Scope Main results Applications Strong Convexity for Risk-Averse Two-Stage Models with Linear Recourse Matthias Claus 1 , Kai Spurkel 1 1 University of Duisburg-Essen, Germany CMS / MMEI March 27-29, 2019, Chemnitz, Germany
Scope Main results Applications Overview Scope 1 Main results 2 Applications 3
Scope Main results Applications Two-Stage Stochastic Programming Philosophy of two-stage stochastic programming: Decide on policy → observe random events → take compensating action. Take all information on the randomness into account to make an optimal decision a-priori (optimal wrt. to some criterion)
Scope Main results Applications Two-Stage Stochastic Programming Philosophy of two-stage stochastic programming: Decide on policy → observe random events → take compensating action. Take all information on the randomness into account to make an optimal decision a-priori (optimal wrt. to some criterion) Our setting: Objective function given as sum of first-stage loss f ( x ) and random recourse costs ϕ ( x , ω ) , i.e. → ˜ f ( x ) + ϕ ( x , ω ) Q ρ ( x ) = ρ [ f ( x ) + ϕ ( x , ω )] with some risk measure ρ .
Scope Main results Applications The model Available policies x → vectors in a finite-dimensional Euclidian space. Randomness given as random vectors z ( ω ) with known distribution Recourse costs ϕ → optimal value of a linear program with randomness on the right-hand side ϕ ( ξ, ω ) = min { q ⊺ y | Wy = z ( ω ) − T ξ, y ≥ 0 }
Scope Main results Applications The model II Goal: Structural properties of the objective in min { ρ [ f ( ξ ) + ϕ ( z ( ω ) − T ξ ) ] : ξ ∈ X } Reformulate as (translation-equivariance !) min { ˆ f ( x ) + Q ρ ( x ) | x ∈ T ( X ) } with reduced function Q ρ ( x ) = ρ [ ϕ ( z ( ω ) − x )] .
Scope Main results Applications The model III Risk-aversion addressed by appropriate choice of risk measure ρ : We consider spectral risk-measures given as weighted averages of the conditional value-at-risk (CVaR): � Q ν g ( x ) = Q α CVaR ( x ) ν g ( d α ) ( ν g prob-measure induced by concave distortion). CVaR has variational representation (due to Rockafellar and Uryasev) η + 1 � � Q α CVaR ( x ) = min α Q EE ( x , η ) η ∈ R where Q EE is the expected excess over threshold η : � � Q EE ( x , η ) = E [ ϕ ( z ( ω ) − x ) − η ] + .
Scope Main results Applications Strong convexity A function on some convex set is called κ -strongly convex with modulus κ > 0 if for all x , y in this set and all 0 < λ < 1 f ( λ x + ( 1 − λ ) y ) ≤ λ f ( x ) + ( 1 − λ ) f ( y ) − κ 2 λ ( 1 − λ ) � x − y � 2 with some constant κ > 0. For continuously differentiable f this is equivalent to (Kachurovskii’s theorem) ( f ′ ( y ) − f ′ ( x ))( y − x ) ≥ κ � y − x � 2 .
Scope Main results Applications Linear Recourse � � ϕ ( t ) = min q ⊺ y | Wy = t , y ≥ 0 . Well defined function under the assumptions A1 For all t there exists some y ≥ 0 with Wy = t (complete recourse). A2 The set { u | W ⊺ u < q } is nonempty (strengthened sufficiently expensive recourse). By LP-duality: d ⊺ ϕ ( t ) = max i t i Linearity regions of ϕ → normal cones to { u | W ⊺ u ≤ q } at its vertices { d i } i ∈ I , denoted by K i
Scope Main results Applications Risk-neutral setting Risk-neutral model ( ρ = E ): � � � � Q E ( x ) = E ω ϕ ( z ( ω ) − x ) = E µ ϕ ( z − x ) Some more assumptions: A3 finite first moments required µ has a density ⇒ Q E is cts. differentiable A4 µ has a density which is bounded by a positive constant some convex, open set V + rB ( r > 0 ) . A1-A4 ⇒ STRONG convexity of Q E on V (due to Schultz, 1994) Strong convexity can be shown by monotonicity of Q ′ E (Kachurovskii).
Scope Main results Applications Main results I Partial strong convexity for Q EE under assumption of an appropriate upper bound for threshhold η (A5) and stronger assumption (A2’) Let V ⊂ R n and W ⊂ R m nonempty and convex. A function f : V × W → R is called partially κ -strongly convex with respect to its first argument if f ( λ ( x 1 , y 1 ) + ( 1 − λ )( x 2 , y 2 )) ≤ λ f ( x 1 , y 1 ) + ( 1 − λ ) f ( x 2 , y 2 ) − κ 2 λ ( 1 − λ ) � x 1 − x 2 � 2 holds for all x 1 , x 2 ∈ V , y 1 , y 2 ∈ W , 0 < λ < 1.
