
A ∈Mm,n(R) 7−→ gsf (A) :=
{‖A‖2∗
‖A‖2F
if A 6= 0,
0 if A = 0.
Theorem (Flores, Ph.D Thesis 2011)
We have:
(G1) gsf (cA) = gsf (A) for c 6= 0.
(G2) 1 ≤ gsf (A) ≤ rank A for A 6= 0.
(G3) If all the non-zero singular values of A are equal, thengsf (A) = rank A.
(G4) rank A = 1 if and only if gsf (A) = 1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -9
Surrogates of the rank function
A proposal of a continuous surrogate of the rank is as follows:
A ∈Mm,n(R) 7−→ gsf (A) :=
{‖A‖2∗
‖A‖2F
if A 6= 0,
0 if A = 0.
Theorem (Flores, Ph.D Thesis 2011)
We have:
(G1) gsf (cA) = gsf (A) for c 6= 0.
(G2) 1 ≤ gsf (A) ≤ rank A for A 6= 0.
(G3) If all the non-zero singular values of A are equal, thengsf (A) = rank A.
(G4) rank A = 1 if and only if gsf (A) = 1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -9
Surrogates of the rank function
Theorem (Lokam, 2001)
For any non-null B ∈Mm,n(R), define
A ∈Mm,n(R 7−→ gB(A) :=
|〈〈A,B〉〉|‖A‖sp‖B‖sp
if A 6= 0,
0 if A = 0.
ThengB(A) ≤ rank A for all A ∈Mm,n(R).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -10
Regularization- approximation of the rank function
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -11
Regularization- approximation of the rank function
Smoothed versions
The underlying idea is as follows: Since
rank A = c[σ1(A), . . . , σp(A)]
=
p∑i=1
θ[σi (A)],
where θ(x) = 1 if x 6= 0, θ(0) = 0 (θ is the counting function on R), wehave to design some smooth approximation of the θ function.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -12
Regularization- approximation of the rank function
Smoothed versions
A first example was proposed by Hiriart-Urruty (2009), it is as following:For ε > 0, let θε be defined as
x ∈ R 7−→ θε(x) := 1− e−x2/ε.
The resulting approximation of the rank function is
A ∈Mm,n(R) 7−→ Rε(A) :=
p∑i=1
[1− e−σ2i (A)/ε].
An alternate expression of the Rε function is
Rε(A) = p − tr(e−ATA/ε).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -13
Regularization- approximation of the rank function
Smoothed versions
A first example was proposed by Hiriart-Urruty (2009), it is as following:For ε > 0, let θε be defined as
x ∈ R 7−→ θε(x) := 1− e−x2/ε.
The resulting approximation of the rank function is
A ∈Mm,n(R) 7−→ Rε(A) :=
p∑i=1
[1− e−σ2i (A)/ε].
An alternate expression of the Rε function is
Rε(A) = p − tr(e−ATA/ε).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -13
Regularization- approximation of the rank function
Smoothed versions
A first example was proposed by Hiriart-Urruty (2009), it is as following:For ε > 0, let θε be defined as
x ∈ R 7−→ θε(x) := 1− e−x2/ε.
The resulting approximation of the rank function is
A ∈Mm,n(R) 7−→ Rε(A) :=
p∑i=1
[1− e−σ2i (A)/ε].
An alternate expression of the Rε function is
Rε(A) = p − tr(e−ATA/ε).
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -13
Regularization- approximation of the rank function
Smoothed versions
Theorem
We have
(i) Rε(A) ≤ rank A for all ε > 0.
(ii) The sequence of functions (Rε)ε>0 increases when ε decreases, andRε(A)→ rank A for all A when ε→ 0.
(iii) If A 6= 0 and r = rank A,
rank A− Rε(A) ≤ ε
r∑i=1
1
σ2i (A)
,
≤ ε2r∑
i=1
1
σ4i (A)
.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -14
Regularization- approximation of the rank function
Smoothed versions
Theorem
We have
(i) Rε(A) ≤ rank A for all ε > 0.
(ii) The sequence of functions (Rε)ε>0 increases when ε decreases, andRε(A)→ rank A for all A when ε→ 0.
(iii) If A 6= 0 and r = rank A,
rank A− Rε(A) ≤ ε
r∑i=1
1
σ2i (A)
,
≤ ε2r∑
i=1
1
σ4i (A)
.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -14
Regularization- approximation of the rank function
Smoothed versions
Theorem
We have
(i) Rε(A) ≤ rank A for all ε > 0.
