Generalized minimizers of convex integral functionals and Pythagorean identities

28/08/2013
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Generalized minimizers of convex integral functionals and Pythagorean identities

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Minimization problem Convex duality Main results Generalized minimizers of convex integral functionals and Pythagorean identities Imre Csisz´ar (Budapest) and Frantiˇsek Mat´uˇs (Prague) GSI’2013, Paris, France, August 28–30, 2013 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48 637–689 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48 637–689 dedicated to the memory of Igor Vajda (1942–2010) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48 637–689 dedicated to the memory of Igor Vajda (1942–2010) arXiv.org > math > arXiv:1202.0666v1 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space γ : R → (−∞, +∞] ..... strictly convex on (0, +∞), Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space γ : R → (−∞, +∞] ..... strictly convex on (0, +∞), γ(0) = limt↓0 γ(t) and γ(t) = +∞ for t < 0 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space γ : R → (−∞, +∞] ..... strictly convex on (0, +∞), γ(0) = limt↓0 γ(t) and γ(t) = +∞ for t < 0 The entropy functional based on γ For a Z-measurable nonnegative function g on Z Hγ(g) = Z γ(g(z)) µ(dz) if the integral exists, finite of not, and Hγ(g) = +∞ otherwise. Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space γ : R → (−∞, +∞] ..... strictly convex on (0, +∞), γ(0) = limt↓0 γ(t) and γ(t) = +∞ for t < 0 The entropy functional based on γ For a Z-measurable nonnegative function g on Z Hγ(g) = Z γ(g(z)) µ(dz) if the integral exists, finite of not, and Hγ(g) = +∞ otherwise. γ ..... autonomous integrand Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space γ : R → (−∞, +∞] ..... strictly convex on (0, +∞), γ(0) = limt↓0 γ(t) and γ(t) = +∞ for t < 0 The entropy functional based on γ For a Z-measurable nonnegative function g on Z Hγ(g) = Z γ(g(z)) µ(dz) if the integral exists, finite of not, and Hγ(g) = +∞ otherwise. γ ..... autonomous integrand γ(t) = t ln t ..... negative Shannon differential entropy (Z = Rn) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints (Z, Z, µ) ..... a σ-finite measure space γ : R → (−∞, +∞] ..... strictly convex on (0, +∞), γ(0) = limt↓0 γ(t) and γ(t) = +∞ for t < 0 The entropy functional based on γ For a Z-measurable nonnegative function g on Z Hγ(g) = Z γ(g(z)) µ(dz) if the integral exists, finite of not, and Hγ(g) = +∞ otherwise. γ ..... autonomous integrand γ(t) = t ln t ..... negative Shannon differential entropy (Z = Rn) γ(t) = − ln t ..... Burg entropy Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R The entropy functional based on β For a Z-measurable nonnegative function g on Z Hβ(g) = Z β(z, g(z)) µ(dz) . Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R The entropy functional based on β For a Z-measurable nonnegative function g on Z Hβ(g) = Z β(z, g(z)) µ(dz) . β ..... nonautonomous integrand Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R The entropy functional based on β For a Z-measurable nonnegative function g on Z Hβ(g) = Z β(z, g(z)) µ(dz) . β ..... nonautonomous integrand For γ, h 0 on Z and β(z, t) in the form Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R The entropy functional based on β For a Z-measurable nonnegative function g on Z Hβ(g) = Z β(z, g(z)) µ(dz) . β ..... nonautonomous integrand For γ, h 0 on Z and β(z, t) in the form h(z)γ(t/h(z)) ..... γ-divergence from h Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R The entropy functional based on β For a Z-measurable nonnegative function g on Z Hβ(g) = Z β(z, g(z)) µ(dz) . β ..... nonautonomous integrand For γ, h 0 on Z and β(z, t) in the form h(z)γ(t/h(z)) ..... γ-divergence from h (in particular, relative entropy) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints β : Z × R → (−∞, +∞] such that β(z, ·) behaves like γ for all z ∈ Z, and β(·, t) is Z-measurable for all t ∈ R The entropy functional based on β For a Z-measurable nonnegative function g on Z Hβ(g) = Z β(z, g(z)) µ(dz) . β ..... nonautonomous integrand For γ, h 0 on Z and β(z, t) in the form h(z)γ(t/h(z)) ..... γ-divergence from h (in particular, relative entropy) γ(t) − γ(h(z)) − γ (h(z))[t − h(z)] ..... γ-Bregman distance from h Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Ga g 0: Z ϕ g dµ = a Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Ga g 0: Z ϕ g dµ = a Minimization of the entropy functionals under given moments Given a ∈ Rd , minimize Hβ(g) subject to g ∈ Ga. Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Ga g 0: Z ϕ g dµ = a Minimization of the entropy functionals under given moments Given a ∈ Rd , minimize Hβ(g) subject to g ∈ Ga. GOALS: decide whether infGa Hβ is finite, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Ga g 0: Z ϕ g dµ = a Minimization of the entropy functionals under given moments Given a ∈ Rd , minimize Hβ(g) subject to g ∈ Ga. GOALS: decide whether infGa Hβ is finite, if yes whether the infimum is attained, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Ga g 0: Z ϕ g dµ = a Minimization of the entropy functionals under given moments Given a ∈ Rd , minimize Hβ(g) subject to g ∈ Ga. GOALS: decide whether infGa Hβ is finite, if yes whether the infimum is attained, if attained construct a minimizer ga, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints ϕ = (ϕ1, . . . , ϕd ): Z → Rd ..... moment mapping (d-tuple of real-valued Z-measurable functions on Z) a = (a1, . . . , ad ) ∈ Rd Z ϕi g dµ = ai , 1 i d ..... the moment constraints on g Ga g 0: Z ϕ g dµ = a Minimization of the entropy functionals under given moments Given a ∈ Rd , minimize Hβ(g) subject to g ∈ Ga. GOALS: decide whether infGa Hβ is finite, if yes whether the infimum is attained, if attained construct a minimizer ga, anyhow, study sequences gn ∈ Ga with Hβ(gn) → infGa Hβ. Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) ϑ1a1 + ϑ2a2 + ϑ3a3 Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) ϑ1a1 + ϑ2a2 + ϑ3a3 − R γ∗(ϑ1 + ϑ2z + ϑ3z2) µ(dz); Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) ϑ1a1 + ϑ2a2 + ϑ3a3 − R γ∗(ϑ1 + ϑ2z + ϑ3z2) µ(dz); minimize over g ∈ Ga and maximize over ϑ; Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) ϑ1a1 + ϑ2a2 + ϑ3a3 − R γ∗(ϑ1 + ϑ2z + ϑ3z2) µ(dz); minimize over g ∈ Ga and maximize over ϑ; or choose clever g and ϑ, tightness suggests Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) ϑ1a1 + ϑ2a2 + ϑ3a3 − R γ∗(ϑ1 + ϑ2z + ϑ3z2) µ(dz); minimize over g ∈ Ga and maximize over ϑ; or choose clever g and ϑ, tightness suggests g(z) = (γ∗) (ϑ1 + ϑ2z + ϑ3z2), which is in Ga for unique ϑ, Minimization problem Convex duality Main results Entropy functional: autonomous integrand Entropy functional: nonautonomous integrand The minimization under moment constraints µ .. Lebesgue measure on Z = R, γ(t) = t ln t, t > 0 Hγ(g) = R g(z) ln g(z) dz ..... negative diff. Shannon entropy ϕ(z) = (1, z, z2) and a = (a1, a2, a3) Ga = g 0: R g(z) dz = a1 , R z g(z) dz = a2 , R z2 g(z) dz = a3 is empty unless a1 = a2 = a3 = 0 or a1 > 0, a3 > 0 To minimize Hγ over Ga, lower bound γ(t) rt −sups>0[rs −γ(s)], holds for t 0 and r ∈ R, the supremum is γ∗(r) = er−1, where γ∗ is the conjugate of γ, the bound is tight iff γ (t) = r iff t = (γ∗) (r); substitute g(z) for t and ϑ1 + ϑ2z + ϑ3z2 for r; integrate for g ∈ Ga and ϑ = (ϑ1, ϑ2, ϑ3) ∈ R3 Hγ(g) ϑ1a1 + ϑ2a2 + ϑ3a3 − R γ∗(ϑ1 + ϑ2z + ϑ3z2) µ(dz); minimize over g ∈ Ga and maximize over ϑ; or choose clever g and ϑ, tightness suggests g(z) = (γ∗) (ϑ1 + ϑ2z + ϑ3z2), which is in Ga for unique ϑ, this ϑ is a maximizer and the corresponding g is a minimizer. Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications cnϕ(µ) = {a ∈ Rd : Ga = ∅} ..... ϕ-cone of µ Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications cnϕ(µ) = {a ∈ Rd : Ga = ∅} ..... ϕ-cone of µ a → Jβ(a) = infGa Hβ ..... value function Jβ Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications cnϕ(µ) = {a ∈ Rd : Ga = ∅} ..... ϕ-cone of µ a → Jβ(a) = infGa Hβ ..... value function Jβ dom(Jβ) = {a ∈ Rd : Jβ(a) < +∞} ..... domain of Jβ Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications cnϕ(µ) = {a ∈ Rd : Ga = ∅} ..... ϕ-cone of µ a → Jβ(a) = infGa Hβ ..... value function Jβ dom(Jβ) = {a ∈ Rd : Jβ(a) < +∞} ..... domain of Jβ dom(Jβ) ⊆ cnϕ(µ) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications cnϕ(µ) = {a ∈ Rd : Ga = ∅} ..... ϕ-cone of µ a → Jβ(a) = infGa Hβ ..... value function Jβ dom(Jβ) = {a ∈ Rd : Jβ(a) < +∞} ..... domain of Jβ dom(Jβ) ⊆ cnϕ(µ) Theorem If dom(Jβ) is nonempty then it is equal to ri(F) over the faces F of cnϕ(µ) with {ϕ/∈cl(F)} β(·, 0) dµ < +∞. Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd here, β∗(z, r) = supt>0 rt − β(z, t) , z ∈ Z, r ∈ R Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd here, β∗(z, r) = supt>0 rt − β(z, t) , z ∈ Z, r ∈ R if Jβ ≡ +∞ then the dual problem amounts biconjugation Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd here, β∗(z, r) = supt>0 rt − β(z, t) , z ∈ Z, r ∈ R if Jβ ≡ +∞ then the dual problem amounts biconjugation thus J∗∗ β (a) supϑ∈Rd ϑ, a − J∗ β(ϑ) where Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd here, β∗(z, r) = supt>0 rt − β(z, t) , z ∈ Z, r ∈ R if Jβ ≡ +∞ then the dual problem amounts biconjugation thus J∗∗ β (a) supϑ∈Rd ϑ, a − J∗ β(ϑ) where J∗ β(ϑ) supa∈Rd ϑ, a − Jβ(a) = Z β∗(z, ϑ, ϕ(z) ) µ(dz) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd here, β∗(z, r) = supt>0 rt − β(z, t) , z ∈ Z, r ∈ R if Jβ ≡ +∞ then the dual problem amounts biconjugation thus J∗∗ β (a) supϑ∈Rd ϑ, a − J∗ β(ϑ) where J∗ β(ϑ) supa∈Rd ϑ, a − Jβ(a) = Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the conjugate of Jβ Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications Jβ(a) = infg∈Ga Hβ(g) ..... the primal problem for a ∈ Rd infGa Hβ supϑ∈Rd ϑ, a − Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the dual problem for a ∈ Rd here, β∗(z, r) = supt>0 rt − β(z, t) , z ∈ Z, r ∈ R if Jβ ≡ +∞ then the dual problem amounts biconjugation thus J∗∗ β (a) supϑ∈Rd ϑ, a − J∗ β(ϑ) where J∗ β(ϑ) supa∈Rd ϑ, a − Jβ(a) = Z β∗(z, ϑ, ϕ(z) ) µ(dz) ..... the conjugate of Jβ Jβ(a) J∗∗ β (a), tight at the points a of lower semicontinuity Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Each dual solution ϑ satisfies the dcq condition, the function g∗ a = β∗ (·, ϑ, ϕ ) does not depend on its choice, Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Each dual solution ϑ satisfies the dcq condition, the function g∗ a = β∗ (·, ϑ, ϕ ) does not depend on its choice, g∗ a ∈ Ga iff the primal solution exists, in which case it equals