Strong Convexity for Risk-Averse Two-Stage Models with Linear - - PowerPoint PPT Presentation

Scope Main results Applications

Strong Convexity for Risk-Averse Two-Stage Models with Linear Recourse

Matthias Claus1, Kai Spurkel1

1University of Duisburg-Essen, Germany

CMS / MMEI March 27-29, 2019, Chemnitz, Germany

Scope Main results Applications

Overview

1

Scope

2

Main results

3

Applications

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) → ˜ Qρ(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) − Tξ, y ≥ 0}

Scope Main results Applications

The model II Goal: Structural properties of the objective in min{ ρ[ f(ξ) + ϕ(z(ω) − Tξ) ] : ξ ∈ X} Reformulate as (translation-equivariance !) min{ˆ f(x) + Qρ(x) | x ∈ T(X)} with reduced function Qρ(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): Qνg(x) =

• Qα CVaR(x) νg(dα)

(νg prob-measure induced by concave distortion). CVaR has variational representation (due to Rockafellar and Uryasev) QαCVaR(x) = min

η∈R

• η + 1

αQEE(x, η)

• where QEE is the expected excess over threshold η:

QEE(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(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y) − κ 2λ(1 − λ)x − y2 with some constant κ > 0. For continuously differentiable f this is equivalent to (Kachurovskii’s theorem) (f ′(y) − f ′(x))(y − x) ≥ κ y − x2.

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: ϕ(t) = max

i

d⊺

i t

Linearity regions of ϕ → normal cones to {u | W ⊺u ≤ q} at its vertices {di}i∈I, denoted by Ki

Scope Main results Applications

Risk-neutral setting Risk-neutral model (ρ = E): QE(x) = Eω

• ϕ(z(ω) − x)
• = Eµ
• ϕ(z − x)
• Some more assumptions:

A3 finite first moments required µ has a density ⇒ QE 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 QE 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 QEE under assumption of an appropriate upper bound for threshhold η (A5) and stronger assumption (A2’) Let V ⊂ Rn and W ⊂ Rm nonempty and convex. A function f : V × W → R is called partially κ-strongly convex with respect to its first argument if f(λ(x1, y1) + (1 − λ)(x2, y2)) ≤λf(x1, y1) + (1 − λ)f(x2, y2) − κ 2λ(1 − λ)x1 − x22 holds for all x1, x2 ∈ V, y1, y2 ∈ W, 0 < λ < 1.

Scope Main results Applications

Main results I Conditions A5 and A2’: Collect i ∈ I with int(KI) = ∅ in I+. Hyperplane {d⊺

i z = η}

intersects extreme rays of Ki in points ˆ yi,j. Denote ˆ yi one with minimal norm. A5 η0 is such that for every i ∈ I+ it holds ˆ yi(η0) < ρ or there exists an index set Ji ⊂ I such that −Ki ⊂

j∈Ji K + j

with ˆ yj(η0) < r for all j ∈ Ji (where r is the one given in A4). A2’ 0 ∈ int

• {W ⊺y ≤ q}
• (second stage costs positive

componentwise) ⇒ Partial strong convexity of QEE wrt. x on Vη0 = V × (−∞, η0]

Scope Main results Applications

Main results I Conditions A5 and A2’: Collect i ∈ I with int(KI) = ∅ in I+. Hyperplane {d⊺

i z = η}

intersects extreme rays of Ki in points ˆ yi,j. Denote ˆ yi one with minimal norm. A5 η0 is such that for every i ∈ I+ it holds ˆ yi(η0) < ρ or there exists an index set Ji ⊂ I such that −Ki ⊂

j∈Ji K + j

with ˆ yj(η0) < r for all j ∈ Ji (where r is the one given in A4). A2’ 0 ∈ int

• {W ⊺y ≤ q}
• (second stage costs positive

componentwise) ⇒ Partial strong convexity of QEE wrt. x on Vη0 = V × (−∞, η0] With a little bit more technical effort - restricted partial strong convexity of QEE 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 QEE given A1-A5 Includes the case of fixed η There is some δ > 0 such that... QEE(λ(x, η1) + (1 − λ)(y, η2)) ≤λf(x, η1) + (1 − λ)f(y, η2) − κ 2λ(1 − λ)x − y2 holds for all 0 < λ < 1 and all (x, η1), (y, η2) ∈ Vη0 with y − x ∈ I+ and η2 − η1 ≥ −δ 3y − x or x − y ∈ I+ and η1 − η2 ≥ −δ 3y − x.