(ii) The sequence of functions (Rε)ε>0 increases when ε decreases, andRε(A)→ rank A for all A when ε→ 0.
(iii) If A 6= 0 and r = rank A,
rank A− Rε(A) ≤ ε
r∑i=1
1
σ2i (A)
,
≤ ε2r∑
i=1
1
σ4i (A)
.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -14
Regularization- approximation of the rank function
Smoothed versions
Another proposal for approximating the rank function, a quite recent one,is due to Zhao. It consists of using, for qll ε > 0, the following evenapproximation of the θ function:
x ∈ R 7−→ τε(x) :=x2
x2 + ε.
The resulting approximation of the rank function is
A ∈Mm,n(R) 7−→ Zε(A) :=
p∑i=1
σ2i (A)
σ2i (A) + ε
.
Alternate expressions of the Zε function are:
Zε(A) = tr[A(ATA + εIn)−1AT ]
= n − εtr(ATA + εIn)−1.
Ref. Y.-B.Zhao, An approximation theory of matrix rank minimization and its
application to quadratic equations, Linear Algebra and Its Applications (2012), 77-93.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -15
Regularization- approximation of the rank function
Smoothed versions
Another proposal for approximating the rank function, a quite recent one,is due to Zhao. It consists of using, for qll ε > 0, the following evenapproximation of the θ function:
x ∈ R 7−→ τε(x) :=x2
x2 + ε.
The resulting approximation of the rank function is
A ∈Mm,n(R) 7−→ Zε(A) :=
p∑i=1
σ2i (A)
σ2i (A) + ε
.
Alternate expressions of the Zε function are:
Zε(A) = tr[A(ATA + εIn)−1AT ]
= n − εtr(ATA + εIn)−1.
Ref. Y.-B.Zhao, An approximation theory of matrix rank minimization and its
application to quadratic equations, Linear Algebra and Its Applications (2012), 77-93.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -15
Regularization- approximation of the rank function
Smoothed versions
Another proposal for approximating the rank function, a quite recent one,is due to Zhao. It consists of using, for qll ε > 0, the following evenapproximation of the θ function:
x ∈ R 7−→ τε(x) :=x2
x2 + ε.
The resulting approximation of the rank function is
A ∈Mm,n(R) 7−→ Zε(A) :=
p∑i=1
σ2i (A)
σ2i (A) + ε
.
Alternate expressions of the Zε function are:
Zε(A) = tr[A(ATA + εIn)−1AT ]
= n − εtr(ATA + εIn)−1.
Ref. Y.-B.Zhao, An approximation theory of matrix rank minimization and its
application to quadratic equations, Linear Algebra and Its Applications (2012), 77-93.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -15
Regularization- approximation of the rank function
Smoothed versions
Theorem (Zhao, 2012)
We have
(i) Zε(A) ≤ rank A for all ε > 0.
(ii) The sequence of functions (Zε)ε>0 increases when ε decreases andZε(A)→ rank A for all A when ε→ 0.
(iii) If A 6= 0 and r = rank A,
rank A− Zε(A) =r∑
i=1
ε
σ2i (A) + ε
≤ εr∑
i=1
1
σ2i (A)
.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -16
Regularization- approximation of the rank function
Smoothed versions
Theorem (Zhao, 2012)
We have
(i) Zε(A) ≤ rank A for all ε > 0.
(ii) The sequence of functions (Zε)ε>0 increases when ε decreases andZε(A)→ rank A for all A when ε→ 0.
(iii) If A 6= 0 and r = rank A,
rank A− Zε(A) =r∑
i=1
ε
σ2i (A) + ε
≤ εr∑
i=1
1
σ2i (A)
.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -16
Regularization- approximation of the rank function
Smoothed versions
Theorem (Zhao, 2012)
We have
(i) Zε(A) ≤ rank A for all ε > 0.
(ii) The sequence of functions (Zε)ε>0 increases when ε decreases andZε(A)→ rank A for all A when ε→ 0.
(iii) If A 6= 0 and r = rank A,
rank A− Zε(A) =r∑
i=1
ε
σ2i (A) + ε
≤ εr∑
i=1
1
σ2i (A)
.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -16
Regularization- approximation of the rank function
Moreau-Yosida approximation (or Moreau envelope)
By definition,
(rankR)λ(A) = infB ∈ Mm,n(R)‖B‖sp≤R
{rank B +
1
2λ‖A− B‖2
F
}
The limiting case, i.e. with R = +∞, is
(rank)λ(A) = infB∈Mm,n(R)
{rank B +
1
2λ‖A− B‖2
F
}
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -17
Regularization- approximation of the rank function
Moreau-Yosida approximation (or Moreau envelope)
Theorem
We have, for all A ∈Mm,n(R) with rank r ≥ 1:
(rank)λ(A) =1
2λ‖A‖2
F −1
2λ
r∑i=1
[σ2i (A)− 2λ]+.
