Weierstrass Institute for Applied Analysis and Stochastics Probabilistic Constraints in Optimization with PDEs R. Henrion (Weierstrass Institute Berlin) Based on joint work with H. Farshbaf Shaker, H. Heitsch, D. Hömberg (WIAS, Berlin), M. Gugat (FAU Erlangen), W.V. Ackooij (EDF R&D, Paris) Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de Nov. 13, 2019
A probabilistic program - standard setting Optimization problem with probabilistic constraint: minimize f ( x ) subject to P ( g i ( x, ξ ) ≤ 0 ( i = 1 , . . . , m )) ≥ p } x ∈ X ⊆ R n ξ : s-dimensional random vector (continuously distributed) chronology: x � ξ Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 2 (19)
A probabilistic program - standard setting Optimization problem with probabilistic constraint: minimize f ( x ) subject to P ( g i ( x, ξ ) ≤ 0 ( i = 1 , . . . , m )) ≥ p } x ∈ X ⊆ R n ξ : s-dimensional random vector (continuously distributed) chronology: x � ξ Perspectives: Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 2 (19)
A probabilistic program - standard setting Optimization problem with probabilistic constraint: minimize f ( x ) subject to P ( g ( x, ξ, t ) ≤ 0 ∀ t ∈ T ) ≥ p } x ∈ X ⊆ R n ξ : s-dimensional random vector (continuously distributed) chronology: x � ξ Perspectives: � infinite inequality system Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 2 (19)
A probabilistic program - standard setting Optimization problem with probabilistic constraint: minimize f ( x ) subject to P ( g ( x, ξ, t ) ≤ 0 ∀ t ∈ T ) ≥ p } x ∈ X ⊆ Banach space ξ : s-dimensional random vector (continuously distributed) chronology: x � ξ Perspectives: � infinite inequality system � infinite dimensional decisions Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 2 (19)
A probabilistic program - standard setting Optimization problem with probabilistic constraint: minimize f ( x ) subject to P ( g ( x, ξ, t ) ≤ 0 ∀ t ∈ T ) ≥ p } x ∈ X ⊆ Banach space ξ : s-dimensional random vector (continuously distributed) chronology: x 1 � ξ 1 � x 2 � ξ 2 � · · · � x s � ξ s Perspectives: � infinite inequality system � infinite dimensional decisions Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 2 (19)
A probabilistic program - standard setting Optimization problem with probabilistic constraint: minimize f ( x ) subject to P ( g ( x 1 , x 2 ( ξ 1 ) , . . . , x s ( ξ s − 1 ) , ξ, t ) ≤ 0 ∀ t ∈ T ) ≥ p } x ∈ X ⊆ Banach space ξ : s-dimensional random vector (continuously distributed) chronology: x 1 � ξ 1 � x 2 � ξ 2 � · · · � x s � ξ s Perspectives: � infinite inequality system � infinite dimensional decisions � dynamic decisions Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 2 (19)
Probabilistic state constraints in PDE constrained optimization 1 Simple PDE arising in shape optimization of mechanical structures or crystal growth: −∇ x · ( κ ( x ) ∇ x y ( x )) = r ( x, ξ ) , x ∈ D n · ( κ ( x ) ∇ x y ( x )) + α y ( x ) = u ( x ) x ∈ ∂D, Probabilistic state constraint: P ( y ( x, ξ ) ≤ ¯ ∀ x ∈ C ⊆ D ) ≥ p y Using control to state operator Y ( r, u ) , probabilistic state constraint turns into a probabilistic constraint on the decision (control) variable: P (¯ y − Y ( r ( x, ξ ) , u ( x )) ≥ 0 ∀ x ∈ C ) ≥ p Motivates to investigate optimization problems min { f ( x ) | P ( g ( x, ξ, t ) ≥ 0 ∀ t ∈ T ) ≥ p } � �� � ϕ ( x ) with X Banach space and T arbitrary (maybe compact) index set. 1 Farshbaf-Shaker, H.. D. Hömberg 2018 Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 3 (19)
for g : X × R s × T → R Semicontinuity of ϕ ( x ) := P ( g ( x, ξ, t ) ≥ 0 ∀ t ∈ T ) Proposition Let X be a Banach space. Assume that g is weakly sequentially upper semicontinuous (w.s.u.s.) in the first two arguments. Then, ϕ : X → R is w.s.u.s. In particular, the probabilistic constraint M := { x ∈ X | ϕ ( x ) ≥ p } is weakly sequentially closed. Proposition Assume that 1. g is weakly sequentially lower semicontinuous (w.s.l.s.). 2. T is compact. 3. Let x ∈ X be such that P ( inf t ∈ T g ( x, ξ, t ) = 0) = 0 Then, ϕ is w.s.l.s. at x . The technical condition 3. may be replaced by the easier to verify conditions � ξ has a density. � g is concave in the second argument. z ∈ R m with g ( x, ¯ � There exists ¯ z, t ) > 0 for all t ∈ T . Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 4 (19)