Scope Main results Applications

Scope Main results Applications

Main results I QEE(x, η) partially strongly convex with respect to x on some set Vη0 = V × (−∞, η0] Extend strong convexity to CVaR Remember that αVaR is minimizer in variational representation of CVaR: ηi = QαVaR(xi) QαCVaR(λx1 + (1 − λ)x2) = min

η∈R

• η + 1

αQEE(λx1 + (1 − λ)x2, η)

• ≤ λη1 + (1 − λ)η2 + 1

αQEE(λx1 + (1 − λ)x2, λη1 + (1 − λ)η2) ≤ λ [η1 + 1 αQEE(x1, η1)] + (1 − λ) [η2 + 1 αQEE(x2, η2)] − κ αλ(1 − λ)x1 − x22 = λ QαCVaR(x1) + (1 − λ) QαCVaR(x2) − κ αλ(1 − λ)x1 − x22

Scope Main results Applications

Main results II Strong convexity of Qα CVaR: Assume A1-A5, A2’ (in particular, there is some η0 > 0 satisfying A5) and the following condition sup

x∈V

QαVaR(x) ≤ η0. Then Qα CVaR is κ

α-strongly convex on V with κ being the

modulus of partial strong convexity for QEE for η ≤ η0.

Scope Main results Applications

Main results II A counterexample: ϕ(t) = max{0, t} and µ uniform distribution on (0, 1). QαCVaR(x) = min

η∈R

• η + 1

2α(1 − x − η)2 − x + 1 2(2 − α) ⇒ QαCVaR not strongly convex on any U ⊂ (0, 1)! A2’ cannot be dropped easily

Scope Main results Applications

Main results III - spectral risk functions Qνg(x) =

• Qα CVaR(x) νg(dα)

with

(0,1]

1 α νg(dα) < ∞

• Assume κ-strong convexity of Qα0CVaR on V for some

0 < α0 < 1 and c = 1 − g(α0) + α0 g′(α0) > 0, where concave g generates νg via νg

• (0, t]
• = g(t) − t g′(t).

⇒ Qνg 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: Mp

s := {µ ∈ P(Rs) |

• Rs tp µ(dt) < ∞}

with p-th order Wasserstein metric Wp(µ, ν) := inf

κ Rs×Rsv − ˜

vp κ(d(v, ˜ v)) | κ ∈ P(Rs × Rs), κ ◦ π−1

1

= µ, π−1

2

= ν 1

p

and Qνg : Rs × Mp

s given by

Qνg(x, µ) =

• (0,1]

QαCVaR(x, µ) νg(dα)

Scope Main results Applications

Implications on stability theory Parametric problem in µ: min

x {ˆ

h(x) + Qνg(x, µ) | x ∈ T(X))} P(µ) with set of optimal solutions Ψg(µ). Theorem: Let X ⊆ Rn be nonempty, closed and convex and let µ0 ∈ Mp

s be such that Qνg(·, µ0) is κ-strongly convex on some

nonempty, open convex set V satisfying Ψg(µ0) ∩ V = ∅. Furthermore, assume that L :=

• (0,1]

1 α

1 p

νg(dα) < ∞. Then there exists a constant r > 0 such that for any µ ∈ Mp

s

satisfying dp(µ, µ0) ≤ r we have Ψg(µ) = ∅ and dH(Ψg(µ0), Ψg(µ)) ≤ 2

• L · maxi∈I di

κ · dp(µ0, µ).

Scope Main results Applications

Future Research Mean-risk models and composite risk functions based on higher moments Partial random data on right-hand side - extension to CVaR-based models

Scope Main results Applications

Literature Claus, Matthias; Schultz, Ruediger; Spuerkel, Kai: Strong convexity in risk-averse stochastic programs with complete recourse Computational Management Science, 15(3), pp. 411-429, (2018) Claus, Matthias; Spuerkel, Kai: Strong Convexity for Risk-Averse Two-Stage Models with Fixed Complete Linear Recourse Preprint: https://arxiv.org/abs/1812.08109

• R. Schultz:

Strong Convexity in Stochastic Programs with Complete Recourse, Journal of Computational and Applied Mathematics 56, pp. 3-22, (1994)

Scope Main results Applications

Thanks for your attention