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Transversality, Subtransversality and Intrinsic Transversality of Pairs of Sets Alexander Kruger Centre for Informatics and Applied Optimization, Faculty of Science and Technology Federation University Australia, Ballarat

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Transversality, Subtransversality and Intrinsic Transversality of Pairs of Sets

Alexander Kruger

Centre for Informatics and Applied Optimization, Faculty of Science and Technology Federation University Australia, Ballarat a.kruger@federation.edu.au

RMIT Optimisation Group, Melbourne, 3 March 2017

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 1 / 34

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Regularity/Transversality

Constraint qualifications Qualification conditions in subdifferential calculus Qualification conditions in convergence analysis Banach open mapping principle Lyusternik–Graves theorem Transversality of sets in differential geometry Separation theorem Optimality/stationarity

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 2 / 34

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Regularity/Transversality

Constraint qualifications Qualification conditions in subdifferential calculus Qualification conditions in convergence analysis Banach open mapping principle Lyusternik–Graves theorem Transversality of sets in differential geometry Separation theorem Optimality/stationarity Regularity

  • f (multi)functions

⇐ ⇒ Transversality

  • f collections of sets

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 2 / 34

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Feasibility Problem and Alternating Projections

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 3 / 34

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Feasibility Problem and Alternating Projections

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 4 / 34

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Feasibility Problem and Alternating Projections

Linear convergence (with rate c ∈ (0, 1)): d(xk+1, ¯ x) ≤ cd(xk, ¯ x)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 4 / 34

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Feasibility Problem and Alternating Projections

Linear convergence (with rate c ∈ (0, 1)): d(xk+1, ¯ x) ≤ cd(xk, ¯ x)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 4 / 34

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Feasibility Problem and Alternating Projections

Linear convergence (with rate c ∈ (0, 1)): d(xk+1, ¯ x) ≤ cd(xk, ¯ x)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 5 / 34

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Feasibility Problem and Alternating Projections

Linear convergence (with rate c ∈ (0, 1)): d(xk+1, ¯ x) ≤ cd(xk, ¯ x) No linear convergence

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 5 / 34

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Outline

1

Transversality/Subtransversality

2

Extremality

3

Dual characterizations of transversality/subtransversality

4

Intrinsic transversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 6 / 34

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Outline

1

Transversality/Subtransversality

2

Extremality

3

Dual characterizations of transversality/subtransversality

4

Intrinsic transversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 7 / 34

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Subtransversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition

{A, B} is subtransversal at ¯ x if ∃α, δ > 0 such that αd (x, A ∩ B) ≤ max{d(x, A), d(x, B)} ∀x ∈ Bδ(¯ x)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 8 / 34

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Subtransversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition

{A, B} is subtransversal at ¯ x if ∃α, δ > 0 such that αd (x, A ∩ B) ≤ max{d(x, A), d(x, B)} ∀x ∈ Bδ(¯ x) (Dolecki, 1982); (Ioffe, 1989); (local) linear regularity (Bauschke, Borwein, 1993); linear estimate, linear coherence (Penot, 1998, 2013); metric inequality (Ngai, Th´ era, 2001); subtransversality (Ioffe, 2015)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 8 / 34

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Subtransversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition

{A, B} is subtransversal at ¯ x if ∃α, δ > 0 such that αd (x, A ∩ B) ≤ max{d(x, A), d(x, B)} ∀x ∈ Bδ(¯ x) (Dolecki, 1982); (Ioffe, 1989); (local) linear regularity (Bauschke, Borwein, 1993); linear estimate, linear coherence (Penot, 1998, 2013); metric inequality (Ngai, Th´ era, 2001); subtransversality (Ioffe, 2015) str[A, B](¯ x) := sup{α in the above condition}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 8 / 34

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Transversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition (K., 2005; K., Luke, Thao, 2016)

