Transversality, Subtransversality and Intrinsic Transversality of - PowerPoint PPT Presentation

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

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

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 Transversality ⇐ ⇒ of (multi)functions of collections of sets Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 2 / 34

Feasibility Problem and Alternating Projections Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 3 / 34

Feasibility Problem and Alternating Projections Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 4 / 34

Feasibility Problem and Alternating Projections Linear convergence (with rate c ∈ (0 , 1)): x ) ≤ cd ( x k , ¯ d ( x k +1 , ¯ x ) Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 4 / 34

Feasibility Problem and Alternating Projections Linear convergence (with rate c ∈ (0 , 1)): x ) ≤ cd ( x k , ¯ d ( x k +1 , ¯ x ) Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 4 / 34

Feasibility Problem and Alternating Projections Linear convergence (with rate c ∈ (0 , 1)): x ) ≤ cd ( x k , ¯ d ( x k +1 , ¯ x ) Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 5 / 34

Feasibility Problem and Alternating Projections Linear convergence No linear convergence (with rate c ∈ (0 , 1)): x ) ≤ cd ( x k , ¯ d ( x k +1 , ¯ x ) Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 5 / 34

Outline Transversality/Subtransversality 1 Extremality 2 Dual characterizations of transversality/subtransversality 3 Intrinsic transversality 4 Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 6 / 34

Outline Transversality/Subtransversality 1 Extremality 2 Dual characterizations of transversality/subtransversality 3 Intrinsic transversality 4 Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 7 / 34

Subtransversality X – normed vector space, A , B ⊂ X , x ∈ A ∩ B ¯ Definition { A , B } is subtransversal at ¯ x if ∃ α, δ > 0 such that α d ( x , A ∩ B ) ≤ max { d ( x , A ) , d ( x , B ) } ∀ x ∈ B δ (¯ x ) Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 8 / 34

Subtransversality X – normed vector space, A , B ⊂ X , x ∈ A ∩ B ¯ Definition { A , B } is subtransversal at ¯ x if ∃ α, δ > 0 such that α d ( x , A ∩ B ) ≤ max { d ( x , A ) , d ( x , B ) } ∀ x ∈ B δ (¯ 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

Subtransversality X – normed vector space, A , B ⊂ X , x ∈ A ∩ B ¯ Definition { A , B } is subtransversal at ¯ x if ∃ α, δ > 0 such that α d ( x , A ∩ B ) ≤ max { d ( x , A ) , d ( x , B ) } ∀ x ∈ B δ (¯ 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) x ) := sup { α in the above condition } str [ A , B ](¯ Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 8 / 34

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 α d ( x , ( A − x 1 ) ∩ ( B − x 2 )) ≤ max { d ( x , A − x 1 ) , d ( x , B − x 2 ) } ∀ x ∈ B δ (¯ x ), x 1 , x 2 ∈ δ B Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 9 / 34

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 α d ( x , ( A − x 1 ) ∩ ( B − x 2 )) ≤ max { d ( x , A − x 1 ) , d ( x , B − x 2 ) } ∀ x ∈ B δ (¯ x ), x 1 , x 2 ∈ δ B 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

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 α d ( x , ( A − x 1 ) ∩ ( B − x 2 )) ≤ max { d ( x , A − x 1 ) , d ( x , B − x 2 ) } ∀ x ∈ B δ (¯ x ), x 1 , x 2 ∈ δ B 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

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

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

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

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

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 ¯ ⇒ { A , B } is subtransversal at ¯ x x Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 12 / 34

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

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

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

Outline Transversality/Subtransversality 1 Extremality 2 Dual characterizations of transversality/subtransversality 3 Intrinsic transversality 4 Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 15 / 34

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 ∃ x 1 , x 2 ∈ ε B such that ( A − x 1 ) ∩ ( B − x 2 ) ∩ B δ (¯ x ) = ∅ Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 16 / 34

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 ∃ x 1 , x 2 ∈ ε B such that ( A − x 1 ) ∩ ( B − x 2 ) ∩ B δ (¯ x ) = ∅ Proposition (K., 2005) { A , B } is locally extremal at ¯ ⇒ { A , B } is not transversal at ¯ x x Alexander Kruger (FedUni, Ballarat) Transversality of Pairs of Sets RMIT, 3 March 2017 16 / 34