Fenchel-Moreau Conjugates of Inf-Transforms and Application to the Stochastic Bellman Equation

Jean-Philippe Chancelier and Michel De Lara CERMICS, Ecole´ des Ponts ParisTech

Sociedad Matem´aticaPeruana XXXV Coloquio SMP2018 IMCA, Lima, 20 de Diciembre de 2018 Examples of inf-transforms in optimization

I Perturbation of constraints (x) gives Y Y inf g(y) inf g(y) y y (x) ∈Y ∈Y and value function

f (x) = inf δ (x)(y) ug(y) y Y   (x,y) K

I Two-stage linear stochastic programming| {z }

fs (x) = inf cs , x u ps , y u δ y 0 , As x+bs +y 0 y h i h i { ≥ ≥ }   Examples of inf-transforms in optimization (continued)

I Product from the left by a (linear) operator L

(Lg)(x) = inf δLy=x g(y) y u  (x,y)  K

I Moreau-Yosida approximation of|g{z }

1 2 f (x) = inf x y ug(y) y αk − k   (x,y) K | {z } I Inf-convolution of g1 and g2

f (x) = inf g1(x y) ug2(y) y −  (x,y)  K | {z } Examples of inf-transforms in optimization (continued)

I Lasso problem

1 2 f (x) = inf x Ay 2 u λ y 1 y 2k − k k k  sparsity, regularization 

I Supervised learning and sparsity | {z }

f (x) = inf l(x, Ay) u λ y 0 y k k  loss function l0 pseudo-norm 

I Bregman “distance” | {z } | {z }

f (x) = inf H(x) H(y) H(x) , x y ug(y) y − − h∇ − i  Bregman “distance” (x,y)  K | {z } Examples of inf-transforms in optimization (continued)

I Upper envelope representations

V (τ, ξ) = inf E(τ, ξ, ξ0) + g(ξ0) ξ0   and Hamilton-Jacobi equation

Question: what about their Fenchel conjugate

f ?(x]) = sup x , x] + f (x) ? − x X · ∈  D E
 (hence what about dual problems?) Main result Two couplings c and d, and an inf-operation with kernel K

c X X♯

f(x) infy Y (x,y) ∔ g(y) ≥ ∈ K  

c+d K ⇒ K ·

c ♯ c+d ♯ ♯ d ♯ f (x ) infy♯ Y♯ (x ,y ) ∔ g− (y ) ≤ ∈ K ·  

d Y Y♯ Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP

Background on couplings and Fenchel-Moreau conjugacy

Fenchel-Moreau conjugation inequality with three couplings

Complements

Conclusion Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP

Background on couplings and Fenchel-Moreau conjugacy

Fenchel-Moreau conjugation inequality with three couplings

Complements

Conclusion Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Basic spaces

I We introduce a first couple of spaces in bilinear duality

n ] n X = R X and X = R X

I and a second couple of spaces in bilinear duality

p n ] q n Y = L (Ω, , P), R X and Y = L (Ω, , P), R X F F n p and q-integrable random
variables with values in R X , where

I (Ω, , P) is a probability space F I 1 p < + and q are such that 1/p + 1/q = 1 ≤ ∞ p n I Random variables, elements of Y = L (Ω, , P), R X will be denoted by bold letters like X F ] q n ]
and elements of Y = L (Ω, , P), R X by X F All Fenchel conjugates will be denoted by I
? ( ?) g , g − Ingredients for a stochastic optimal control problem

I Let time t = 0, 1,..., T be discrete, with T N∗ ∈ I Consider a stochastic optimal control problem with n I state space X = R X n I control space U = R U I white noise process Wt t=1,...,T { } n taking values in uncertainty space W = R W and defined over the probability space (Ω, , P) F I For each time t = 0, 1,..., T 1, we have − I dynamics Ft : X U W X × × → I instantaneous costs Lt : X U W [0, + ] × × → ∞ I final cost K : X [0, + ] → ∞ We introduce the Bellman functions