{A, B} is transversal at ¯ x if ∃α, δ > 0 such that αd (x, (A − x1) ∩ (B − x2)) ≤ max{d(x, A − x1), d(x, B − x2)} ∀x ∈ Bδ(¯ x), x1, x2 ∈ δB

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 9 / 34

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Transversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition (K., 2005; K., Luke, Thao, 2016)

{A, B} is transversal at ¯ x if ∃α, δ > 0 such that αd (x, (A − x1) ∩ (B − x2)) ≤ max{d(x, A − x1), d(x, B − x2)} ∀x ∈ Bδ(¯ x), x1, x2 ∈ δB tr[A, B](¯ x) := sup{α in the above definition}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 9 / 34

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Transversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition (K., 2005; K., Luke, Thao, 2016)

{A, B} is transversal at ¯ x if ∃α, δ > 0 such that αd (x, (A − x1) ∩ (B − x2)) ≤ max{d(x, A − x1), d(x, B − x2)} ∀x ∈ Bδ(¯ x), x1, x2 ∈ δB tr[A, B](¯ x) := sup{α in the above definition} 0 ≤ tr[A, B](¯ x) ≤ str[A, B](¯ x) Transversality = ⇒ Subtransversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 9 / 34

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Transversality vs Subtransversality

Transversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 10 / 34

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Transversality vs Subtransversality

Transversality Subtransversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 10 / 34

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Transversality vs Subtransversality

Transversality Subtransversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 11 / 34

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Subtransversality vs Alternating Projections

X – Euclidean space, A, B ⊂ X closed and convex, ¯ x ∈ A ∩ B

Theorem (Bauschke, Borwein, 1993)

{A, B} is subtransversal at ¯ x ⇒ alternating projections converge linearly for any starting point

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 12 / 34

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Subtransversality vs Alternating Projections

X – Euclidean space, A, B ⊂ X closed and convex, ¯ x ∈ A ∩ B

Theorem (Bauschke, Borwein, 1993)

{A, B} is subtransversal at ¯ x ⇒ alternating projections converge linearly for any starting point

Theorem (Luke, Thao, Teboulle, to appear)

Alternating projections converge linearly for any starting point close to ¯ x ⇒ {A, B} is subtransversal at ¯ x

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 12 / 34

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Subtransversality vs Alternating Projections

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 13 / 34

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Transversality vs Alternating Projections

X – Euclidean space, A, B ⊂ X closed, ¯ x ∈ A ∩ B

Theorem (Drusvyatskiy, Ioffe, Lewis, 2015)

{A, B} is transversal at ¯ x ⇒ alternating projections converge linearly for any starting point close to ¯ x

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 14 / 34

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Transversality vs Alternating Projections

X – Euclidean space, A, B ⊂ X closed, ¯ x ∈ A ∩ B

Theorem (Drusvyatskiy, Ioffe, Lewis, 2015)

{A, B} is transversal at ¯ x ⇒ alternating projections converge linearly for any starting point close to ¯ x Lewis & Malick, 2008; Lewis, Luke & Malick, 2009

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 14 / 34

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Outline

1

Transversality/Subtransversality

2

Extremality

3

Dual characterizations of transversality/subtransversality

4

Intrinsic transversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 15 / 34

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Transversality vs Extremality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition (K., Mordukhovich, 1980; K., 1981)

{A, B} is locally extremal at ¯ x if ∃δ > 0 such that ∀ε > 0 ∃x1, x2 ∈ εB such that (A − x1) ∩ (B − x2) ∩ Bδ(¯ x) = ∅

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 16 / 34

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Transversality vs Extremality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition (K., Mordukhovich, 1980; K., 1981)

{A, B} is locally extremal at ¯ x if ∃δ > 0 such that ∀ε > 0 ∃x1, x2 ∈ εB such that (A − x1) ∩ (B − x2) ∩ Bδ(¯ x) = ∅

Proposition (K., 2005)

