Reduced-Order Approaches for PDE-Constrained Multiobjective - PowerPoint PPT Presentation

Reduced-Order Approaches for PDE-Constrained Multiobjective Optimization joint work with U Dubrovnik / TU Eindhoven / U Erlangen / TU München / U Paderborn Stefan Volkwein University of Konstanz, Chair Numerical Optimization New Trends in PDE-Constrained Optimization, (RICAM, Linz), October 16, 2019 ,

Outline of the Talk 1 Multiobjective Optimization 2 The Reference Point Method 3 ROM for Nonsmooth Optimization 4 Greedy Controllability of Parametrized Linear Systems 5 Conclusions 2 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

1 Multiobjective Optimization 1 Multiobjective Optimization [Banholzer / Beermann / Makarov / Reichle / Spura] [M. Dellnitz / B. Gebken / S. Peitz] 3 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

1 Multiobjective Optimization Multiobjective / Multicriterial Optimization [e.g., Ehrgott’05] Competing goals in many applications : e.g. – production (quality ← → production costs) – transport (travel costs ← → comfort ← → energy consumption) Multiobjective optimization : ˆ J ∈ C 1 ( U ad , R k ) , U Hilbert space ( ˆ subject to ( ˆ P ) min ˆ J 1 ( u ) ,..., ˆ ) J ( u ) = J k ( u ) u ∈ U ad ⊂ U Pareto optimal points : ¯ u ∈ U ad Pareto optimal, if there is no u ∈ U ad such that J j ( u ) ≤ ˆ ˆ u ) for 1 ≤ j ≤ k and J j ( u ) < ˆ ˆ u ) for at least one j ∈ { 1 ,..., k } J j ( ¯ J j ( ¯ Sets : Pareto set P s = � � ¯ u Pareto optimal , Pareto front P f = ˆ { } J ( P s ) ⊂ R k u ∈ U ad ¯ Pareto set Pareto front 40 35 35 30 30 25 25 ← J 1 (u)=6 (u−0.5) 2 − 3 20 20 J 2 axis 15 15 J 2 (u)=5 (u+0.75) 2 − 4 → 10 10 5 Pareto set 5 0 ← Pareto front 0 −5 −10 −5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −5 0 5 10 15 20 25 30 35 u axis J 1 axis 4 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

1 Multiobjective Optimization Optimality Conditions and Optimization Methods Karush-Kuhn-Tucker [Kuhn/Tucker’51] : ¯ u ∈ U ad Pareto optimal, then there is ¯ µ k ) ∈ R k µ = ( ¯ µ 1 ,..., ¯ ⟨ k ⟨ ˆ ⟩ k k J ′ ⟩ µ i ˆ J ′ ≥ 0 for all u ∈ U ad µ i ≤ 1 , µ i = 1 , µ i 0 ≤ ¯ ∑ ¯ ∑ ¯ i ( ¯ u ) , u − ¯ u U = ∑ ¯ i ( ¯ u ) , u − ¯ u i = 1 i = 1 i = 1 U k → first-order (necessary) optimality conditions for min ˆ µ i ˆ G ( u ; ¯ µ ) = J i ( u ) ∑ ¯ u ∈ U ad i = 1 k Weighted sum method : solve ¯ G ( u ; µ ) with 0 ≤ µ i ≤ 1 and ˆ u µ = argmin µ i = 1 ∑ i = 1 u ∈ U ad 2 Euclidean reference point method [Wierzbicki’79] : min F T ( u ) = 1 ˆ 2 ∥ T − ˆ 2 for reference point T = ( T 1 ,..., T k ) ⊤ J ( u ) ∥ u ∈ U ad Sets : Pareto set P s = � � ¯ u Pareto optimal , Pareto front P f = ˆ { } J ( P s ) ⊂ R k u ∈ U ad ¯ Pareto set Pareto front Reference point method 40 35 12 35 10 30 30 8 25 25 ← J 1 (u)=6 (u−0.5) 2 − 3 6 20 20 J 2 axis J 2 axis ← computed efficient point 4 15 15 J 2 (u)=5 (u+0.75) 2 − 4 → ← new efficient point 10 2 10 Point T 5 0 Pareto set 5 ← Pareto front 0 −2 ← Pareto front 0 −5 −4 −10 −5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −5 0 5 10 15 20 25 30 35 −5 0 5 10 15 u axis J 1 axis J 1 axis 5 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