I We define Bellman functions by, for all x X and t = T 1,..., 0, ∈ −

VT (x) = K(x) T 1 − Vt (x) = inf E Ls (Xs , Us , Ws+1) u K(XT ) X,U s=t  X  where Xt = x X, Xs+1 = Fs (Xs , Us , Ws+1) and ∈ σ(Us ) σ(Xs ), for s = t,..., T 1 ⊂ − I If the Bellman functions are measurable, they satisfy the backward Bellman inequation, for t = T 1,..., 0 − Bellman function

Vt (x) inf E Lt (x, u, Wt+1) uVt+1 Ft (x, u, Wt+1) ≥ u U ∈ z }| { instantaneous dynamics  cost
 | {z } | {z } Fenchel conjugates of the Bellman functions

Theorem The Bellman functions satisfy the backward inequalities

( ?) Vt (x) inf inf (x, u, ) − (X) u E Vt+1(X) ≥ X u U − H ·  ∈  
   for t = T 1,..., 0, where the Hamiltonian is defined by − H ] ] (x, u, X ) = E Lt (x, u, Wt+1) Ft (x, u, Wt+1) , X H u  D E  Moreover, letting V ? be the Fenchel conjugates of the t t=0,1,...,T Bellman functions, we have, for all x] X] and t = T 1,..., 0,  ∈ −

? ] ] ? ] ? ] Vt (x ) inf sup ( , u, X ) (x ) u E Vt+1(X ) ≤ X] H ·  u U  ∈     Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Obtaining upper and lower estimates in approximations of Bellman functions (I)

I Suppose that the Bellman functions

Vt t=0,1,...,T satisfy the Bellman equation and{ } are convex l.s.c. and proper, that is,

?? Vt = V , t = 0, 1,..., T t ∀ I This is the case in Stochastic Dual Dynamic Programming (SDDP), when

I the dynamics Ft are jointly linear in state and control I the instantaneous costs Lt are jointly convex in state and control I the final cost K is convex I together with technical assumptions Obtaining upper and lower estimates in approximations of Bellman functions (II)

? I The Fenchel conjugates Vt t=0,1,...,T of the Bellman functions{ } are convex l.s.c. and proper, by construction

I Suppose that they satisfy a “Bellman like” equation

? ] ] ? ] ? ] Vt (x )= inf sup ( , u, X ) (x ) u E Vt+1(X ) X] H ·  u U  ∈     for t = T 1,..., 0 − Obtaining upper and lower estimates in approximations of Bellman functions (III)

I With the Bellman operators deduced from the Bellman equation and “Bellman like” equation, one can produce (by an adequate algorithm like the SDDP algorithm) lower bound functions

V V V k t,(k) t,(k+1) t N ≤ ≤ ? ∀ ∈ (V t,(k) V t,(k+1) Vt ≤ ≤ that are piecewise affine e e

I Since the Bellman functions Vt t=0,1,...,T are convex l.s.c. and proper,{ we} deduce that

?? ? ? V V Vt = V V V t,(k) ≤ t,(k+1) ≤ t ≤ t,(k+1) ≤ t,(k) I Thus, we can control the evolution of thee SDDP algorithme Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP

Background on couplings and Fenchel-Moreau conjugacy

Fenchel-Moreau conjugation inequality with three couplings

Complements

Conclusion Main result Two couplings c and d, and an inf-operation with kernel K

c X X♯

f(x) infy Y (x,y) ∔ g(y) ≥ ∈ K  

c+d K ⇒ K ·

c ♯ c+d ♯ ♯ d ♯ f (x ) infy♯ Y♯ (x ,y ) ∔ g− (y ) ≤ ∈ K ·  

d Y Y♯ The Fenchel conjugacy

Definition Two vector spaces X and X], paired by a bilinear product , , (in the sense of convex analysis), h i give rise to the classic Fenchel conjugacy

? ] ] ] ] f (x ) = sup x , x + f (x) , x X − ∀ ∈ x X · ∈  D E
 for any function f : X R → Fenchel conjugate Fourier transform sup + + → → × ] ] supx X x , x + f (x) x exp( x , x )f (x)dx ∈ − X · ∈ 
 R Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Moreau lower and upper additions

I The Moreau lower addition extends the usual addition with

(+ ) + ( ) = ( ) + (+ ) = ∞ −∞ −∞ ∞ −∞ · · I The Moreau upper addition extends the usual addition with