Scope Main results Applications Main results I Conditions A5 and A2’: Collect i ∈ I with int ( K I ) � = ∅ in I + . Hyperplane { d ⊺ i z = η } y i one with intersects extreme rays of K i in points ˆ y i , j . Denote ˆ minimal norm. A5 η 0 is such that for every i ∈ I + it holds � ˆ y i ( η 0 ) � < ρ or there j ∈ J i K + exists an index set J i ⊂ I such that − K i ⊂ � with j y j ( η 0 ) � < r for all j ∈ J i (where r is the one given in A4). � ˆ � � A2’ 0 ∈ int { W ⊺ y ≤ q } (second stage costs positive componentwise) ⇒ Partial strong convexity of Q EE wrt. x on V η 0 = V × ( −∞ , η 0 ]
Scope Main results Applications Main results I Conditions A5 and A2’: Collect i ∈ I with int ( K I ) � = ∅ in I + . Hyperplane { d ⊺ i z = η } y i one with intersects extreme rays of K i in points ˆ y i , j . Denote ˆ minimal norm. A5 η 0 is such that for every i ∈ I + it holds � ˆ y i ( η 0 ) � < ρ or there j ∈ J i K + exists an index set J i ⊂ I such that − K i ⊂ � with j y j ( η 0 ) � < r for all j ∈ J i (where r is the one given in A4). � ˆ � � A2’ 0 ∈ int { W ⊺ y ≤ q } (second stage costs positive componentwise) ⇒ Partial strong convexity of Q EE wrt. x on V η 0 = V × ( −∞ , η 0 ] With a little bit more technical effort - restricted partial strong convexity of Q EE without A2’
Scope Main results Applications Figure: Truncated cones
Scope Main results Applications Figure: Even more truncated cones
Scope Main results Applications Main results I Restricted partial strong convexity for Q EE given A1-A5 Includes the case of fixed η There is some δ > 0 such that... Q EE ( λ ( x , η 1 ) + ( 1 − λ )( y , η 2 )) ≤ λ f ( x , η 1 ) + ( 1 − λ ) f ( y , η 2 ) − κ 2 λ ( 1 − λ ) � x − y � 2 holds for all 0 < λ < 1 and all ( x , η 1 ) , ( y , η 2 ) ∈ V η 0 with y − x ∈ I + and η 2 − η 1 ≥ − δ 3 � y − x � or x − y ∈ I + and η 1 − η 2 ≥ − δ 3 � y − x � .
Scope Main results Applications
Scope Main results Applications Main results I Q EE ( x , η ) partially strongly convex with respect to x on some set V η 0 = V × ( −∞ , η 0 ] Extend strong convexity to CVaR Remember that α VaR is minimizer in variational representation of CVaR: η i = Q α VaR ( x i ) η + 1 � � Q α CVaR ( λ x 1 + ( 1 − λ ) x 2 ) = min α Q EE ( λ x 1 + ( 1 − λ ) x 2 , η ) η ∈ R ≤ λη 1 + ( 1 − λ ) η 2 + 1 α Q EE ( λ x 1 + ( 1 − λ ) x 2 , λη 1 + ( 1 − λ ) η 2 ) ≤ λ [ η 1 + 1 α Q EE ( x 1 , η 1 )] + ( 1 − λ ) [ η 2 + 1 α Q EE ( x 2 , η 2 )] − κ αλ ( 1 − λ ) � x 1 − x 2 � 2 = λ Q α CVaR ( x 1 ) + ( 1 − λ ) Q α CVaR ( x 2 ) − κ αλ ( 1 − λ ) � x 1 − x 2 � 2
Scope Main results Applications Main results II Strong convexity of Q α CVaR : Assume A1-A5, A2’ (in particular, there is some η 0 > 0 satisfying A5) and the following condition Q α VaR ( x ) ≤ η 0 . sup x ∈ V Then Q α CVaR is κ α -strongly convex on V with κ being the modulus of partial strong convexity for Q EE for η ≤ η 0 .
Scope Main results Applications Main results II A counterexample: ϕ ( t ) = max { 0 , t } and µ uniform distribution on ( 0 , 1 ) . η + 1 2 α ( 1 − x − η ) 2 � � Q α CVaR ( x ) = min η ∈ R − x + 1 2 ( 2 − α ) ⇒ Q α CVaR not strongly convex on any U ⊂ ( 0 , 1 ) ! A2’ cannot be dropped easily
Scope Main results Applications Main results III - spectral risk functions � � � 1 � Q ν g ( x ) = Q α CVaR ( x ) ν g ( d α ) α ν g ( d α ) < ∞ with ( 0 , 1 ] Assume κ -strong convexity of Q α 0 CVaR on V for some 0 < α 0 < 1 and c = 1 − g ( α 0 ) + α 0 g ′ ( α 0 ) > 0 , � � where concave g generates ν g via ν g ( 0 , t ] = g ( t ) − t g ′ ( t ) . ⇒ Q ν g strongly convex on V with modulus κ c
Scope Main results Applications Implications on stability theory Setting for stability/perturbation analysis of risk-averse two-stage programs: � R s � t � p µ ( d t ) < ∞} M p s := { µ ∈ P ( R s ) | with p-th order Wasserstein metric � � v )) | κ ∈ P ( R s × R s ) , v � p κ ( d ( v , ˜ R s × R s � v − ˜ W p ( µ, ν ) := inf κ � 1 κ ◦ π − 1 = µ, π − 1 p = ν 1 2 and Q ν g : R s × M p s given by � Q ν g ( x , µ ) = Q α CVaR ( x , µ ) ν g ( d α ) ( 0 , 1 ]
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