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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 parametric optimization to two-stage SP Take a parametric mixed-integer program min x,y { c ( x ) + q ( y ) | x ∈ X, y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } , ( P z ) M. Claus Conclusions from classical parametric integer programming January 4, 2016 2 / 17

From parametric optimization to two-stage SP Take a parametric mixed-integer program min x,y { c ( x ) + q ( y ) | x ∈ X, y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } , ( P z ) add an information constraint decide x − → observe z − → decide y M. Claus Conclusions from classical parametric integer programming January 4, 2016 2 / 17

From parametric optimization to two-stage SP Take a parametric mixed-integer program min x,y { c ( x ) + q ( y ) | x ∈ X, y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } , ( P z ) add an information constraint decide x − → observe z − → decide y and assume purely exogenous stochastic uncertainty z = z ( ω ) . M. Claus Conclusions from classical parametric integer programming January 4, 2016 2 / 17

From parametric optimization to two-stage SP Take a parametric mixed-integer program min x,y { c ( x ) + q ( y ) | x ∈ X, y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } , ( P z ) add an information constraint decide x − → observe z − → 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 ∈ R m 1 × Z m 2 } | x ∈ X } � �� � =: f ( x,z ( ω )) M. Claus Conclusions from classical parametric integer programming January 4, 2016 2 / 17

From parametric optimization to two-stage SP Take a parametric mixed-integer program min x,y { c ( x ) + q ( y ) | x ∈ X, y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } , ( P z ) add an information constraint decide x − → observe z − → 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 ∈ R m 1 × Z m 2 } | x ∈ X } � �� � =: f ( x,z ( ω )) Task: Pick an ”optimal” random variable taking into account risk aversion. M. Claus Conclusions from classical parametric integer programming January 4, 2016 2 / 17

From parametric optimization to two-stage SP Take a parametric mixed-integer program min x,y { c ( x ) + q ( y ) | x ∈ X, y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } , ( P z ) add an information constraint decide x − → observe z − → 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 ∈ R m 1 × Z m 2 } | x ∈ X } � �� � =: f ( x,z ( ω )) Task: Pick an ”optimal” random variable taking into account risk aversion. → Mean risk models: min { ρ ( f ( x, z ( ω ))) | x ∈ X } M. Claus Conclusions from classical parametric integer programming January 4, 2016 2 / 17

Stability in two-stage SP 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? M. Claus Conclusions from classical parametric integer programming January 4, 2016 3 / 17

Stability in two-stage SP 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: min x ∈ Z E [ χ { 0 } ( z ( ω ))( x 2 + λ )] , where λ ∈ R is fixed and P ( z ( ω ) = 0) = 1 . M. Claus Conclusions from classical parametric integer programming January 4, 2016 3 / 17

Stability in two-stage SP 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: min x ∈ Z E [ χ { 0 } ( z ( ω ))( x 2 + λ )] , where λ ∈ R is fixed and P ( z ( ω ) = 0) = 1 . → Unique optimal solution x = 0 yields the value λ . M. Claus Conclusions from classical parametric integer programming January 4, 2016 3 / 17

Stability in two-stage SP 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: min x ∈ Z E [ χ { 0 } ( z ( ω ))( x 2 + 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 z ǫ ( · ) defined by P ( z ǫ ( ω ) = ǫ ) = 1 and solve min x ∈ Z E [ χ { 0 } ( z ǫ ( ω ))( x 2 + λ )] . M. Claus Conclusions from classical parametric integer programming January 4, 2016 3 / 17

Stability in two-stage SP 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: min x ∈ Z E [ χ { 0 } ( z ( ω ))( x 2 + 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 z ǫ ( · ) defined by P ( z ǫ ( ω ) = ǫ ) = 1 and solve min x ∈ Z E [ χ { 0 } ( z ǫ ( ω ))( x 2 + λ )] . → For ǫ � = 0 , every integer is an optimal solution with value 0 . M. Claus Conclusions from classical parametric integer programming January 4, 2016 3 / 17