(+ ) ( ) = ( ) (+ ) = + ∞ u −∞ −∞ u ∞ ∞ Background on couplings and Fenchel-Moreau conjugacies

] I Let be given two sets X (“primal”) and X (“dual”) ] I Consider a coupling function c : X X R = [ , + ] × → −∞ ∞ c ] I We also use the notation X X for a coupling ↔ Definition The c-Fenchel-Moreau conjugate of a function f : X R, with respect to the coupling c, → c is the function f : X] R defined by → c ] ] ] ] f (x ) = sup c(x, x ) + f (x) , x X − ∀ ∈ x X · ∈ 


Fenchel-Moreau conjugate (max, +) Kernel transform (+, ) × ] ] supx X c(x, x ) + f (x) x c(x, x )f (x)dx ∈ − X · ∈ 
 R Background on couplings and Fenchel-Moreau conjugacies

With the coupling c, we associate the reverse coupling c0 ] ] ] ] ] c0 : X X R , c0(x , x) = c(x, x ) , (x , x) X X × → ∀ ∈ ×

] I The c0-Fenchel-Moreau conjugate of a function g : X R, c0 → with respect to the coupling c0, is the function g : X R → c0 ] ] g (x) = sup c(x, x ) + g(x ) , x X ] ] − ∀ ∈ x X · ∈ 
 cc I The c-Fenchel-Moreau biconjugate f : X R → of a function f : X R is given by → cc ] c ] f (x) = sup c(x, x ) + f (x ) , x X ] ] − ∀ ∈ x X · ∈ 
 The( c)-Fenchel-Moreau conjugate of g : X R is given by − → c ] ] ] ] g − (x ) = sup c(x, x ) + g(x) , x X − − ∀ ∈ x X · ∈ 

 Fenchel inequality with a general coupling

I Conjugacies are special cases of dualites, that make it possible to obtain dual problems

c ] c ] sup f (x ) + g − (x ) inf f (x) u g(x) ] ] − − ≤ x X x X · ∈ ∈ 

   I In particular, optimization under constraints x X gives ∈ c ] c ] sup f (x ) + δX− (x ) inf f (x) u δX (x) ] ] − − ≤ x X x X · ∈ ∈ 

   0 if x X where δX (x) = ∈ (+ if x X ∞ 6∈ I Hence, the issue is to find a coupling c c c that gives nice expressions for f and δX− Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Couplings for discrete convexity A present for Kazuo Murota

“Primal” “Dual” Coupling n n R R ?(x, x]) = x , x] n n Z Z (z, p) = z , p n n • h i Z R (?)(z, x]) = z , x] n n n ] ] Z R R c z, (x, x ) = z , x + δ (x)(z) × − N ·
n
where integer neighbours (x) of x R are such that N ∈ n n z (z) Z , z Z { } ⊂ N ⊂ ∀ ∈ Conjugacies for discrete convexity

n I For any function f : Z R, we define → n f •(p) = sup z , p + f (z) p Z z Zn h i − ∀ ∈ ∈  ·  (?) ] ]
] n f (x ) = sup z , x + f (z) x R z Zn − ∀ ∈ ∈  D E ·  c ] ]
] n n f (x, x ) = sup z , x + f (z) (x, x ) R R − ∀ ∈ × z (x) · ∈N  D E
 I and we have the following relations

(?) n f • = f on Z (?) ? n f = f u δZn on R c (?) n sup f (x, ) = f
on R x Rn · ∈ Biconjugacies for discrete convexity

n I For any function f : Z R, we define → n f ••(z) = sup z , p + f •(p) z Z p Zn h i − ∀ ∈ ∈  ·  (?)(?) ] (?
) ] n f (z) = sup z , x + f (x ) z Z x] Rn − ∀ ∈ ∈  D E ·  cc ]
c ] n f (z) = sup sup z , x + f (x, x ) z Z −1 ] n − ∀ ∈ x (z) x R · ∈N ∈  D E
 I and we have the following relations

?0 ? n f •• = f u δZn u δZn on Z (?)(?) (?)?0 ?? n f = f = f
u δZn f •• on Z cc ≥ n f f •• on Z ≥
Convex extensible functions

n I For any function f : Z R, we define →n the convex closure f : R R by → ] ] n f (x) = sup x , x + α z , x + α f (z) , z Z x] Rn,α R ≤ ∀ ∈ ∈ ∈ nD E D E o