{A, B} is locally extremal at ¯ x ⇒ {A, B} is not transversal at ¯ x

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 16 / 34

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Examples: Extremality

¯ x A B

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 17 / 34

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Examples: Extremality

A B

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 17 / 34

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Examples: Extremality

¯ x A B ¯ x A B ¯ x A B ¯ x A

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 17 / 34

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Examples: Extremality

¯ x A B ¯ x A B ¯ x A B ¯ x A Subtransversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 17 / 34

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Outline

1

Transversality/Subtransversality

2

Extremality

3

Dual characterizations of transversality/subtransversality

4

Intrinsic transversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 18 / 34

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Dual Characterizations: Normals

X – normed vector space, A ⊂ X, ¯ x ∈ A Fr´ echet normal cone to A at ¯ x: NA(¯ x) :=

  • x∗ ∈ X ∗ |

lim sup

a→¯ x, a∈A\{¯ x}

x∗, a − ¯ x a − ¯ x ≤ 0

  • Alexander Kruger (FedUni, Ballarat)

Transversality of Pairs of Sets RMIT, 3 March 2017 19 / 34

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Dual Characterizations: Normals

X – normed vector space, A ⊂ X, ¯ x ∈ A Fr´ echet normal cone to A at ¯ x: NA(¯ x) :=

  • x∗ ∈ X ∗ |

lim sup

a→¯ x, a∈A\{¯ x}

x∗, a − ¯ x a − ¯ x ≤ 0

  • dim X < ∞

Limiting normal cone to A at ¯ x: NA(¯ x) := Lim sup

a→¯ x, a∈A

NA(a) :=

  • x∗ = lim

k→∞ x∗ k | x∗ k ∈ NA(ak), ak ∈ A, ak → ¯

x

  • Alexander Kruger (FedUni, Ballarat)

Transversality of Pairs of Sets RMIT, 3 March 2017 19 / 34

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Dual Characterizations: Normals

X – normed vector space, A ⊂ X, ¯ x ∈ A Fr´ echet normal cone to A at ¯ x: NA(¯ x) :=

  • x∗ ∈ X ∗ |

lim sup

a→¯ x, a∈A\{¯ x}

x∗, a − ¯ x a − ¯ x ≤ 0

  • dim X < ∞

Limiting normal cone to A at ¯ x: NA(¯ x) := Lim sup

a→¯ x, a∈A

NA(a) :=

  • x∗ = lim

k→∞ x∗ k | x∗ k ∈ NA(ak), ak ∈ A, ak → ¯

x

  • X – Euclidean space,

A – closed Proximal normal cone to A at ¯ x: Np

A(¯

x) := cone

  • P−1

A (¯

x) − ¯ x

  • Alexander Kruger (FedUni, Ballarat)

Transversality of Pairs of Sets RMIT, 3 March 2017 19 / 34

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Dual Characterizations: Normals

X – Euclidean space, A – closed, ¯ x ∈ A NA(¯ x) =

  • v ∈ X |

lim sup

a→¯ x, a∈A\{¯ x}

v, a − ¯ x a − ¯ x ≤ 0

  • Np

A(¯

x) = cone

  • P−1

A (¯

x) − ¯ x

  • NA(¯

x) = Lim sup

a→¯ x, a∈A

Np

A(a) = Lim sup a→¯ x, a∈A

NA(a) Np

A(¯

x) ⊆ NA(¯ x) ⊆ NA(¯ x)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 20 / 34

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Dual Characterizations: Transversality

X – Asplund space, A, B – closed, ¯ x ∈ A ∩ B

Theorem (K., 2005)

{A, B} is transversal at ¯ x ⇐ ⇒ ∃α, δ > 0 such that x∗

1 + x∗ 2 > α ∀a ∈ A ∩ Bδ(¯

x), b ∈ B ∩ Bδ(¯ x), x∗

1 ∈ NA(a),

x∗

2 ∈ NB(b), satisfying x∗ 1 + x∗ 2 = 1

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 21 / 34

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Dual Characterizations: Transversality

X – Asplund space, A, B – closed, ¯ x ∈ A ∩ B

Theorem (K., 2005)

