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Conclusions from classical parametric integer programming for stochastic optimization Matthias Claus University of Duisburg-Essen January 4, 2016 M. Claus Conclusions from classical parametric integer programming January 4, 2016 1 / 17 From
University of Duisburg-Essen
Conclusions from classical parametric integer programming January 4, 2016 1 / 17
Take a parametric mixed-integer program (Pz) minx,y{c(x) + q(y) | x ∈ X, y ∈ C(x, z), y ∈ Rm1 × Zm2},
Conclusions from classical parametric integer programming January 4, 2016 2 / 17
Take a parametric mixed-integer program (Pz) minx,y{c(x) + q(y) | x ∈ X, y ∈ C(x, z), y ∈ Rm1 × Zm2}, add an information constraint decide x − →
− → decide y
Conclusions from classical parametric integer programming January 4, 2016 2 / 17
Take a parametric mixed-integer program (Pz) minx,y{c(x) + q(y) | x ∈ X, y ∈ C(x, z), y ∈ Rm1 × Zm2}, add an information constraint decide x − →
− → decide y and assume purely exogenous stochastic uncertainty z = z(ω).
Conclusions from classical parametric integer programming January 4, 2016 2 / 17
Take a parametric mixed-integer program (Pz) minx,y{c(x) + q(y) | x ∈ X, y ∈ C(x, z), y ∈ Rm1 × Zm2}, add an information constraint decide x − →
− → decide y and assume purely exogenous stochastic uncertainty z = z(ω). → Two-stage-formulation: min{c(x) + min {q(y) | y ∈ C(x, z(ω)), y ∈ Rm1 × Zm2}
| x ∈ X}
Conclusions from classical parametric integer programming January 4, 2016 2 / 17
Take a parametric mixed-integer program (Pz) minx,y{c(x) + q(y) | x ∈ X, y ∈ C(x, z), y ∈ Rm1 × Zm2}, add an information constraint decide x − →
− → decide y and assume purely exogenous stochastic uncertainty z = z(ω). → Two-stage-formulation: min{c(x) + min {q(y) | y ∈ C(x, z(ω)), y ∈ Rm1 × Zm2}
| x ∈ X} Task: Pick an ”optimal” random variable taking into account risk aversion.
Conclusions from classical parametric integer programming January 4, 2016 2 / 17
Take a parametric mixed-integer program (Pz) minx,y{c(x) + q(y) | x ∈ X, y ∈ C(x, z), y ∈ Rm1 × Zm2}, add an information constraint decide x − →
− → decide y and assume purely exogenous stochastic uncertainty z = z(ω). → Two-stage-formulation: min{c(x) + min {q(y) | y ∈ C(x, z(ω)), y ∈ Rm1 × Zm2}
| x ∈ X} Task: Pick an ”optimal” random variable taking into account risk aversion. → Mean risk models: min{ρ(f(x, z(ω))) | x ∈ X}
Conclusions from classical parametric integer programming January 4, 2016 2 / 17
min{ρ(f(x, z(ω))) | x ∈ X} is a parametric problem w.r.t. the distribution of z. → Stability of optimal values, solution sets?
Conclusions from classical parametric integer programming January 4, 2016 3 / 17
min{ρ(f(x, z(ω))) | x ∈ X} is a parametric problem w.r.t. the distribution of z. → Stability of optimal values, solution sets? Example: minx∈Z E[χ{0}(z(ω))(x2 + λ)], where λ ∈ R is fixed and P(z(ω) = 0) = 1.
Conclusions from classical parametric integer programming January 4, 2016 3 / 17
min{ρ(f(x, z(ω))) | x ∈ X} is a parametric problem w.r.t. the distribution of z. → Stability of optimal values, solution sets? Example: minx∈Z E[χ{0}(z(ω))(x2 + λ)], where λ ∈ R is fixed and P(z(ω) = 0) = 1. → Unique optimal solution x = 0 yields the value λ.