g∗ a , Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Each dual solution ϑ satisfies the dcq condition, the function g∗ a = β∗ (·, ϑ, ϕ ) does not depend on its choice, g∗ a ∈ Ga iff the primal solution exists, in which case it equals g∗ a , Hβ(g) = Jβ(a) + Bβ(g, g∗ a ) + Cβ(g) for g ∈ Ga , Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Each dual solution ϑ satisfies the dcq condition, the function g∗ a = β∗ (·, ϑ, ϕ ) does not depend on its choice, g∗ a ∈ Ga iff the primal solution exists, in which case it equals g∗ a , Hβ(g) = Jβ(a) + Bβ(g, g∗ a ) + Cβ(g) for g ∈ Ga , gn g∗ a for each sequence gn ∈ Ga with Hβ(gn) → Jβ(a). Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Each dual solution ϑ satisfies the dcq condition, the function g∗ a = β∗ (·, ϑ, ϕ ) does not depend on its choice, g∗ a ∈ Ga iff the primal solution exists, in which case it equals g∗ a , Hβ(g) = Jβ(a) + Bβ(g, g∗ a ) + Cβ(g) for g ∈ Ga , gn g∗ a for each sequence gn ∈ Ga with Hβ(gn) → Jβ(a). When the dcq fails: (NEW) Minimization problem Convex duality Main results Domain of the value function Primal and dual problems Assuming constraint qualifications CONSTRAINT QUALIFICATIONS a ∈ ri(dom(Jβ)) and Jβ(a) finite ..... primal (pcq) ∃ϑ ∈ Rd s.t. Z β∗(z, ϑ, ϕ(z) ) µ(dz) < +∞ and r → β∗(z, r) is finite around ϑ, ϕ(z) for µ-a.a.z ∈ Z ..... dual (dcq) Proposition (assuming pcq ) The primal and dual values coincide, Jβ(a) = J∗∗ β (a) = supϑ∈Rd ..., and the dual one is attained by some ϑ ∈ Rd . When the dcq holds: Each dual solution ϑ satisfies the dcq condition, the function g∗ a = β∗ (·, ϑ, ϕ ) does not depend on its choice, g∗ a ∈ Ga iff the primal solution exists, in which case it equals g∗ a , Hβ(g) = Jβ(a) + Bβ(g, g∗ a ) + Cβ(g) for g ∈ Ga , gn g∗ a for each sequence gn ∈ Ga with Hβ(gn) → Jβ(a). When the dcq fails: (NEW) gn for each sequence gn ∈ Ga with Hβ(gn) → Jβ(a). Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) ..... Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) ..... in particular ˜ga is constructed s.t. Hβ(g) = Jβ(a) + Bβ(g, ˜ga) + Cβ(g) for g ∈ Ga. Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) ..... in particular ˜ga is constructed s.t. Hβ(g) = Jβ(a) + Bβ(g, ˜ga) + Cβ(g) for g ∈ Ga. Bβ(·, ·) ..... Bregman distance based on β Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) ..... in particular ˜ga is constructed s.t. Hβ(g) = Jβ(a) + Bβ(g, ˜ga) + Cβ(g) for g ∈ Ga. Bβ(·, ·) ..... Bregman distance based on β Cβ 0 ..... explicit correction functional Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) ..... in particular ˜ga is constructed s.t. Hβ(g) = Jβ(a) + Bβ(g, ˜ga) + Cβ(g) for g ∈ Ga. Bβ(·, ·) ..... Bregman distance based on β Cβ 0 ..... explicit correction functional (vanishes if β essentially smooth) Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Theorem (assuming a ∈ Rd with Jβ(a) finite only) Denote by F the unique face of the ϕ-cone cnϕ(µ) with a ∈ ri(F). Restrict µ to ϕ−1(cl(F)) and modify the story. ..... ..... (‘weakened’ dcq) ..... in particular ˜ga is constructed s.t. Hβ(g) = Jβ(a) + Bβ(g, ˜ga) + Cβ(g) for g ∈ Ga. Bβ(·, ·) ..... Bregman distance based on β Cβ 0 ..... explicit correction functional (vanishes if β essentially smooth) ˜ga ..... generalized primal solution Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem given h Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem given h Theorem covers inf Bβ(·, h) subject to moment constraints Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem given h Theorem covers inf Bβ(·, h) subject to moment constraints