{A, B} is transversal at ¯ x ⇐ ⇒ ∃α, δ > 0 such that x∗

1 + x∗ 2 > α ∀a ∈ A ∩ Bδ(¯

x), b ∈ B ∩ Bδ(¯ x), x∗

1 ∈ NA(a),

x∗

2 ∈ NB(b), satisfying x∗ 1 + x∗ 2 = 1

tr[A, B](¯ x) = sup{α in the above condition}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 21 / 34

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Dual Characterizations: Transversality

dim X < ∞, A, B – closed, ¯ x ∈ A ∩ B {A, B} is transversal at ¯ x ⇐ ⇒ ∃α > 0 such that x∗

1 + x∗ 2 > α

∀x∗

1 ∈ NA(¯

x), x∗

2 ∈ NB(¯

x), satisfying x∗

1 + x∗ 2 = 1

tr[A, B](¯ x) = sup{α in the above condition}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 22 / 34

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Dual Characterizations: Transversality

dim X < ∞, A, B – closed, ¯ x ∈ A ∩ B {A, B} is transversal at ¯ x ⇐ ⇒ ∃α > 0 such that x∗

1 + x∗ 2 > α

∀x∗

1 ∈ NA(¯

x), x∗

2 ∈ NB(¯

x), satisfying x∗

1 + x∗ 2 = 1

tr[A, B](¯ x) = sup{α in the above condition} {A, B} is transversal at ¯ x ⇐ ⇒ NA(¯ x) ∩ (−NB(¯ x)) = {0}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 22 / 34

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Dual Characterizations: Transversality

dim X < ∞, A, B – closed, ¯ x ∈ A ∩ B {A, B} is transversal at ¯ x ⇐ ⇒ ∃α > 0 such that x∗

1 + x∗ 2 > α

∀x∗

1 ∈ NA(¯

x), x∗

2 ∈ NB(¯

x), satisfying x∗

1 + x∗ 2 = 1

tr[A, B](¯ x) = sup{α in the above condition} {A, B} is transversal at ¯ x ⇐ ⇒ NA(¯ x) ∩ (−NB(¯ x)) = {0} Transversality (Clarke et al, 1998); normal qualification condition (Mordukhovich, 2006; Penot, 2013); regular intersection (Lewis & Malick, 2008); linearly regular intersection (Lewis et al, 2009); alliedness property (Penot, 2013); transversal intersection (Ioffe, 2015) . . .

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 22 / 34

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Dual Characterizations: Extremality

¯ x A B

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 23 / 34

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Dual Characterizations: Extremality

¯ x A B ¯ x A B ¯ x A B ¯ x A

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 23 / 34

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Dual Characterizations: Extremality

¯ x A B ¯ x A B ¯ x A B ¯ x A Extremal principle – generalized separabilty (K., Mordukhovich, 1980; K., 1981; Mordukhovich, Shao, 1996)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 23 / 34

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Dual Characterizations: Subtransversality

X – Asplund space, A, B – closed, ¯ x ∈ A ∩ B

Theorem (K., Luke, Thao, 2016)

{A, B} is subtransversal at ¯ x if ∃α, δ > 0 such that x∗

1 + x∗ 2 > α

∀a ∈ (A \ B) ∩ Bδ(¯ x), b ∈ (B \ A) ∩ Bδ(¯ x), x ∈ Bδ(¯ x) with x = a, x = b, 1 − δ < x−a

x−b < 1 + δ, and all nonzero x∗ 1 ∈ NA(a),

x∗

2 ∈ NB(b) satisfying x∗ 1 + x∗ 2 = 1,

x∗

1, x − a

x∗

1 x − a > 1 − δ,

x∗

2, x − b

x∗

2 x − b > 1 − δ

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 24 / 34

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Dual Characterizations: Subtransversality

X – Asplund space, A, B – closed, ¯ x ∈ A ∩ B

Theorem (K., Luke, Thao, 2016)