I Convex closure and Fenchel biconjugate are related by

?? (?)?0 n f (x) = f δ n (x) = f (x) , x R u Z ∀ ∈
Definition n We say that the function f : Z R is convex extensible if → n f (z) = f (z) , z Z ∀ ∈ We introduce a suitable coupling (?) for which convex extensible functions = (?)-convex functions

n n I Integer space Z coupled with real space R by the bilinear coupling( ?) = , h· ·i (?) n n I The conjugate f : R R of f : Z R is given by → → (?) ] ? ] ] ] n f (x ) = f uδZn (x ) = sup z , x + f (z) , x R n − ∀ ∈ z Z ·
∈  D E
 (?)(?) n I The biconjugate f : Z R is given by → (?)(?) ] (?) ] n f (z) = sup z , x + f (x ) , z Z ] n − ∀ ∈ x R · ∈  D E
 Proposition

f is convex extensible f = f (?)(?) ⇐⇒ Local convex extension

n I For any function f : Z R, we define → n the local convex extension f : R R by → f (x) = sup x , x] e+ α z , x] + α f (z) , z (x) x] Rn,α R ≤ ∀ ∈ N ∈ ∈ nD E D E o

e n where integer neighbours (x) of x R are such that N ∈ n n z (z) Z , z Z { } ⊂ N ⊂ ∀ ∈

I The local convex extension is larger than the convex extension:

n f (x) f (x) , x R ≥ ∀ ∈ I When integer neighbourse are few, the local convex extension coincides with the original function on the integers:

n n z = (z) , z Z f (z) = f (z) , z Z { } N ∀ ∈ ⇒ ∀ ∈ e We introduce a suitable coupling

n n n I Integer space Z coupled with real space R R by localization of the bilinear coupling , × n h· ·i w.r.t. neighbours (x) Z : N ⊂ ] ] ] c z, (x, x ) = z , x + δ (x)(z) = z , x + δ −1(z)(x) − N − N D E · D E ·
c n n
n
I The c-conjugate f : R R R of f : Z R is × → → c ] ] ] n n f (x, x ) = sup z , x + f (z) , (x, x ) R R − ∀ ∈ × z (x) · ∈N  D E
 cc n I The c-biconjugate f : Z R is → cc ] c ] n f (z) = sup sup z , x + f (x, x ) , z Z −1 ] n − ∀ ∈ x (z) x R · ∈N ∈  D E
 Integrally convex functions

I We have that f c (x, x]) f (?)(x]) ≤ I The local convex extension satisfies

0 f (x) = sup x , x] + f c (x, x]) f (?)? (x) = f (x) ] n − ≥ x R · ∈  D E
 e cc n I The c-biconjugate f : Z R satisfies → cc n f (z) = sup f (x) , z Z x −1(z) ∀ ∈ ∈N e Definition n We say that the function f : Z R is integrally convex if → (?)?0 n f (x) = f (x) = f (x) , x R ∀ ∈ e When integer neighbours are few, all functions are c-convex functions!

n I From z = (z) , z Z and { } N ∀ ∈ cc n f (z) sup f (x) = f (z) f (z) = f (z) , z Z ≤ x −1(z) ≤ ∀ ∈ ∈N wee deduce that e e

cc n f (z) = sup f (x) = f (z) = f (z) , z Z x −1(z) ∀ ∈ ∈N e e n I Therefore, if the function f : Z R is integrally convex: → n f (z) = f (z) = sup f (x) = f (z) , z R x −1(z) ∀ ∈ ∈N e e Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Optimal transport

The optimal transport problem is

inf c(x, y)dπ(x, y) π Π(µ,ν) ∈ ZX ×Y where

I the sets and are two Polish spaces X Y I we denote by ( ), ( ) and ( ) the correspondingP X probability × Y P X spaces (rectangleP Y and marginals)