One element in Proxλ(rank)(A), is provided by B := UΣBV , where
• U and V are orthogonal matrices such that A = UΣAV , withΣA = diagm,n[σi (A), . . . , σr (A), 0, . . . , 0] (a singular valuedecomposition of A with σ1(A) ≥ · · · ≥ σr (A) > 0;)
•
σi (B) =
σi (A) if σi (A) >
√2λ
0 or σi (A) if σi (A) =√
2λ
0 if σi (A) <√
2λ.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -18
Regularization- approximation of the rank function
Moreau-Yosida approximation (or Moreau envelope)
Theorem
We have, for all A ∈Mm,n(R) with rank r ≥ 1:
(rank)λ(A) =1
2λ‖A‖2
F −1
2λ
r∑i=1
[σ2i (A)− 2λ]+.
One element in Proxλ(rank)(A), is provided by B := UΣBV , where
• U and V are orthogonal matrices such that A = UΣAV , withΣA = diagm,n[σi (A), . . . , σr (A), 0, . . . , 0] (a singular valuedecomposition of A with σ1(A) ≥ · · · ≥ σr (A) > 0;)
•
σi (B) =
σi (A) if σi (A) >
√2λ
0 or σi (A) if σi (A) =√
2λ
0 if σi (A) <√
2λ.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -18
Regularization- approximation of the rank function
Moreau-Yosida approximations (or Moreau’s envelope)
Theorem
We have, for all A ∈Mm,n(R) of rank r ≥ 1:
(rankR)λ(A) = 12λ‖A‖
2F −
12λ
∑ri=1{σ2
i (A)− [(σi (A)− R)+]2 − 2λ}+.
One element in Proxλ(rankR)(A), is provided by B := UΣBV withΣB = diagm,n[σ1(B), . . . , σp(B)], where
• If√
2λ ≥ R, σi (B) :=
R if σi (A) > 2λ+λ2
2λ ,
0 or R if σi (A) = 2λ+λ2
2λ ,
0 if σi (A) < 2λ+λ2
2λ .
• If√
2λ < R, σi (B) :=
R if σi (A) > R,
σi (A) if√
2λ < σi (A) ≤ R,
0 or σi (A) if√
2λ = σi (A),
0 if σi (A) <√
2λ.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -19
Regularization- approximation of the rank function
Moreau-Yosida approximations (or Moreau’s envelope)
Theorem
We have, for all A ∈Mm,n(R) of rank r ≥ 1:
(rankR)λ(A) = 12λ‖A‖
2F −
12λ
∑ri=1{σ2
i (A)− [(σi (A)− R)+]2 − 2λ}+.
One element in Proxλ(rankR)(A), is provided by B := UΣBV withΣB = diagm,n[σ1(B), . . . , σp(B)], where
• If√
2λ ≥ R, σi (B) :=
R if σi (A) > 2λ+λ2
2λ ,
0 or R if σi (A) = 2λ+λ2
2λ ,
0 if σi (A) < 2λ+λ2
2λ .
• If√
2λ < R, σi (B) :=
R if σi (A) > R,
σi (A) if√
2λ < σi (A) ≤ R,
0 or σi (A) if√
2λ = σi (A),
0 if σi (A) <√
2λ.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -19
Regularization- approximation of the rank function
Moreau-Yosida approximation
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -20
Regularization- approximation of the rank function
Moreau-Yosida approximation
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -21
The generalized subdifferentials of the rank function
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -22
The generalized subdifferentials of the rank function
The generalized subdifferentials of the rank function
Which one?All the generalized subdifferentials of the rank function coincide. We note∂(rank) for the common subdifferential. For A ∈Mm,n(R), ∂(rank)(A) isconstructed as follows:
Consider the matrices U ∈ O(m) and V ∈ O(n) such that
U.Diagm,n(σ(A)).V = A
(in other words, we collect all the orthogonal matrices U and V whichgive a singular value decomposition of A).
Consider the ”diagonal” matrices Diagm,n(x∗), where x∗ ∈ Rp is suchthat x∗i = 0 for all i = 1, . . . , r (recall that r = rank A).