{A, B} is subtransversal at ¯ x if ∃α, δ > 0 such that x∗

1 + x∗ 2 > α

∀a ∈ (A \ B) ∩ Bδ(¯ x), b ∈ (B \ A) ∩ Bδ(¯ x), x ∈ Bδ(¯ x) with x = a, x = b, 1 − δ < x−a

x−b < 1 + δ, and all nonzero x∗ 1 ∈ NA(a),

x∗

2 ∈ NB(b) satisfying x∗ 1 + x∗ 2 = 1,

x∗

1, x − a

x∗

1 x − a > 1 − δ,

x∗

2, x − b

x∗

2 x − b > 1 − δ

str[A, B](¯ x) ≥ sup{α in the above theorem}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 24 / 34

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Outline

1

Transversality/Subtransversality

2

Extremality

3

Dual characterizations of transversality/subtransversality

4

Intrinsic transversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 25 / 34

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Intrinsic Transversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition

{A, B} is intrinsically transversal at ¯ x if ∃α, δ > 0 such that x∗

1 + x∗ 2 > α ∀a ∈ (A \ B) ∩ Bδ(¯

x), b ∈ (B \ A) ∩ Bδ(¯ x), x ∈ Bδ(¯ x) with x = a, x = b, 1 − δ < x−a

x−b < 1 + δ, and all nonzero

x∗

1 ∈ NA(a), x∗ 2 ∈ NB(b) satisfying x∗ 1 + x∗ 2 = 1,

x∗

1, x − a

x∗

1 x − a > 1 − δ,

x∗

2, x − b

x∗

2 x − b > 1 − δ

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 26 / 34

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Intrinsic Transversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition

{A, B} is intrinsically transversal at ¯ x if ∃α, δ > 0 such that x∗

1 + x∗ 2 > α ∀a ∈ (A \ B) ∩ Bδ(¯

x), b ∈ (B \ A) ∩ Bδ(¯ x), x ∈ Bδ(¯ x) with x = a, x = b, 1 − δ < x−a

x−b < 1 + δ, and all nonzero

x∗

1 ∈ NA(a), x∗ 2 ∈ NB(b) satisfying x∗ 1 + x∗ 2 = 1,

x∗

1, x − a

x∗

1 x − a > 1 − δ,

x∗

2, x − b

x∗

2 x − b > 1 − δ

itr[A, B](¯ x) := sup{α in the above condition}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 26 / 34

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Intrinsic Transversality

X – normed vector space, A, B ⊂ X, ¯ x ∈ A ∩ B

Definition

{A, B} is intrinsically transversal at ¯ x if ∃α, δ > 0 such that x∗

1 + x∗ 2 > α ∀a ∈ (A \ B) ∩ Bδ(¯

x), b ∈ (B \ A) ∩ Bδ(¯ x), x ∈ Bδ(¯ x) with x = a, x = b, 1 − δ < x−a

x−b < 1 + δ, and all nonzero

x∗

1 ∈ NA(a), x∗ 2 ∈ NB(b) satisfying x∗ 1 + x∗ 2 = 1,

x∗

1, x − a

x∗

1 x − a > 1 − δ,

x∗

2, x − b

x∗

2 x − b > 1 − δ

itr[A, B](¯ x) := sup{α in the above condition} X – Asplund space: 0 ≤ tr[A, B](¯ x) ≤ itr[A, B](¯ x) ≤ str[A, B](¯ x) Transversality = ⇒ Intrinsic transversality = ⇒ Subtransversality

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 26 / 34

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Relative Limiting Normals

dim X < ∞, A, B ⊂ X, ¯ x ∈ A ∩ B Cone of pairs of relative limiting normals to {A, B} at ¯ x: NA,B(¯ x) :=

  • (x∗

1, x∗ 2) | x∗ 1k ∈ NA(ak) \ {0}, x∗ 2k ∈ NB(bk) \ {0},

ak ∈ A \ B, bk ∈ B \ A, xk = ak, xk = bk, ak → ¯ x, bk → ¯ x, xk → ¯ x, xk − ak xk − bk → 1 x∗