Convexity of the probabilistic constraint M := { x ∈ X | ϕ ( x ) ≥ p } As before, let ϕ ( x ) := P ( g ( x, ξ, t ) ≥ 0 ∀ t ∈ T ) . Theorem (Prekopa) Assume that � g is quasiconcave in the first two variables simultaneously. � ξ has a log-concave density (e.g. Gaussian etc.) Then, the probabilistic constraint defines a convex set M for all p ∈ [0 , 1] . All these properties may be verified by imposing standard assumptions for the simple PDE displayed before. = ⇒ existence of solutions, convex optimization problem (along with convex objective) Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 5 (19)
Spheric-radial decomposition of a Gaussian random vector in R m Let ξ ∼ N (0 , R ) with R = LL T . Then, � P ( ξ ∈ M ) = µ η ( { r ≥ 0 : rLv ∈ M } ) dµ ζ ( v ) , v ∈ S m − 1 where µ η , µ ζ are the laws of η ∼ χ ( m ) and of the uniform distribution on S m − 1 . Sampling of uniform distribution on the sphere M 𝑤 𝑀𝑤 Application to parameter-dependent inequality systems: 2 , 𝑇 𝑛−1 � ϕ ( x ) := P ( g ( x, ξ, t ) ≤ 0 ∀ t ∈ T ) = µ η ( { r ≥ 0 : g ( x, rLv, t ) ≤ 0 ∀ t ∈ T } ) dµ ζ ( v ) v ∈ S m − 1 Obtain ∇ ϕ as another spherical integral by differentiating under the integral (if allowed!) Apply nonlinear programming method to solve optimization problem. 2 Deák (1980,2000), Royset/Polak (2004,2007), W.v. Ackooij, H. (2014,2017) Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 6 (19)
ϕ ( x )= P ( g i ( x, ξ ) ≤ 0 ( i =1 , . . . , m )) Differentiability of Theorem Assume that � X - ref.+sep. B-space, g ∈ C 1 ( X × R m , R p ) and g i ( x, · ) convex. � ϕ (¯ x ) > 0 . 5 at a point of interest ¯ x . � ∃ l > 0 : �∇ x g i ( x, z ) � ≤ le � z � ∀ x ∈ B 1 /l (¯ x ) ∀ z : � z � ≥ l ∀ i = 1 , . . . , m . � rank {∇ z g i (¯ x, z ) , ∇ z g j (¯ x, z ) } = 2 ∀ i � = j ∈ I ( z ) ∀ z : g (¯ x, z ) ≤ 0 , where, I ( z ) := { i | g i (¯ x, z ) = 0 } . Then, ϕ is strictly differentiable at ¯ x and the gradient formula � χ ( ρ (¯ x, v )) � ∇ x g i ∗ ( v ) (¯ ∇ ϕ (¯ x ) = − x, ρ (¯ x, v ) Lv ) dµ ζ ( v ) � ∇ z g i ∗ ( v ) (¯ x, ρ (¯ x, v ) Lv ) , Lv v ∈ S m − 1 𝜍 2 (𝑦, 𝑤 2 ) holds true. Here, i ∗ ( v ) := { i | ρ (¯ x, v ) } . 2 (𝑦, 𝑨) ≤ 0 x, v ) = ρ i (¯ 𝜍(𝑦, 𝑤) 𝑀𝑤 𝜍 1 (𝑦, 𝑤 1 ) μ 1 (𝑦, 𝑨) ≤ 0 Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 7 (19)
Optimal Neumann boundary control for vibrating string 3 For given initial conditions y 0 ∈ H 1 (0 , 1) , y 1 ∈ L 2 (0 , 1) , solve min � u � 2 subject to cost function L 2 (0 ,T ) y (0 , x ) = y 0 ( x ) , y t (0 , x ) = y 1 ( x ) , x ∈ (0 , 1) initial conditions y ( t, 0) = 0 , y x ( t, 1) = u ( t ) , t ∈ (0 , T ) boundary conditions y tt ( t, x ) = c 2 y xx ( t, x ) , ( t, x ) ∈ (0 , T ) × (0 , 1) wave equation y ( T, x ) = 0 , y t ( T, x ) = 0 , x ∈ (0 , 1) terminal conditions 3 Farshbaf-Shaker, Gugat, Heitsch, H. 2019 Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 8 (19)
Analytical solutions Control-to-state operator u �→ y given analytically by y ( t, x ) = � ∞ n =0 α n ( t ) ϕ n ( x ) , where √ �� π � x 2 � √ ϕ n ( x ) := sin 2 + nπ L L �� � �� � α 0 + α 1 1 α n ( t ) := n cos √ λ n c sin λ n c t λ n c t n � t �� � + c 2 ϕ n ( L ) 1 λ n c ( t − s ) √ λ n c u ( s ) sin ds 0 Theorem (Gugat 2015) Let T ≥ 2 , k := max { n ∈ N : 2 n ≤ T } and ∆ := T − 2 k .... For t ∈ [0 , 2) , let � k + 1 , t ∈ (0 , ∆] , d ( t ) := t ∈ (∆ , 2) . k, Then the optimal control u 0 that solves (NEC) is 4–periodic, with � � 1 y ′ 0 (1 − t ) − y 1 (1 − t ) , t ∈ (0 , 1) , 2 d ( t ) u 0 ( t ) = � � 1 y ′ 0 ( t − 1) + y 1 ( t − 1) t ∈ (1 , 2) . , 2 d ( t ) Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 9 (19)
Solution of the deterministic problem For initial conditions y 0 ( x ) = x, y 1 ( x ) = 0 one gets a bang-bang solution: u ( t ) 0.2 0.1 t 1 2 3 4 - 0.1 - 0.2 Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 10 (19)
Solution of the deterministic problem For initial conditions y 0 ( x ) = x, y 1 ( x ) = 0 one gets a bang-bang solution: u ( t ) 0.2 0.1 t 1 2 3 4 - 0.1 - 0.2 How to adapt the problem when initial conditions are random? Epsilon = 0.100, N = 100, T = 4, S = 256 Out-of-sample: Initial State 1 [Initial State y(0,x)] 0.5 0 0 0.25 0.5 0.75 1 [Position x] Probabilistic Constraints in Optimization with PDEs · Nov. 13, 2019 · Page 10 (19)
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