Then, collect all the matrices of the form UDiagm,n(x∗)V .
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -23
The generalized subdifferentials of the rank function
The generalized subdifferentials of the rank function
In a single formula,
∂(rank)(A)= {UDiag(x∗)V | U ∈ O(m),V ∈ O(n) such that UΣAV = A
and x∗i = 0 for all i = 1, . . . , r}.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -24
The generalized subdifferentials of the rank function
The generalized subdifferentials of the rank function
Alternate expression of ∂(rank)(A)
∂(rank)(A) = N(AT )⊗ N(A)
={∑
i ,j aijαiβTj | (αi ) is a basis of N(AT )
(βj) is a basis of N(A)} .
withN(AT ) = {u ∈ Rm| ATu = 0}
N(A) = {v ∈ Rn| Av = 0}.
Dimension of ∂(rank)(A)
∂(rank)(A) is a vector space of dimension (m− r)(n− r) with r = rank A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -25
The generalized subdifferentials of the rank function
The generalized subdifferentials of the rank function
Alternate expression of ∂(rank)(A)
∂(rank)(A) = N(AT )⊗ N(A)
={∑
i ,j aijαiβTj | (αi ) is a basis of N(AT )
(βj) is a basis of N(A)} .
withN(AT ) = {u ∈ Rm| ATu = 0}
N(A) = {v ∈ Rn| Av = 0}.
Dimension of ∂(rank)(A)
∂(rank)(A) is a vector space of dimension (m− r)(n− r) with r = rank A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -25
The generalized subdifferentials of the rank function
Illustration
For m = n = 2, we have
If A = 0 then ∂(rank)(0) =M2,2(R).
If rank A = 2 then ∂(rank)(A) = {0}.If rank A = 1 then ∂(rank)(A) is a vector space of dimension 1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -26
Further notions related to the rank
Contents
1 The rank in linear algebra or matricial calculus
2 The rank from the topological and semi-algebraic viewpoint
3 Best approximation in terms of rank
4 Global minimization of the rank
5 Relaxed form of the rank function
6 Surrogates of the rank function
7 Regularization- approximation of the rank function
8 The generalized subdifferentials of the rank function
9 Further notions related to the rank
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -27
Further notions related to the rank
The spark of a matrix
Definition
Given A ∈Mm,n(R), the spark of A is the smallest positive integer k suchthat there exists a set of k columns of A which are linearly dependent.
If A is of full column rank, we should adopt +∞ as for the spark of A.
If one column of A is a zero-column: then spark A = 1.
If A does not contain any zero-column and is not of full column rank,
2 ≤ sparkA ≤ rank A + 1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -28
Further notions related to the rank
The spark of a matrix
Definition
Given A ∈Mm,n(R), the spark of A is the smallest positive integer k suchthat there exists a set of k columns of A which are linearly dependent.
If A is of full column rank, we should adopt +∞ as for the spark of A.
If one column of A is a zero-column: then spark A = 1.
If A does not contain any zero-column and is not of full column rank,
2 ≤ sparkA ≤ rank A + 1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -28
Further notions related to the rank
The matrix rigidity
Definition
The rigidity of A ∈Mm,n(R), denoted by RA(k) is the smallest number ofentries that need to be changed in order to reduce the rank of A below k .
A variational formulation of RA(k) can easily be proposed as a rankminimization problem:
RA(k) = minM∈A+Sk
c(M),
where c(M) denotes the number of nonzero entries in the matrix M andSk = {S ∈Mn(R)| rank S ≤ k}
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -29
Further notions related to the rank
The matrix rigidity
Definition
The rigidity of A ∈Mm,n(R), denoted by RA(k) is the smallest number ofentries that need to be changed in order to reduce the rank of A below k .
A variational formulation of RA(k) can easily be proposed as a rankminimization problem:
RA(k) = minM∈A+Sk
c(M),
where c(M) denotes the number of nonzero entries in the matrix M andSk = {S ∈Mn(R)| rank S ≤ k}
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -29
Further notions related to the rank
The cp-rank of matrix
A matrix M ∈ Sn(R) is said to be copositive if
〈Mx , x〉 ≥ 0 for all x ∈ Rn+.
A matrix A ∈ Sn(R) is then called completely positive if
〈〈A,M〉〉 ≥ 0 for all copositive matrices M.