1k → x∗ 1, x∗ 2k → x∗ 2,

x∗

1k, xk − ak

x∗

1k xk − ak → 1,

x∗

2k, xk − bk

x∗

2k xk − bk → 1

  • Alexander Kruger (FedUni, Ballarat)

Transversality of Pairs of Sets RMIT, 3 March 2017 27 / 34

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Relative Limiting Normals

dim X < ∞, A, B ⊂ X, ¯ x ∈ A ∩ B Cone of pairs of relative limiting normals to {A, B} at ¯ x: NA,B(¯ x) :=

  • (x∗

1, x∗ 2) | x∗ 1k ∈ NA(ak) \ {0}, x∗ 2k ∈ NB(bk) \ {0},

ak ∈ A \ B, bk ∈ B \ A, xk = ak, xk = bk, ak → ¯ x, bk → ¯ x, xk → ¯ x, xk − ak xk − bk → 1 x∗

1k → x∗ 1, x∗ 2k → x∗ 2,

x∗

1k, xk − ak

x∗

1k xk − ak → 1,

x∗

2k, xk − bk

x∗

2k xk − bk → 1

  • NA,B(¯

x) ⊂ NA(¯ x) × NB(¯ x)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 27 / 34

slide-54
SLIDE 54

Intrinsic Transversality

dim X < ∞, A, B – closed, ¯ x ∈ A ∩ B

Theorem (K., Luke, Thao, 2016)

{A, B} is intrinsically transversal at ¯ x ⇐ ⇒ ∃α > 0 such that x∗

1 + x∗ 2 > α ∀(x∗ 1, x∗ 2) ∈ NA,B(¯

x) satisfying x∗

1 + x∗ 2 = 1.

itr[A, B](¯ x) = min

(x∗

1 ,x∗ 2 )∈NA,B(¯

x) x∗

1 +x∗ 2 =1

x∗

1 + x∗ 2

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 28 / 34

slide-55
SLIDE 55

Intrinsic Transversality

dim X < ∞, A, B – closed, ¯ x ∈ A ∩ B

Theorem (K., Luke, Thao, 2016)

{A, B} is intrinsically transversal at ¯ x ⇐ ⇒ ∃α > 0 such that x∗

1 + x∗ 2 > α ∀(x∗ 1, x∗ 2) ∈ NA,B(¯

x) satisfying x∗

1 + x∗ 2 = 1.

itr[A, B](¯ x) = min

(x∗

1 ,x∗ 2 )∈NA,B(¯

x) x∗

1 +x∗ 2 =1

x∗

1 + x∗ 2

Corollary

{A, B} is intrinsically transversal at ¯ x ⇐ ⇒

  • x∗ ∈ X ∗ | (x∗, −x∗) ∈ NA,B(¯

x)

  • ⊂ {0}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 28 / 34

slide-56
SLIDE 56

Intrinsic Transversality

X – Euclidean space, A, B ⊂ X, ¯ x ∈ A ∩ B

Theorem

{A, B} is intrinsically transversal at ¯ x ⇐ ⇒ ∃α, δ > 0 such that v1 + v2 > α ∀x ∈ X, a ∈ A \ B, b ∈ B \ A with x − ¯ x < δ, 0 < x − a < δ, 0 < x − b < δ, 1 − δ < x−a

x−b < 1 + δ, and all

nonzero v1 ∈ Np

A(a), v2 ∈ Np B(b) satisfying v1 + v2 = 1

v1, x − a v1 x − a > 1 − δ, v2, x − b v2 x − b > 1 − δ itr[A, B](¯ x) = sup{α in the above condition}