Conclusions from classical parametric integer programming January 4, 2016 3 / 17
min{ρ(f(x, z(ω))) | x ∈ X} is a parametric problem w.r.t. the distribution of z. → Stability of optimal values, solution sets? Example: minx∈Z E[χ{0}(z(ω))(x2 + λ)], where λ ∈ R is fixed and P(z(ω) = 0) = 1. → Unique optimal solution x = 0 yields the value λ. Consider the random variables zǫ(·) defined by P(zǫ(ω) = ǫ) = 1 and solve minx∈Z E[χ{0}(zǫ(ω))(x2 + λ)].
Conclusions from classical parametric integer programming January 4, 2016 3 / 17
min{ρ(f(x, z(ω))) | x ∈ X} is a parametric problem w.r.t. the distribution of z. → Stability of optimal values, solution sets? Example: minx∈Z E[χ{0}(z(ω))(x2 + λ)], where λ ∈ R is fixed and P(z(ω) = 0) = 1. → Unique optimal solution x = 0 yields the value λ. Consider the random variables zǫ(·) defined by P(zǫ(ω) = ǫ) = 1 and solve minx∈Z E[χ{0}(zǫ(ω))(x2 + λ)]. → For ǫ = 0, every integer is an optimal solution with value 0.
Conclusions from classical parametric integer programming January 4, 2016 3 / 17
min{ρ(f(x, z(ω))) | x ∈ X} is a parametric problem w.r.t. the distribution of z. → Stability of optimal values, solution sets? Example: minx∈Z E[χ{0}(z(ω))(x2 + λ)], where λ ∈ R is fixed and P(z(ω) = 0) = 1. → Unique optimal solution x = 0 yields the value λ. Consider the random variables zǫ(·) defined by P(zǫ(ω) = ǫ) = 1 and solve minx∈Z E[χ{0}(zǫ(ω))(x2 + λ)]. → For ǫ = 0, every integer is an optimal solution with value 0. Conclusion: No stability in two-stage SP if the underlying deterministic mixed-integer problem is not ”well behaved”.
Conclusions from classical parametric integer programming January 4, 2016 3 / 17
Question: What has to be assumed of f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}?
Conclusions from classical parametric integer programming January 4, 2016 4 / 17
Question: What has to be assumed of f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}? Sufficient: f is defined by a MILP f(x, z) = c⊤x + min{q⊤y | Wy = z − Tx, y ∈ Rm1
≥0 × Zm2 ≥0},
the matrix W is rational and (A1) W(Rm1
≥0 × Zm2 ≥0) = Rs,
(A2) {u ∈ Rs | W ⊤u ≤ q} = ∅.
Conclusions from classical parametric integer programming January 4, 2016 4 / 17
Question: What has to be assumed of f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}? Sufficient: f is defined by a MILP f(x, z) = c⊤x + min{q⊤y | Wy = z − Tx, y ∈ Rm1
≥0 × Zm2 ≥0},
the matrix W is rational and (A1) W(Rm1
≥0 × Zm2 ≥0) = Rs,
(A2) {u ∈ Rs | W ⊤u ≤ q} = ∅. → Schultz, Tiedemann (2006), R¨
Conclusions from classical parametric integer programming January 4, 2016 4 / 17
Question: What has to be assumed of f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}? Sufficient: f is defined by a MILP f(x, z) = c⊤x + min{q⊤y | Wy = z − Tx, y ∈ Rm1
≥0 × Zm2 ≥0},
the matrix W is rational and (A1) W(Rm1
≥0 × Zm2 ≥0) = Rs,
(A2) {u ∈ Rs | W ⊤u ≤ q} = ∅. → Schultz, Tiedemann (2006), R¨
Improvement by Claus, Kr¨ atschmer, Schultz (2015): Assume that f is continuous almost everywhere and fulfills a growth condition: (G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs.