Stability in two-stage SP 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: min x ∈ Z E [ χ { 0 } ( z ( ω ))( x 2 + 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 z ǫ ( · ) defined by P ( z ǫ ( ω ) = ǫ ) = 1 and solve min x ∈ Z E [ χ { 0 } ( z ǫ ( ω ))( x 2 + 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”. M. Claus Conclusions from classical parametric integer programming January 4, 2016 3 / 17

Stability in two-stage SP Question: What has to be assumed of f ( x, z ) = c ( x ) + min { q ( y ) | y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } ? M. Claus Conclusions from classical parametric integer programming January 4, 2016 4 / 17

Stability in two-stage SP Question: What has to be assumed of f ( x, z ) = c ( x ) + min { q ( y ) | y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } ? Sufficient: f is defined by a MILP f ( x, z ) = c ⊤ x + min { q ⊤ y | Wy = z − Tx, y ∈ R m 1 ≥ 0 × Z m 2 ≥ 0 } , the matrix W is rational and (A1) W ( R m 1 ≥ 0 × Z m 2 ≥ 0 ) = R s , (A2) { u ∈ R s | W ⊤ u ≤ q } � = ∅ . M. Claus Conclusions from classical parametric integer programming January 4, 2016 4 / 17

Stability in two-stage SP Question: What has to be assumed of f ( x, z ) = c ( x ) + min { q ( y ) | y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } ? Sufficient: f is defined by a MILP f ( x, z ) = c ⊤ x + min { q ⊤ y | Wy = z − Tx, y ∈ R m 1 ≥ 0 × Z m 2 ≥ 0 } , the matrix W is rational and (A1) W ( R m 1 ≥ 0 × Z m 2 ≥ 0 ) = R s , (A2) { u ∈ R s | W ⊤ u ≤ q } � = ∅ . → Schultz, Tiedemann (2006), R¨ omisch, Vigerske (2008), . . . M. Claus Conclusions from classical parametric integer programming January 4, 2016 4 / 17

Stability in two-stage SP Question: What has to be assumed of f ( x, z ) = c ( x ) + min { q ( y ) | y ∈ C ( x, z ) , y ∈ R m 1 × Z m 2 } ? Sufficient: f is defined by a MILP f ( x, z ) = c ⊤ x + min { q ⊤ y | Wy = z − Tx, y ∈ R m 1 ≥ 0 × Z m 2 ≥ 0 } , the matrix W is rational and (A1) W ( R m 1 ≥ 0 × Z m 2 ≥ 0 ) = R s , (A2) { u ∈ R s | W ⊤ u ≤ q } � = ∅ . → Schultz, Tiedemann (2006), R¨ omisch, Vigerske (2008), . . . 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 : R n → (0 , ∞ ) and a constant γ > 0 such that | f ( x, z ) | ≤ η ( x )( � z � γ + 1) for all ( x, z ) ∈ R n × R s . M. Claus Conclusions from classical parametric integer programming January 4, 2016 4 / 17

Back to parametric mixed integer programming f ( x, z ) = c ⊤ x + min { q ⊤ y | Wy = z − Tx, y ∈ R m 1 ≥ 0 × Z m 2 ≥ 0 } M. Claus Conclusions from classical parametric integer programming January 4, 2016 5 / 17

Back to parametric mixed integer programming f ( x, z ) = c ⊤ x + min { q ⊤ y | Wy = z − Tx, y ∈ R m 1 ≥ 0 × Z m 2 ≥ 0 } Theorem (Blair, Jeroslow 1977) Assume (A1), (A2) and the rationality of W . Then (i) f is real valued and lower semicontinuous on R n × R s . (ii) f is continuous on ( R n × R s ) \ A , where the ( n + s ) -dim. Lebesgue measure of A = ( − T, I ) − 1 (bd W ( R m 1 ≥ 0 × Z m 2 ≥ 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 ′ ) ∈ R n × R s . M. Claus Conclusions from classical parametric integer programming January 4, 2016 5 / 17