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 29 / 34

slide-57
SLIDE 57

Intrinsic Transversality

X – Euclidean space, A, B – closed, ¯ x ∈ A ∩ B itr2[A, B](¯ x) := − min

(v1,v2)∈NA,B(¯ x) v1=v2=1

v1, v2

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 30 / 34

slide-58
SLIDE 58

Intrinsic Transversality

X – Euclidean space, A, B – closed, ¯ x ∈ A ∩ B itr2[A, B](¯ x) := − min

(v1,v2)∈NA,B(¯ x) v1=v2=1

v1, v2 itr2[A, B](¯ x) + 2(itr[A, B](¯ x))2 = 1

Proposition

{A, B} is intrinsically transversal at ¯ x ⇐ ⇒ itr2[A, B](¯ x) < 1

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 30 / 34

slide-59
SLIDE 59

Intrinsic Transversality

X – Euclidean space, A, B – closed, ¯ x ∈ A ∩ B

  • itr[A, B](¯

x) := lim inf

a∈A\B, b∈B\A a→¯ x, b→¯ x

max

  • d

b − a a − b, NA(a)

  • , d

a − b a − b, NB(b)

  • (Drusvyatskiy, Ioffe, Lewis, 2015)

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 31 / 34

slide-60
SLIDE 60

Intrinsic Transversality

X – Euclidean space, A, B – closed, ¯ x ∈ A ∩ B

  • itr[A, B](¯

x) := lim inf

a∈A\B, b∈B\A a→¯ x, b→¯ x

max

  • d

b − a a − b, NA(a)

  • , d

a − b a − b, NB(b)

  • (Drusvyatskiy, Ioffe, Lewis, 2015)
  • itr[A, B](¯

x) = 0 ⇐ ⇒ itr[A, B](¯ x) = 0

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 31 / 34

slide-61
SLIDE 61

Intrinsic Transversality

X – Euclidean space, A, B – closed, ¯ x ∈ A ∩ B

  • itr[A, B](¯

x) := lim inf

a∈A\B, b∈B\A a→¯ x, b→¯ x

max

  • d

b − a a − b, NA(a)

  • , d

a − b a − b, NB(b)

  • (Drusvyatskiy, Ioffe, Lewis, 2015)
  • itr[A, B](¯

x) = 0 ⇐ ⇒ itr[A, B](¯ x) = 0

Proposition

{A, B} is intrinsically transversal at ¯ x ⇐ ⇒

  • itr[A, B](¯

x) > 0

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 31 / 34

slide-62
SLIDE 62

References

  • A. Y. Kruger and B. S. Mordukhovich, Extremal points and the

Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24:8 (1980), 684–687, in Russian.

  • B. S. Mordukhovich and Y. Shao, Extremal characterizations of

Asplund spaces, Proc. Amer. Math. Soc. 124 (1996), 197–205.

  • A. Y. Kruger, Stationarity and regularity of set systems, Pacif.
  • J. Optimiz. 1 (2005), 101–126.
  • A. Y. Kruger, About regularity of collections of sets, Set-Valued
  • Anal. 14 (2006), 187–206.
  • A. Y. Kruger, About stationarity and regularity in variational

analysis, Taiwanese J. Math. 13 (2009), 1737–1785.

  • A. Y. Kruger and N. H. Thao, About uniform regularity of

collections of sets, Serdica Math. J. 39 (2013), 287–312.

Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 32 / 34

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SLIDE 63

References

  • D. Drusvyatskiy, A. D. Ioffe and A. S. Lewis, Transversality and

alternating projections for nonconvex sets. Found. Comput.

  • Math. 15 (2015), 1637–1651.
  • A. Y. Kruger and N. H. Thao, Regularity of collections of sets

and convergence of inexact alternating projections, J. Convex

  • Anal. 23 (2016), 823–847.
  • A. Y. Kruger, D. R. Luke and N. H. Thao, Set regularities and

feasibility problems, Math. Program., Ser. B (2016), DOI 10.1007/s10107-016-1039-x.

  • A. Y. Kruger, D. R. Luke and N. H. Thao, About

subtransversality of collections of sets, arXiv:1611.04787.

  • A. Y. Kruger, About intrinsic transversality of pairs of sets,

arXiv:1701.08246.

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SLIDE 64

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