Conclusions from classical parametric integer programming January 4, 2016 4 / 17
f(x, z) = c⊤x + min{q⊤y | Wy = z − Tx, y ∈ Rm1
≥0 × Zm2 ≥0}
Conclusions from classical parametric integer programming January 4, 2016 5 / 17
f(x, z) = c⊤x + min{q⊤y | Wy = z − Tx, y ∈ Rm1
≥0 × Zm2 ≥0}
Assume (A1), (A2) and the rationality of W. Then (i) f is real valued and lower semicontinuous on Rn × Rs. (ii) f is continuous on (Rn × Rs) \ A, where the (n + s)-dim. Lebesgue measure of A = (−T, I)−1(bd W(Rm1
≥0 × Zm2 ≥0))
is equal to zero. (iii) There exist constants C, D ≥ 0 such that |f(x, z) − f(x′, z′)| ≤ C(x, z) − (x′, z′) + D for all (x, z), (x′, z′) ∈ Rn × Rs.
Conclusions from classical parametric integer programming January 4, 2016 5 / 17
(G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs.
Conclusions from classical parametric integer programming January 4, 2016 6 / 17
(G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs. MILP case: |f(x, z)| ≤ |f(x, z) − f(0, 0)| + |f(0, 0)| ≤ C(x, z) + D + |f(0, 0)|
Conclusions from classical parametric integer programming January 4, 2016 6 / 17
(G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs. MILP case: |f(x, z)| ≤ |f(x, z) − f(0, 0)| + |f(0, 0)| ≤ C(x, z) + D + |f(0, 0)| → (G) holds with η(x) := Cx + D + |f(0, 0)| + 1.
Conclusions from classical parametric integer programming January 4, 2016 6 / 17
(G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs. MILP case: |f(x, z)| ≤ |f(x, z) − f(0, 0)| + |f(0, 0)| ≤ C(x, z) + D + |f(0, 0)| → (G) holds with η(x) := Cx + D + |f(0, 0)| + 1. Observation: (x, z) can be replaced with h(x, z) if the growth condition is fulfilled for the mapping (x, z) → |h(x, z)|. In this case, γf = γh. → This is especially the case if h is H¨
Conclusions from classical parametric integer programming January 4, 2016 6 / 17
(G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs. MILP case: |f(x, z)| ≤ |f(x, z) − f(0, 0)| + |f(0, 0)| ≤ C(x, z) + D + |f(0, 0)| → (G) holds with η(x) := Cx + D + |f(0, 0)| + 1. Observation: (x, z) can be replaced with h(x, z) if the growth condition is fulfilled for the mapping (x, z) → |h(x, z)|. In this case, γf = γh. → This is especially the case if h is H¨
Conclusion: The proposed growth condition is more general than the standard MILP setting.
Conclusions from classical parametric integer programming January 4, 2016 6 / 17
(G) There is a locally bounded mapping η : Rn → (0, ∞) and a constant γ > 0 such that |f(x, z)| ≤ η(x)(zγ + 1) for all (x, z) ∈ Rn × Rs. MILP case: |f(x, z)| ≤ |f(x, z) − f(0, 0)| + |f(0, 0)| ≤ C(x, z) + D + |f(0, 0)| → (G) holds with η(x) := Cx + D + |f(0, 0)| + 1. Observation: (x, z) can be replaced with h(x, z) if the growth condition is fulfilled for the mapping (x, z) → |h(x, z)|. In this case, γf = γh. → This is especially the case if h is H¨
Conclusion: The proposed growth condition is more general than the standard MILP setting. → Which other (mixed-)integer parametric problems are covered?
Conclusions from classical parametric integer programming January 4, 2016 6 / 17
Assume that (i) A is an integral matrix, (ii) Ay ≤ b has an integral solution and (iii) min{q⊤y | Ay ≤ b} exists. Then d∞(argmin {q⊤y | Ay ≤ b}, argmin {q⊤y | Ay ≤ b, y ∈ Zm}) ≤ m∆(A). Here, ∆(A) denotes the maximum of the absolute values of the determinants of square submatrices of A.
Conclusions from classical parametric integer programming January 4, 2016 7 / 17
Assume that (i) A is an integral matrix, (ii) Ay ≤ b has an integral solution and (iii) min{q⊤y | Ay ≤ b} exists. Then d∞(argmin {q⊤y | Ay ≤ b}, argmin {q⊤y | Ay ≤ b, y ∈ Zm}) ≤ m∆(A). Here, ∆(A) denotes the maximum of the absolute values of the determinants of square submatrices of A. → This proximity result can be generalized to quadratic integer problems.