I the setΠ( µ, ν) is made of probabilities π ( ) on the rectangle, whose marginals∈ are P Xµ × Y ( ) and ν ( ) ∈ P X ∈ P Y I the measurable cost function c : [0, + ], where c(x, y) represents X × Y → ∞ the cost to move from x towards y ∈ X ∈ Y We introduce a suitable coupling between probabilities and functions

0 0 I We denote by Cb ( ) and Cb ( ) the spaces of continuousX boundedY functions

I We introduce the bilinear coupling

( ) ( ) β C 0( ) C 0( ) P X × P Y ←→ b X × b Y β (µ, ν); (ψ, φ) = φ(y)dν(y) ψ(x)dµ(x) − Z Z
Y X Conjugacy properties in optimal transport

C(µ, ν) = inf c(x, y)dπ(x, y) π Π(µ,ν) ∈ ZX ×Y c D(ψ, φ)= sup φ(y) ψ(x) c(x, y) = sup φ(y) + ψ− (y) x ,y − − y ∈X ∈Y   ∈Y   We have the following conjugacy equalities and inequalities

0 C β(ψ, φ) = D(ψ, φ) = Dβ β(ψ, φ) 0 0 C(µ, ν) C ββ (µ, ν) = Dβ (µ, ν) ≥ = sup φ(y)dν(y) ψ(x)dµ(x) D(ψ, φ) ψ,φ − −  ZY ZX  sup φ(y)dν(y) ψ(x)dµ(x) ≥ φ ψ c − − ≤  ZY ZX  Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP

Background on couplings and Fenchel-Moreau conjugacy

Fenchel-Moreau conjugation inequality with three couplings

Complements

Conclusion We introduce a sum coupling

I Let be given two “primal” sets X, Y ] ] I and two “dual” sets X , Y , I together with two coupling functions

] ] c : X X R , d : Y Y R × → × → We define the sum coupling c + d — coupling the “primal” product set X Y with the “dual”· product set X] Y] — by × × ] ] c + d : X Y X Y R , × × × → · (x, y), (x], y ]) c(x, x]) + d(y, y ])

7→ ·
A kernel , two couplings c and d and a newK kernel c+d K · c X X♯

c+d K K ·

d Y Y♯ We introduce the conjugate of a kernel bivariate function w.r.t. a sum coupling

With any kernel bivariate function

: X Y R , K × →

defined on the “primal” product set X Y, we associate the conjugate, with respect× to the coupling c + d, defined on the “dual” product set X] Y], by · × c+d (x], y ])= sup c(x, x]) + d(y, y ]) + (x, y) K · x X,y Y − K ∈ ∈ · · ] ] ] ]
 (x , y ) X Y ∀ ∈ × Main result: Fenchel-Moreau conjugation inequalities with three couplings

Theorem For any bivariate function : X Y R and univariate functions K × → f : X R and g : Y R, all defined→ on the “primal”→ sets, we have that

f (x) inf (x, y) u g(y) , x X ≥ y Y K ∀ ∈ ⇒ ∈  

c ] c+d ] ] d ] ] ] f (x ) inf (x , y ) u g − (y ) , x X ≤ y ] Y] K · ∀ ∈ ∈   I The left hand side assumption is a primal inequality, which is rather weak (upper bound for an infimum)

I whereas the right hand side conclusion is a dual inequality, which is rather strong (lower bound for an infimum)

Main result Two couplings c and d, and an inf-operation with kernel K

f (x) inf (x, y) u g(y) ≥ y Y K ⇒ ∈  

c ] c+d ] ] d ] f (x ) inf (x , y ) u g − (y ) ≤ y ] Y] K · c Fenchel- ∈  c+d Fenchel-  ( d) Fenchel- Moreau− conjugate Moreau− conjugate Moreau− − conjugate of f · of of g | {z } | {zK } | {z } Main result Two couplings c and d, and an inf-operation with kernel K

f (x) inf (x, y) u g(y) ≥ y Y K ⇒ ∈  

c ] c+d ] ] d ] f (x ) inf (x , y ) u g − (y ) ≤ y ] Y] K · c Fenchel- ∈  c+d Fenchel-  ( d) Fenchel- Moreau− conjugate Moreau− conjugate Moreau− − conjugate of f · of of g | {z } | {zK } | {z }

I The left hand side assumption is a primal inequality, which is rather weak (upper bound for an infimum)

