Introduction Cost Robustness Applications Optimization over Integers with Robustness in Cost and Few Constraints Kai-Simon Goetzmann (Joint work with Sebastian Stiller and Claudio Telha) WAOA 2011 – Sept 09 Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Classical Optimization over Integers. x 2 max x 1 e.g. totally unimodular IPs, Unbounded Knapsack Problem, IPs with two variables per inequality, . . . Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Classical Optimization over Integers. But what if. . . ? x 2 x 1 e.g. totally unimodular IPs, Unbounded Knapsack Problem, IPs with two variables per inequality, . . . Kai-Simon Goetzmann Robust Optimization over Integers
◆ Introduction Cost Robustness Applications Cost Robust Counterpart Given set of feasible solutions X ⊆ ❩ n cost vector c ∈ ◗ n find x ∈ X minimizing the cost c T x Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Cost Robust Counterpart Given set of feasible solutions X ⊆ ❩ n cost vector c ∈ ◗ n vector of possible cost changes d ∈ ◆ n maximal number of changes Γ ∈ [ n ] find x ∈ X minimizing the cost c T x Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Cost Robust Counterpart Given set of feasible solutions X ⊆ ❩ n cost vector c ∈ ◗ n vector of possible cost changes d ∈ ◆ n maximal number of changes Γ ∈ [ n ] find x ∈ X minimizing the worst case cost � c T x + max | d i x i | S ⊆ [ n ] i ∈ S | S |≤ Γ Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Cost Robust Counterpart Given set of feasible solutions X ⊆ ❩ n cost vector c ∈ ◗ n vector of possible cost changes d ∈ ◆ n maximal number of changes Γ ∈ [ n ] find x ∈ X minimizing the worst case cost � c T x + max | d i x i | S ⊆ [ n ] i ∈ S | S |≤ Γ This defines the ( d, Γ) -CRC of P = ( c, X ) . Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Introduction 1 General Result for Cost Robust IPs 2 Applications of the General Result 3 Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Introduction 1 General Result for Cost Robust IPs 2 Applications of the General Result 3 Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications To solve CRC of P : Solve the ( α, c ′ ) -MMin of P . n � c j ( x j ) = c j +max { c ′ j x j − α, 0 } +max {− c ′ min � c j ( x j ) , � j x j − α, 0 } x ∈ X j =1 c j ( x j ) � − α / c ′ j x j α / c ′ j Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Theorem Let X ⊆ ❩ n , c ∈ ◗ n , d ∈ ◆ n , Γ ∈ [ n ] . If there is a ρ -approximation algorithm for the ( α, d ) -MMin of P for all α ≥ 0 bounds u j on the absolute value of x j can be computed in polynomial time, then there is a pseudopolynomial ρ -approximation algorithm for the ( d, Γ) -CRC of P = ( c, X ) . Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Proof (Sketch): � � � c T x + max min | d j x j | x ∈ X S ⊆ [ n ] j ∈ S | S |≤ Γ Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Proof (Sketch): � � � c T x + max min | d j x j | x ∈ X S ⊆ [ n ] j ∈ S | S |≤ Γ ⇓ duality � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } x ∈ X j =1 ϑ ≥ 0 Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Proof (Sketch): � � � c T x + max min | d j x j | x ∈ X S ⊆ [ n ] j ∈ S | S |≤ Γ ⇓ duality � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } x ∈ X j =1 ϑ ≥ 0 Γ 0 | d j x j | Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Proof (Sketch): � � � c T x + max min | d j x j | x ∈ X S ⊆ [ n ] j ∈ S | S |≤ Γ ⇓ duality � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } x ∈ X j =1 ϑ ≥ 0 Γ Γ 0 ϑ | d j x j | Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Proof (Sketch): � � � c T x + max min | d j x j | x ∈ X S ⊆ [ n ] j ∈ S | S |≤ Γ ⇓ duality � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } x ∈ X j =1 ϑ ≥ 0 � c j ( x j ) � �� � n � = min Γ ϑ +min � c j ( x j ) − ϑ / d j ϑ ≥ 0 x ∈ X x j ϑ / d j j =1 Kai-Simon Goetzmann Robust Optimization over Integers