Conclusions from classical parametric integer programming January 4, 2016 7 / 17
f(x, z) = c(x) + min{y⊤Qy + q⊤y | Ay ≤ h(x, z), y ∈ Zm}
Conclusions from classical parametric integer programming January 4, 2016 8 / 17
f(x, z) = c(x) + min{y⊤Qy + q⊤y | Ay ≤ h(x, z), y ∈ Zm}
Assume that (i) A is integral, rank A = m and Q is a positive definite diagonal matrix, (ii) {y ∈ Zm | Ay ≤ h(x, z)} = 0 for all (x, z) ∈ Rn × Rs and (iii) min{y⊤Qy + q⊤y | Ay ≤ b} exists. Then f is finite, the infimum is attained and there exists constants C, D ≥ 0 such that |f(x, z)| ≤ C∆(x, z)(x, z) + D holds true for all (x, z) ∈ Rn × Rs. Here, ∆(x, z) denotes the maximum of the absolute values of the determinants of square submatrices of A h(x, z) −sQ A⊤ q
Conclusions from classical parametric integer programming January 4, 2016 8 / 17
∆(x, z) = max{| det B| | B is a square sub-matrix of A h(x, z) −sQ A⊤ q
Conclusions from classical parametric integer programming January 4, 2016 9 / 17
∆(x, z) = max{| det B| | B is a square sub-matrix of A h(x, z) −sQ A⊤ q
→ Laplace expansion yields a constant E such that ∆(x, z) ≤ E(h(x, z) + 1) for all (x, z) ∈ Rn × Rs.
Conclusions from classical parametric integer programming January 4, 2016 9 / 17
∆(x, z) = max{| det B| | B is a square sub-matrix of A h(x, z) −sQ A⊤ q
→ Laplace expansion yields a constant E such that ∆(x, z) ≤ E(h(x, z) + 1) for all (x, z) ∈ Rn × Rs. → If the growth condition holds for (x, z) → |h(x, z)| and c, then it also holds for f with γf = max{γc, 2γh}.
Conclusions from classical parametric integer programming January 4, 2016 9 / 17
∆(x, z) = max{| det B| | B is a square sub-matrix of A h(x, z) −sQ A⊤ q
→ Laplace expansion yields a constant E such that ∆(x, z) ≤ E(h(x, z) + 1) for all (x, z) ∈ Rn × Rs. → If the growth condition holds for (x, z) → |h(x, z)| and c, then it also holds for f with γf = max{γc, 2γh}. → If c and h are continuous and h−1(Rk \ (R \ Z)k) has Lebesgue measure zero, then the Lebesgue measure of the set of discontinuities of f is equal to zero.
Conclusions from classical parametric integer programming January 4, 2016 9 / 17
∆(x, z) = max{| det B| | B is a square sub-matrix of A h(x, z) −sQ A⊤ q
→ Laplace expansion yields a constant E such that ∆(x, z) ≤ E(h(x, z) + 1) for all (x, z) ∈ Rn × Rs. → If the growth condition holds for (x, z) → |h(x, z)| and c, then it also holds for f with γf = max{γc, 2γh}. → If c and h are continuous and h−1(Rk \ (R \ Z)k) has Lebesgue measure zero, then the Lebesgue measure of the set of discontinuities of f is equal to zero. Conclusion: Our approach allows to derive stability for two-stage SPs with QIP recourse problems.
Conclusions from classical parametric integer programming January 4, 2016 9 / 17
Consider the case where f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2} is defined by C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)}, q is convex and g = (g1, . . . , gk)⊤ is such that gi is convex and epi gi is closed for every i = 1, . . . , k.
Conclusions from classical parametric integer programming January 4, 2016 10 / 17
Consider the case where f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2} is defined by C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)}, q is convex and g = (g1, . . . , gk)⊤ is such that gi is convex and epi gi is closed for every i = 1, . . . , k. → Helpful fact: Convex functions are Lipschitz continuous on bounded subsets.
