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Topics in Convex Analysis Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Topics in Convex Analysis Tim Hoheisel (McGill University, Montreal) Spring School on Variational Analysis Paseky, May 19–25, 2019 Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Outline 1 Fundamentals from Convex Analysis Convex sets and functions Subdifferentiation and conjugacy of convex functions Infimal convolution and the Attouch-Brezis´ Theorem Consequences of Attouch-Brezis´ 2 Conjugacy of composite functions via K-convexity and inf-convolution K-convexity Composite functions and scalarization Conjugacy results Applications 3 A new class of matrix support functionals The generalized matrix-fractional function The closed convex hull of D(A; B) with applications Applications of the GMF Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals 1. Fundamentals from Convex Analysis Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) B"(x) = x + "B pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) pos S := R+S (conical hull) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions The Euclidean setting and Minkowski notation In what follows E will be a Euclidean space, i.e. a real-vector space equipped with an inner product h·; ·i : E × E ! R of dimension κ < 1: Examples E = Rn ; hx; yi := xT y; κ = n E = Rm×n ; hA; Bi := tr (A T B); κ = mn Minkowski addition/multiplication: Let A ⊂ E A + B := fa + b j a 2 A; b 2 B g (B ⊂ E) A + x := A + fxg (x 2 E) Λ · A := fλa j a 2 A; λ 2 Λ g (Λ ⊂ R) λA := fλg · A (λ 2 R) Examples: U; V ⊂ E subspaces. Then U + V = span (U [ V) B"(x) = x + "B pos S := R+S (conical hull) S ⊂ E is said to be convex if λS + (1 − λ)S ⊂ S (λ 2 (0; 1)); 0 a cone if λS ⊂ S (λ ≥ 0). Note that K ⊂ E is a convex cone iff K + K ⊂ K. Figure: Convex set/non-convex cone Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions Convex sets and cones ”The great watershed in optimization is not between linearity and nonlinearity, but convexity and nonconvexity.” (R.T. Rockafellar, *1935) Fundamentals from Convex Analysis Conjugacy of composite functions via K-convexity and inf-convolution A new class of matrix support functionals Convex sets and functions Convex sets and cones ”The great watershed in optimization is not between linearity and nonlinearity, but convexity and nonconvexity.” (R.T.
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