2 The Reference Point Method 2 The Reference Point Method [Banholzer / Beermann / Makarov / Reichle / Spura] 6 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

2 The Reference Point Method Scalarization by Introduction of a Distance Function T k ) ⊤ ∈ R k with ˆ Ideal vector : ˆ J i ( u ) for i = 1 ,..., k T = ( ˆ T 1 ,..., ˆ ˆ T i = min u ∈ U ad Define : g : R k → R , where, e.g., g ( s ) = 1 2 ∥ s ∥ 2 2 or g ( s ) = ∥ s ∥ ∞ { ˜ Reference point problem : for chosen T = ( T 1 ,..., T k ) ⊤ ∈ R k T ∈ R k | ˜ T ≤ ˆ } consider T = T ≤ ˆ subject to ( ˆ P T ) ˆ ( T − ˆ ) F T ( u ) = g J ( u ) u ∈ U ad min u ∈ U ad Weighted Sum Method global 4 ERPM with a-priori reference points; global RPM with p = 10; global RPM Maximum Norm; global 4 4 4 Objective Space Reference points Reference points Reference points Pareto points Pareto points Pareto points Pareto points 2 2 2 2 Objective Space Objective Space Objective Space Example : 0 0 0 0 J 1 ( u ) = u 2 ˆ 10 − 2 u 5 − 23 -2 -2 -2 -2 5 -4 -4 -4 -4 J 2 ( u ) = u 4 40 + 2 u 3 15 − u 2 ˆ 4 -6 -6 -6 -6 -8 -8 -8 -8 -6 -4 -2 0 2 -6 -4 -2 0 2 -6 -4 -2 0 2 -6 -4 -2 0 2 7 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

2 The Reference Point Method Euclidean Reference Point Method � ¯ Sets : Pareto set P s = { � u Pareto optimal } u ∈ U ad ¯ Sets : Pareto front P f = ˆ J ( P s ) ⊂ R k Reference point : choose T = ( T 1 ,..., T k ) ⊤ ∈ P f + R k z + x | z ∈ P f and R k ∋ x ≤ 0 { } ≤ 0 = Reference point problem : consider k � 2 � 2 F T ( u ) = 1 ˆ � � T − ˆ � 2 = 1 � T i − ˆ � � subject to ( ˆ P T ) J ( u ) J i ( u ) u ∈ U ad min ∑ 2 2 u ∈ U ad i = 1 k First derivative : ˆ F ′ ( ˆ J i ( u ) − T i ) ⟨ ˆ J ′ T ( u ) = ∑ i ( u ) , •⟩ U ∈ L ( U , R ) i = 1 First-order optimality conditions : If ¯ u T solves ( ˆ P T ) we have ( ˆ )⟨ ˆ k J ′ ⟩ for all u ∈ U ad ∑ J i ( ¯ u T ) − T i i ( ¯ u T ) , u − ¯ u T U ≥ 0 i = 1 8 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

2 The Reference Point Method Computation of the Pareto Set and Pareto Front Sets : Pareto set P s = { � � ¯ u Pareto optimal } , Pareto front P f = ˆ J ( P ) ⊂ R k u ∈ U ad ¯ Algorithm 1 (Euclidean reference point method) ) ⊤ and set i = 1 ; 1: Choose a reference point T ( 1 ) = ( T ( 1 ) ,..., T ( 1 ) 1 k 2: Compute for the distance function k � 2 ≥ 0 � T ( i ) − ˆ � 2 � T ( i ) F ( i ) ( u ) = 1 ˆ � � 2 = 1 � − ˆ � J ( u ) ∑ J j ( u ) 2 2 j j = 1 the solution to the scalar reference point problem { } u ( i ) F ( i ) P ( i ) ˆ � ( ˆ T ) ¯ T = argmin T ( u ) � u ∈ U ad 3: Choose a new reference point T ( i + 1 ) and go back to step 2. J i = ˆ u ( i ) Variation of the reference points : for ˆ T ) set J ( ¯ ϕ || ϕ ⊥ T ( i + 1 ) : = ˆ J i + h || · ∥ ϕ || ∥ + h ⊥ · for i ≥ 1 ∥ ϕ ⊥ ∥ and choose h || , h ⊥ > 0 to control the coarseness of P f points 9 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