Conclusions from classical parametric integer programming January 4, 2016 10 / 17
Consider the case where f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2} is defined by C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)}, q is convex and g = (g1, . . . , gk)⊤ is such that gi is convex and epi gi is closed for every i = 1, . . . , k. → Helpful fact: Convex functions are Lipschitz continuous on bounded subsets. Assumptions: (C1) C(0, 0) is bounded and (C2) C(x, z) ∩ (Rm1 × Zm2) = ∅ for all (x, z) ∈ Rn × Rs.
Conclusions from classical parametric integer programming January 4, 2016 10 / 17
Assume (C1) and (C2), then for every R > 0 there exists a constant K(R), such that d∞(C(x, z), C(x′, z′)) ≤ K(R)h(x, z) − h(x′, z′) whenever (x, z) − (x′, z′) ≤ R. Set Θ(x, z, y) := (max{g1(y) − h1(x, z), 0}, . . . , max{gk(y) − hk(x, z), 0})⊤ ∈ Rk, then K(R) = sup(x,z,y):h(x,z)≤R, y /
∈C(x,z) dC(x,z)(y) Θ(x,z,y)∞ < ∞
holds.
Conclusions from classical parametric integer programming January 4, 2016 11 / 17
Assume (C1) and (C2), then for every R > 0 there exists a constant K(R), such that d∞(C(x, z), C(x′, z′)) ≤ K(R)h(x, z) − h(x′, z′) whenever (x, z) − (x′, z′) ≤ R. Set Θ(x, z, y) := (max{g1(y) − h1(x, z), 0}, . . . , max{gk(y) − hk(x, z), 0})⊤ ∈ Rk, then K(R) = sup(x,z,y):h(x,z)≤R, y /
∈C(x,z) dC(x,z)(y) Θ(x,z,y)∞ < ∞
holds. → C(x, z) is compact for all (x, z) ∈ Rn × Rs.
Conclusions from classical parametric integer programming January 4, 2016 11 / 17
Assume (C1) and (C2), then for every R > 0 there exists a constant K(R), such that d∞(C(x, z), C(x′, z′)) ≤ K(R)h(x, z) − h(x′, z′) whenever (x, z) − (x′, z′) ≤ R. Set Θ(x, z, y) := (max{g1(y) − h1(x, z), 0}, . . . , max{gk(y) − hk(x, z), 0})⊤ ∈ Rk, then K(R) = sup(x,z,y):h(x,z)≤R, y /
∈C(x,z) dC(x,z)(y) Θ(x,z,y)∞ < ∞
holds. → C(x, z) is compact for all (x, z) ∈ Rn × Rs. → The continuous relaxation is solvable.
Conclusions from classical parametric integer programming January 4, 2016 11 / 17
Setting f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}, C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)}
Conclusions from classical parametric integer programming January 4, 2016 12 / 17
Setting f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}, C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)} → For y2 ∈ Zm2 define Cy2(x, z) := {(y1, y2) ∈ C(x, z)}. Then C(x, z) ∩ (Rm1 × Zm2) =
y2∈Zm2 Cy2(x, z).
Conclusions from classical parametric integer programming January 4, 2016 12 / 17
Setting f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}, C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)} → For y2 ∈ Zm2 define Cy2(x, z) := {(y1, y2) ∈ C(x, z)}. Then C(x, z) ∩ (Rm1 × Zm2) =
y2∈Zm2 Cy2(x, z).
→ C(x, z) is compact, hence Z(x, z) := {y2 ∈ Zm2 | Cy2 = ∅} is finite.
Conclusions from classical parametric integer programming January 4, 2016 12 / 17
Setting f(x, z) = c(x) + min {q(y) | y ∈ C(x, z), y ∈ Rm1 × Zm2}, C(x, z) = {y ∈ Rm1 × Rm2 | g(y) ≤ h(x, z)} → For y2 ∈ Zm2 define Cy2(x, z) := {(y1, y2) ∈ C(x, z)}. Then C(x, z) ∩ (Rm1 × Zm2) =
y2∈Zm2 Cy2(x, z).
→ C(x, z) is compact, hence Z(x, z) := {y2 ∈ Zm2 | Cy2 = ∅} is finite. Consequence: If c is real-valued c, then so is f(x, z) = c(x) + miny2∈Z(x,z) miny1∈Cy2(x,z) q(y).