I whereas the right hand side conclusion is a dual inequality, which is rather strong (lower bound for an infimum) Main result (second formulation) Three couplings c, d and K

f g −K 0 u ≥ ⇒ ( ) Fenchel- Moreau−K − conjugate |of{zg}

c d c+d f + (g − )−K 0 · ≤ c Fenchel- · c+d − ( ) Fenchel- Moreau conjugate −K − of f Moreau· conjugate |{z} | of{zg −d } Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP

Background on couplings and Fenchel-Moreau conjugacy

Fenchel-Moreau conjugation inequality with three couplings

Complements

Conclusion Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion The duality equality case

The duality equality case is the property that

f (x) = inf (x, y) u g(y) , x X y Y K ∀ ∈ ⇒ ∈  

c ] c+d ] ] d ] ] ] f (x ) = inf (x , y ) u g − (y ) , x X y ] Y] K · ∀ ∈ ∈   Sufficient conditions for the duality equality case

Corollary Consider any bivariate function : X Y R K × → and univariate functions f : X R and g : Y R, all defined on the “primal” sets.→ → The equality case holds true when ( d)( d) 1. g − − = g 2. the following function has a saddle point (or no duality gap)

] ] (x, y), y (X Y) Y ∈ × × 7→ ] ] d ]
c(x, x ) + (x, y) + d(y, y ) g − (y ) − K u · · 
 3. the two coupling functions c : X X] R and × → d : Y Y] R, and the kernel : X Y R all take× finite→ values K × → Sufficient conditions for the duality equality case

Corollary Consider any bivariate function : X Y R K × → and univariate functions f : X R and g : Y R, all defined on the “primal” sets.→ We define →

c ] ] x] (y) = ( , y) (x ) = inf c(x, x ) u (x, y) K − K · x X − K ∈   
 The equality case holds true when

d ] d ] sup x] (y) + g(y) = inf x] (y ) u g − (y ) y − K − y ] Y] K Y  ·  ∈   ∈

Sufficient conditions for the duality equality case

Corollary Consider any bivariate function : X Y R K × → and univariate functions f : X R and g : Y ] , + ], all defined on the “primal” sets.→ → − ∞ ∞ The equality case holds true when 1. the coupling d : Y Y] R is the duality bilinear form , × → h i between Y and its algebraic dual Y] 2. the function g is a proper convex function (the function g never takes the value and is not identically equal to + ), −∞ ∞ ] ] 3. for any x X , the function x] is a proper convex function ∈ K 4. for any x] X], the function g is continuous at some point ∈ where ] is finite Kx Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Perturbation + Fenchel-Moreau duality

I To design a dual problem to the original problem

inf h(y) u g(y) y Y ∈
I take a bivariate function : X Y R , K × → where X is a perturbation set, such that (0, y) = h(y) K I then define f (x) = infy Y (x, y) u g(y) , x X ∈ K ∀ ∈ I then take two couplings c and d, and obtain

c ] c+d ] ] d ] ] ] f (x ) inf (x , y ) u g − (y ) , x X ≤ y ] Y] K · ∀ ∈ ∈   I and finally obtain the dual problem

inf h(y) u g(y) = f (0) y∈Y ≥
cc c+d ] c −d ] f (0) sup ( , y ) (0) u g (y ) ] ] ≥ y ∈Y K · · − 
 dual problem | {z } Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP Fenchel conjugates of Bellman functions Application to SDDP

Background on couplings and Fenchel-Moreau conjugacy Background on couplings and Fenchel-Moreau conjugacies Couplings for discrete convexity Couplings for optimal transport

Fenchel-Moreau conjugation inequality with three couplings

Complements The duality equality case Design of dual problems Fenchel-Moreau conjugate of generalized inf-convolution

Conclusion Fenchel conjugate of inf-convolution

The classic inf-convolution

g1g2 (x) = inf g1(y1) u g2(y2) y1+y2=x  
satisfies ? ? ? g1g2 = g1 + g2 ·
The classic inf-convolution corresponds to (y1, x, y2) = δx (y1 + y2): I

g1g2 (x) = infy1+y2=x g1(y1) u g2(y2)  