Definition
The smallest number of columns of a matrix B in a factorizationA = BBT of a completely positive matrix A is called the cp-rank of A anddenoted as cp-rank A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -30
Further notions related to the rank
The cp-rank of matrix
A matrix M ∈ Sn(R) is said to be copositive if
〈Mx , x〉 ≥ 0 for all x ∈ Rn+.
A matrix A ∈ Sn(R) is then called completely positive if
〈〈A,M〉〉 ≥ 0 for all copositive matrices M.
Definition
The smallest number of columns of a matrix B in a factorizationA = BBT of a completely positive matrix A is called the cp-rank of A anddenoted as cp-rank A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -30
Further notions related to the rank
The cp-rank of matrix
A matrix M ∈ Sn(R) is said to be copositive if
〈Mx , x〉 ≥ 0 for all x ∈ Rn+.
A matrix A ∈ Sn(R) is then called completely positive if
〈〈A,M〉〉 ≥ 0 for all copositive matrices M.
Definition
The smallest number of columns of a matrix B in a factorizationA = BBT of a completely positive matrix A is called the cp-rank of A anddenoted as cp-rank A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -30
Further notions related to the rank
The cp-rank of matrix
For the zero matrix (which indeed lies in the cone CPn(R)), we adoptby convention that its cp-rank is 0.
For matrices A ∈ Sn(R) which are not in CPn(R), we posecp-rank A = +∞.
If one wishes to compare the rank and the cp-rank of a matrix, one gets atthe following inequality:
rank A ≤ cp-rank A,
for all completely positive matrices, hence for all symmetric matrices.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -31
Further notions related to the rank
The cp-rank of matrix
For the zero matrix (which indeed lies in the cone CPn(R)), we adoptby convention that its cp-rank is 0.
For matrices A ∈ Sn(R) which are not in CPn(R), we posecp-rank A = +∞.
If one wishes to compare the rank and the cp-rank of a matrix, one gets atthe following inequality:
rank A ≤ cp-rank A,
for all completely positive matrices, hence for all symmetric matrices.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -31
Further notions related to the rank
The cp-rank of a matrix
Theorem
(i) If A and B are completely positive,
cp-rank (A + B) ≤ cp-rank A + cp-rank B.
(ii) If A is completely positive and c > 0,
cp-rank (cA) = cp-rank A.
(iii) The cp-rank is a lower-semicontinuous function: If (Aν)ν is asequence of completely positive matrices converging to A, then A iscompletely positive and
lim infν→+∞
cp-rank Aν ≥ cp-rank A.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -32
Further notions related to the rank
Upper bounds for the cp-rank
Theorem (Barioli and Berman, 2003)
(i) For every completely positive matrix of rank k ≥ 2,
cp-rank A ≤ k(k + 1)
2− 1.
(ii) For every integer k ≥ 2, there exists a completely positive matrix A
whose rank is k and whose cp-rank is k(k+1)2 − 1.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -33
Further notions related to the rank
Upper bounds for the cp-rank
The DJL Conjecture (1994)
If A is an n × n completely positive matrix, n ≥ 4, then
cp-rank A ≤ n2
4.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -34
Further notions related to the rank
The convex relaxed form
For R > 0, consider
A ∈ Sn(R) 7→ cp-rankR(A) :=
{cp-rank A if A is CP and ‖A‖∗ ≤ R;
+∞ otherwise.
Theorem
We have
A ∈ Sn(R) 7→ co(cp-rankR)(A) :=
{1R ‖A‖∗ if A is CP and ‖A‖∗ ≤ R;+∞ otherwise.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -35
Further notions related to the rank
References
J.-B.Hiriart-Urruty, When only global optimization matters, J. of GlobalOptimization (DOI: 10.1007/s10898-011-9826-7) (2012).
J.-B.Hiriart-Urruty and J.Malick, A fresh variational analysis look at theworld of the positive semidefinite matrices, J. of Optimization Theory andApplications, Vol 153(3) (2012), 551-577.
J.-B.Hiriart-Urruty and H.Y.Le, Convexifying the set of matrices of boundedrank: applications to the quasiconvexification and convexification of therank function, Optimization Letters 6(5): 841-849 (2012).
H.Y.Le, Confexifying the Counting Function on Rp for Convexifying theRank Function on Mm,n(R), J. of Convex Analysis Vol 19(2) (2012).
H.Y.Le, The generalized subdifferentials of the rank function, Optimization
Letters (2012), DOI: 10.1007/s11590-012-0456-x.
JBHU–LE (IMT, Toulouse) A variational approach of the rank function. Septembre 2012 -36