Conclusions from classical parametric integer programming January 4, 2016 12 / 17
Additional assumptions: (C3) There exist constants β1, L1 > 0 such that L(r) ≤ L1rβ1 for all r > 0, where L(r) denotes the (minimal) Lipschitz constant for q on Br(0). (C4) There exist constants β2, L2 > 0 such that K(R) ≤ L2Rβ2 for all R > 0.
Conclusions from classical parametric integer programming January 4, 2016 13 / 17
Additional assumptions: (C3) There exist constants β1, L1 > 0 such that L(r) ≤ L1rβ1 for all r > 0, where L(r) denotes the (minimal) Lipschitz constant for q on Br(0). (C4) There exist constants β2, L2 > 0 such that K(R) ≤ L2Rβ2 for all R > 0.
Assume that the growth condition is fulfilled for c and (x, z) → |h(x, z)| and that (C1) - (C4) hold true. Then the growth condition is also fulfilled for f with γf = max{γc, (β1 + β2 + 1)γh}.
Conclusions from classical parametric integer programming January 4, 2016 13 / 17
Additional assumptions: (C3) There exist constants β1, L1 > 0 such that L(r) ≤ L1rβ1 for all r > 0, where L(r) denotes the (minimal) Lipschitz constant for q on Br(0). (C4) There exist constants β2, L2 > 0 such that K(R) ≤ L2Rβ2 for all R > 0.
Assume that the growth condition is fulfilled for c and (x, z) → |h(x, z)| and that (C1) - (C4) hold true. Then the growth condition is also fulfilled for f with γf = max{γc, (β1 + β2 + 1)γh}. Conclusion: Under a compactness condition, our approach allows to derive stability for two-stage SPs with recourse problems from a fairly general class.
Conclusions from classical parametric integer programming January 4, 2016 13 / 17
min{ρ(f(x, z(ω))) | x ∈ X}
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f.
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant.
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant. Examples:
Every convex risk measure, especially every coherent one
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant. Examples:
Every convex risk measure, especially every coherent one (Expectation)
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant. Examples:
Every convex risk measure, especially every coherent one (Expectation) Upper semideviation of order q ≥ 1 from target: E[(X − t)q
+]
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant. Examples:
Every convex risk measure, especially every coherent one (Expectation) Upper semideviation of order q ≥ 1 from target: E[(X − t)q
+]
Upper semideviation of order q ≥ 1: E[X] + λE[(X − E[X])q
+]
1 q , λ ∈ (0, 1)
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant. Examples:
Every convex risk measure, especially every coherent one (Expectation) Upper semideviation of order q ≥ 1 from target: E[(X − t)q
+]
Upper semideviation of order q ≥ 1: E[X] + λE[(X − E[X])q
+]
1 q , λ ∈ (0, 1)
Conditional Value-at-risk: Expectation of the (1 − α) ∗ 100% worst outcomes
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
min{ρ(f(x, z(ω))) | x ∈ X} Assumption on the recourse problem: The growth condition is fulfilled for f. Assumption on the risk measure: The risk measure ρ is induced by a mapping that is convex, nondecreasing w.r.t. the P-almost sure partial order and law invariant. Examples:
Every convex risk measure, especially every coherent one (Expectation) Upper semideviation of order q ≥ 1 from target: E[(X − t)q
+]
Upper semideviation of order q ≥ 1: E[X] + λE[(X − E[X])q
+]
1 q , λ ∈ (0, 1)
Conditional Value-at-risk: Expectation of the (1 − α) ∗ 100% worst outcomes Every conic combination of covered risk functionals
Conclusions from classical parametric integer programming January 4, 2016 14 / 17
Reformulation of the objective function: ρ(f(x, z(ω)) = Q(x, ν) = Rρ
, where ν = P ◦ z−1
Conclusions from classical parametric integer programming January 4, 2016 15 / 17
Reformulation of the objective function: ρ(f(x, z(ω)) = Q(x, ν) = Rρ
, where ν = P ◦ z−1
Let M ⊆ Mγp
s
be a locally uniformly · γp
s,2-integrating subset, and let Df
denote the set of discontinuity points of f. If x ∈ Rn and ν ∈ M satisfy δx ⊗ ν(Df) = 0, then under the growth condition (G) the mapping Q|Rn×M is continuous at (x, ν) with respect to the product topology of the standard topology on Rn and the relative topology of weak convergence on M.