Definition of generalized inf-convolution Definition

I Let be given three sets X, Y1 and Y2 I For any trivariate convoluting function

: Y1 X Y2 R , I × × → we define the -inf-convolution I of two functions g1 : Y1 R and g2 : Y2 R by → →

I g1g2 (x) = inf g1(y1) u (y1, x, y2) ug2(y2) y1 Y1,y2 Y2 I ∈ ∈  convoluting 
function | {z } Definition of generalized inf-convolution Definition

I Let be given three sets X, Y1 and Y2 I For any trivariate convoluting function

: Y1 X Y2 R , I × × → we define the -inf-convolution I of two functions g1 : Y1 R and g2 : Y2 R by → →

I g1g2 (x) = inf g1(y1) u (y1, x, y2) ug2(y2) y1 Y1,y2 Y2 I ∈ ∈  convoluting 
function | {z } The classic inf-convolution corresponds to (y1, x, y2) = δx (y1 + y2): I

g1g2 (x) = infy1+y2=x g1(y1) u g2(y2)  
Fenchel-Moreau conjugate of generalized inf-convolution Proposition Let be given three “primal” sets X, Y1, Y2 and three “dual” sets ] ] ] X , Y1, Y2, together with three coupling functions

c ] d1 ] d2 ] X X , Y1 Y , Y2 Y ↔ ↔ 1 ↔ 2 For any univariate functions f : X R, → g1 : Y1 R and g2 : Y2 R, all defined→ on the “primal”→ sets, we have that

I f (x) g1 g2 (x) , x X ≥  ∀ ∈ ⇒ ]
c ] ( d1) ( d2) ] ] ] f (x ) g − I g − (x ) , x X , ≤ 1  2 ∀ ∈ where the convoluting function ] on the “dual”
sets is given by I ] = c+d1+d2 I I · · The -inf-convolution is minus the Fenchel-MoreauI conjugate of a sum

Proposition The -inf-convolution is given by I

I g1 g2 = (g1 g2)I  − u ? ? ? This generalizes g1g2 = g1 + g2 ·

Fenchel-Moreau conjugate of generalized inf-convolution Proposition If there exist two coupling functions

] ] Γ1 : X Y1 R , Γ2 : X Y2 R , × → × → such that the partial c-Fenchel-Moreau conjugate of the convoluting function splits as I c ] ] ] (y1, , y2) (x ) = Γ1(x , y1) + Γ2(x , y2) , I · · then the c-Fenchel-Moreau conjugate of the inf-convolution I g1g2 is given by a sum as

I c Γ1 Γ2 g1g2 = g1 + g2 ·
Fenchel-Moreau conjugate of generalized inf-convolution Proposition If there exist two coupling functions

] ] Γ1 : X Y1 R , Γ2 : X Y2 R , × → × → such that the partial c-Fenchel-Moreau conjugate of the convoluting function splits as I c ] ] ] (y1, , y2) (x ) = Γ1(x , y1) + Γ2(x , y2) , I · · then the c-Fenchel-Moreau conjugate of the inf-convolution I g1g2 is given by a sum as

I c Γ1 Γ2 g1g2 = g1 + g2 ·
? ? ? This generalizes g1g2 = g1 + g2 ·
Outline of the presentation

Fenchel conjugates of Bellman functions and application to SDDP

Background on couplings and Fenchel-Moreau conjugacy

Fenchel-Moreau conjugation inequality with three couplings

Complements

Conclusion Conclusion

I We have proven a new Fenchel-Moreau conjugation inequality with three couplings (and given sufficient conditions for the equality case)

I We have provided a general method to design dual problems by means of one kernel and two couplings

I We have introduced a generalized inf-convolution, and have provided formulas for Fenchel-Moreau conjugates

I We have shown that Fenchel conjugates of Bellman functions satisfy a “Bellman like” inequation, and we have sketched an application to the SDDP algorithm Thank you:-)

c X X♯

f(x) infy Y (x,y) ∔ g(y) ≥ ∈ K  

c+d K ⇒ K ·

c ♯ c+d ♯ ♯ d ♯ f (x ) infy♯ Y♯ (x ,y ) ∔ g− (y ) ≤ ∈ K ·  

d Y Y♯