because Bβ(·, h) = Hβ for some β Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem given h Theorem covers inf Bβ(·, h) subject to moment constraints because Bβ(·, h) = Hβ for some β ... (generalized) Bregman projections, dcq always holds Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem given h Theorem covers inf Bβ(·, h) subject to moment constraints because Bβ(·, h) = Hβ for some β ... (generalized) Bregman projections, dcq always holds ... exotic Pythagorean identities Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem given h Theorem covers inf Bβ(·, h) subject to moment constraints because Bβ(·, h) = Hβ for some β ... (generalized) Bregman projections, dcq always holds ... exotic Pythagorean identities ..... Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) K∗ β (a) = supϑ∈Rd ϑ, a − Kβ(a) ..... dual value Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) K∗ β (a) = supϑ∈Rd ϑ, a − Kβ(a) ..... dual value Theorem (existence only) Assuming the dcq, for every a ∈ Rd with K∗ β (a) finite there exists a unique Z-measurable function ha such that K∗ β (a) − ϑ, a − Kβ(ϑ) Bβ(ha, β∗ (·, ϑ, ϕ )) for all ϑ satisfying the dcq condition. Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) K∗ β (a) = supϑ∈Rd ϑ, a − Kβ(a) ..... dual value Theorem (existence only) Assuming the dcq, for every a ∈ Rd with K∗ β (a) finite there exists a unique Z-measurable function ha such that K∗ β (a) − ϑ, a − Kβ(ϑ) Bβ(ha, β∗ (·, ϑ, ϕ )) for all ϑ satisfying the dcq condition. Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) K∗ β (a) = supϑ∈Rd ϑ, a − Kβ(a) ..... dual value Theorem (existence only) Assuming the dcq, for every a ∈ Rd with K∗ β (a) finite there exists a unique Z-measurable function ha such that K∗ β (a) − ϑ, a − Kβ(ϑ) Bβ(ha, β∗ (·, ϑ, ϕ )) for all ϑ satisfying the dcq condition. ha ..... generalized ‘dual solution’, Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) K∗ β (a) = supϑ∈Rd ϑ, a − Kβ(a) ..... dual value Theorem (existence only) Assuming the dcq, for every a ∈ Rd with K∗ β (a) finite there exists a unique Z-measurable function ha such that K∗ β (a) − ϑ, a − Kβ(ϑ) Bβ(ha, β∗ (·, ϑ, ϕ )) for all ϑ satisfying the dcq condition. ha ..... generalized ‘dual solution’, if Jβ(a) = K∗ β (a) then ha = ˜ga Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem Kβ(a) Z β∗(z, ϑ, ϕ(z) ) µ(dz) K∗ β (a) = supϑ∈Rd ϑ, a − Kβ(a) ..... dual value Theorem (existence only) Assuming the dcq, for every a ∈ Rd with K∗ β (a) finite there exists a unique Z-measurable function ha such that K∗ β (a) − ϑ, a − Kβ(ϑ) Bβ(ha, β∗ (·, ϑ, ϕ )) for all ϑ satisfying the dcq condition. ha ..... generalized ‘dual solution’, if Jβ(a) = K∗ β (a) then ha = ˜ga For the integrand t ln t, this is MLE in EF’s an explicit construction of ha is available in Cs&M (2008) Probab. Th. Rel. F. Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem 12 EXAMPLES in Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem 12 EXAMPLES in Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem 12 EXAMPLES in Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48 637–689 Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem 12 EXAMPLES in Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48 637–689 dedicated to the memory of Igor Vajda (1942–2010) Minimization problem Convex duality Main results Dispensing with pcq Bregman projections The dual problem 12 EXAMPLES in Csisz´ar, I. and Mat´uˇs, F., Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48 637–689 dedicated to the memory of Igor Vajda (1942–2010) arXiv.org > math > arXiv:1202.0666v1