Conclusions from classical parametric integer programming January 4, 2016 15 / 17
Reformulation of the objective function: ρ(f(x, z(ω)) = Q(x, ν) = Rρ
, where ν = P ◦ z−1
Let M ⊆ Mγp
s
be a locally uniformly · γp
s,2-integrating subset, and let Df
denote the set of discontinuity points of f. If x ∈ Rn and ν ∈ M satisfy δx ⊗ ν(Df) = 0, then under the growth condition (G) the mapping Q|Rn×M is continuous at (x, ν) with respect to the product topology of the standard topology on Rn and the relative topology of weak convergence on M. Key idea: Considering Ψ−weak topologies on suitable subclasses of Borel probability measures.
Conclusions from classical parametric integer programming January 4, 2016 15 / 17
ϕ(ν) := inf{Q(x, ν) | x ∈ X}
Conclusions from classical parametric integer programming January 4, 2016 16 / 17
ϕ(ν) := inf{Q(x, ν) | x ∈ X}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X. Then ϕ|M is upper semicontinuous in ν with respect to the relative topology of weak convergence.
Conclusions from classical parametric integer programming January 4, 2016 16 / 17
ϕ(ν) := inf{Q(x, ν) | x ∈ X}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X. Then ϕ|M is upper semicontinuous in ν with respect to the relative topology of weak convergence. Φ(ν) := {x ∈ X | Q(x, ν) = ϕ(ν)}
Conclusions from classical parametric integer programming January 4, 2016 16 / 17
ϕ(ν) := inf{Q(x, ν) | x ∈ X}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X. Then ϕ|M is upper semicontinuous in ν with respect to the relative topology of weak convergence. Φ(ν) := {x ∈ X | Q(x, ν) = ϕ(ν)}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X and X be compact. Then ϕ|M is continuous on M and Φ|M is upper semicontinuous on M with respect to the relative topology of weak convergence.
Conclusions from classical parametric integer programming January 4, 2016 16 / 17
ϕ(ν) := inf{Q(x, ν) | x ∈ X}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X. Then ϕ|M is upper semicontinuous in ν with respect to the relative topology of weak convergence. Φ(ν) := {x ∈ X | Q(x, ν) = ϕ(ν)}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X and X be compact. Then ϕ|M is continuous on M and Φ|M is upper semicontinuous on M with respect to the relative topology of weak convergence.
Upper semicontinuity: For any µ0 ∈ M and any open set O ⊆ Rn such that Φ|M(µ0) ⊆ O there exists a neighborhood N of µ0 with respect to the topology of weak convergence such that Φ|M(µ) ⊆ O for all µ ∈ N.
Conclusions from classical parametric integer programming January 4, 2016 16 / 17
ϕ(ν) := inf{Q(x, ν) | x ∈ X}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X. Then ϕ|M is upper semicontinuous in ν with respect to the relative topology of weak convergence. Φ(ν) := {x ∈ X | Q(x, ν) = ϕ(ν)}
In addition to the previous assumptions, let δx ⊗ ν(Df) = 0 hold for all x ∈ X and X be compact. Then ϕ|M is continuous on M and Φ|M is upper semicontinuous on M with respect to the relative topology of weak convergence.
Upper semicontinuity: For any µ0 ∈ M and any open set O ⊆ Rn such that Φ|M(µ0) ⊆ O there exists a neighborhood N of µ0 with respect to the topology of weak convergence such that Φ|M(µ) ⊆ O for all µ ∈ N. → Interpretation: The solution set does not ”explode” under small perturbations.
Conclusions from classical parametric integer programming January 4, 2016 16 / 17
Conclusions from classical parametric integer programming January 4, 2016 17 / 17