❩ Introduction Cost Robustness Applications Finding the optimal ϑ � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } x ∈ X j =1 ϑ ≥ 0 ϑ 0 Kai-Simon Goetzmann Robust Optimization over Integers
❩ Introduction Cost Robustness Applications Finding the optimal ϑ � � n � � � c T x +Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } � �� � x ∈ X j =1 ϑ ≥ 0 =0 for ϑ ≥ d j u j ϑ ∗ ≤ max j { d j u j } ϑ max j { d j u j } 0 Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Finding the optimal ϑ � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } ���� � �� � x ∈ X j =1 ϑ ≥ 0 ∈ ❩ ∈ ❩ ϑ ∗ ≤ max j { d j u j } ϑ ∗ ∈ ❩ ϑ max j { d j u j } 0 Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Finding the optimal ϑ � � n � � � c T x + Γ ϑ + min max { d j x j − ϑ, 0 } + max {− d j x j − ϑ, 0 } x ∈ X j =1 ϑ ≥ 0 ϑ ∗ ≤ max j { d j u j } ϑ ∗ ∈ ❩ ϑ max j { d j u j } 0 ⇒ only max j { d j u j } + 1 possible values for ϑ ∗ , can enumerate in pseudopolynomial time. � Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Theorem (Extension 1) Let X ⊆ ❩ n , c ∈ ◗ n , d ∈ ◆ n , Γ ∈ [ n ] . If there is a ρ -approximation algorithm for the ( α, d ) -MMin of P for all α ≥ 0 bounds u j on the absolute value of x j can be computed in polynomial time, then there is a pseudopolynomial ρ -approximation algorithm for the ( d, Γ) -CRC of P = ( c, X ) . Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Theorem (Extension 1) Let X ⊆ ❩ n , c ∈ ◗ n , d ∈ ◆ n , Γ ∈ [ n ] . If there is a ρ -approximation algorithm for the ( α, d ) -MMin of P for all α ≥ 0 bounds u j on the absolute value of x j can be computed in polynomial time, then there is a pseudopolynomial ρ -approximation algorithm for the ( d, Γ) -CRC of P = ( c, X ) . If ρ = 1 , and if the optimal values of the ( α, d ) -MMin of P are convex in α , then there is an exact polynomial time algorithm for the ( d, Γ) -CRC of P . Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Theorem (Extension 1) Let X ⊆ ❩ n , c ∈ ◗ n , d ∈ ◆ n , Γ ∈ [ n ] . If there is a ρ -approximation algorithm for the ( α, d ) -MMin of P for all α ≥ 0 bounds u j on the absolute value of x j can be computed in polynomial time, then there is a pseudopolynomial ρ -approximation algorithm for the ( d, Γ) -CRC of P = ( c, X ) . If ρ = 1 , and if the optimal values of the ( α, d ) -MMin of P are convex in α , then there is an exact polynomial time algorithm for the ( d, Γ) -CRC of P . Proof (Idea): Binary search for ϑ ∗ . Kai-Simon Goetzmann Robust Optimization over Integers
◆ Introduction Cost Robustness Applications Theorem (Extensions 2 and 3) Let X ⊆ ❩ n , c ∈ ◗ n , d ∈ ◆ n , Γ ∈ [ n ] . If there is a ρ -approximation algorithm for the ( α, d ) -MMin of P for all α ≥ 0 bounds u j on the absolute value of x j can be computed in polynomial time, then there is a pseudopolynomial ρ -approximation algorithm for the ( d, Γ) -CRC of P = ( c, X ) . Kai-Simon Goetzmann Robust Optimization over Integers
Introduction Cost Robustness Applications Theorem (Extensions 2 and 3) Let X ⊆ ❩ n , c ∈ ◗ n , d ∈ ◆ n , Γ ∈ [ n ] . If there is a ρ -approximation algorithm for the ( α, d ) -MMin of P for all α ≥ 0 bounds u j on the absolute value of x j can be computed in polynomial time, then there is a pseudopolynomial ρ -approximation algorithm for the ( d, Γ) -CRC of P = ( c, X ) . If X ⊆ ◆ n and c ≥ 0 , then for all ε > 0 there is a polynomial time ρ (1 + ε ) -approximation algorithm for the ( d, Γ) -CRC of P for minimization problems. Kai-Simon Goetzmann Robust Optimization over Integers
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