2 The Reference Point Method Heat Equation with Convection Bicriterial optimal control problem : minimize ∫ t f   ∫ � 2 d x d t � y ( t , x ) − y d ( t , x ) � � 2 ( ) ∥ y − y d ∥ J ( y , u ) = 1  = 1 0 Ω   L 2 ( Q )   ∫ t f m 2   2 � 2 d t ∥ u ∥ 2 ∑ � �  � u i ( t ) U 0 i = 1 subject to the parabolic convection-diffusion PDE m ( t , x ) ∈ Q = ( 0 , t f ) × Ω , Ω = Ω 1 ˙ ∪ ... ˙ y t ( t , x ) − ∆ y ( t , x )+ b ( t , x ) · ∇ y ( t , x ) = ∑ u i ( t ) χ Ω i ( x ) , ∪ Ω m i = 1 ( SE ) ∂ y ∂ n ( t , x )+ α i y ( t , x ) = α i y a ( t ) , ( t , x ) ∈ Σ i = ( 0 , t f ) × Γ i , 1 ≤ i ≤ r x ∈ Ω ⊂ R d , d ∈ { 1 , 2 , 3 } y ( 0 , x ) = y ◦ ( x ) , and the bilateral control constraints u ∈ U ad = { u ∈ U = L 2 ( 0 , t f ; R m ) � u a ( t ) ≤ u ( t ) ≤ u b ( t ) in [ 0 , t f ] � } Assumptions : y d ∈ L 2 ( 0 , t f ; L 2 ( Ω )) , b bounded, χ Ω i ∈ L 2 ( Ω ) , α i ≥ 0 , y a ∈ L 2 ( 0 , t f ) , y ◦ ∈ L 2 ( Ω ) , u a ≤ u b in U Bilinear form for ( SE ): for ϕ , ψ ∈ H 1 ( Ω ) and t ∈ [ 0 , t f ] define r r ∫ ∫ ∫ ( ) ∑ ∑ a ( t ; ϕ , ψ ) = Ω ∇ ϕ · ∇ ψ + b ( t , · ) ∇ ϕ ψ d x + α i ϕψ d x , ⟨ g a ( t ) , ϕ ⟩ = α i y a ( t ) ϕ d x Γ i Γ i i = 1 i = 1 10 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein

2 The Reference Point Method Reduced Formulation for the Multiobjective Problem Bilinear form for ( SE ): for ϕ , ψ ∈ H 1 ( Ω ) and t ∈ [ 0 , t f ] define r r ∫ ∫ ∫ ( ) ∑ ∑ a ( t ; ϕ , ψ ) = Ω ∇ ϕ · ∇ ψ + b ( t , · ) · ∇ ϕ ψ d x + α i ϕψ d x , ⟨ g a ( t ) , ϕ ⟩ = α i y a ( t ) ϕ d x Γ i Γ i i = 1 i = 1 Control-to-state operator : y = S ( u ) solves for given control u ∈ U = L 2 ( 0 , t f ; R m ) m d ∫ ϕ d x for all ϕ ∈ H 1 ( Ω ) and ∑ d t ⟨ y ( t ) , ϕ ⟩ L 2 ( Ω ) + a ( t ; y ( t ) , ϕ ) = ⟨ g a ( t ) , ϕ ⟩ + u i ( t ) y ( 0 ) = y ◦ Ω i i = 1 ( ˆ 2 ( ) ( ) ∥ S ( u ) − y d ∥ ) J 1 ( S ( u ) , u ) J 1 ( u ) = 1 L 2 ( Q ) Reduced cost functional : ˆ for u ∈ U J ( u ) = = ˆ J 2 ( u ) J 2 ( S ( u ) , u ) 2 ∥ u ∥ 2 U Reduced multicriterial optimal control problem : min ˆ subject to { � � u a ≤ u ≤ u b in [ 0 , t f ] } ( ˆ P ) J ( u ) u ∈ U ad = u ∈ U 11 / 30 PDE-Constrained Multiobjective Optimization by ROM Stefan Volkwein