Inverse Tension Problems Rectilinear and Chebyshev Distances ¸ i˘ C gdem G¨ uler gueler@mathematik.uni-kl.de University of Kaiserslautern Cologne Twente Workshop 2008 - Gargnano Italy – p. 1/23
Outline Introduction to Inverse Optimization Tension Problems on Networks Inverse Minimum Cost Tension Problem under L 1 Norm Inverse Minimum Cost Tension Problem under L ∞ Norm Inverse Maximum Tension Problem under L 1 Norm Conclusions and Future Research Cologne Twente Workshop 2008 - Gargnano Italy – p. 2/23
Inverse Optimization Definition 1. Given an optimization problem and a feasible solution to it, the inverse optimization problem is to find a minimal adjustment of the parameters of the problem (costs, capacities,...) such that the given solution becomes optimum. ⇒ Optimization problem Forward problem = ⇒ Inverse optimization problem Backward problem = Cologne Twente Workshop 2008 - Gargnano Italy – p. 3/23
Inverse Optimization - Motivation Geographical Sciences: Predicting the transmission time of the seismic waves in order to model earthquake movements Cologne Twente Workshop 2008 - Gargnano Italy – p. 4/23
Inverse Optimization - Motivation Geographical Sciences: Predicting the transmission time of the seismic waves in order to model earthquake movements Medical Imaging: In X-ray tomography to estimate the dimension of the body parts Cologne Twente Workshop 2008 - Gargnano Italy – p. 4/23
Inverse Optimization - Motivation Geographical Sciences: Predicting the transmission time of the seismic waves in order to model earthquake movements Medical Imaging: In X-ray tomography to estimate the dimension of the body parts Traffic Equilibrium: Imposing tolls to change the travel costs so that system optimal flow will be equal to the user equilibrium flow Cologne Twente Workshop 2008 - Gargnano Italy – p. 4/23
Tension Problems on Networks Given G = ( N, A ) a connected digraph θ ∈ R A is a tension on graph G with potential π ∈ R N such that θ ij = π j − π i ∀ ( i, j ) ∈ A (1) Cologne Twente Workshop 2008 - Gargnano Italy – p. 5/23
Tension Problems on Networks Given G = ( N, A ) a connected digraph θ ∈ R A is a tension on graph G with potential π ∈ R N such that θ ij = π j − π i ∀ ( i, j ) ∈ A (1) Properties of tensions [Pla (1971), Rockafellar (1984)]: For all cycles C, � a ij ∈ C + θ ij − � a ij ∈ C − θ ij = 0 . Any linear combination of tensions is a tension. A tension is orthogonal to any circulation. Cologne Twente Workshop 2008 - Gargnano Italy – p. 5/23
Tension Problems on Networks Minimum cost tension problem (MCT): � min c ij θ ij (2) a ij ∈ A subject to t ij ≤ θ ij ≤ T ij ∀ a ij ∈ A θ is a tension where t ij ∈ R ∪ {−∞} and T ij ∈ R ∪ { + ∞} are lower and upper bounds. Cologne Twente Workshop 2008 - Gargnano Italy – p. 6/23
Tension Problems on Networks Minimum cost tension problem (MCT): � min c ij θ ij (2) a ij ∈ A subject to t ij ≤ θ ij ≤ T ij ∀ a ij ∈ A θ is a tension where t ij ∈ R ∪ {−∞} and T ij ∈ R ∪ { + ∞} are lower and upper bounds. Maximum tension problem (MaxT): G contains 2 special nodes, s and t , and an arc a st ∈ A with bounds ( t st , T st ) = ( −∞ , ∞ ) . max θ st (3) subject to t ij ≤ θ ij ≤ T ij ∀ a ij ∈ A θ is a tension Cologne Twente Workshop 2008 - Gargnano Italy – p. 6/23
Inverse Tensions - Motivation Inverse network flows have been thoroughly analyzed = ⇒ Can we extend the results to tensions?? Cologne Twente Workshop 2008 - Gargnano Italy – p. 7/23
Inverse Tensions - Motivation Inverse network flows have been thoroughly analyzed = ⇒ Can we extend the results to tensions?? Can we find a generalization for linear programs with totally unimodular matrices? Cologne Twente Workshop 2008 - Gargnano Italy – p. 7/23
Inverse Tensions - Motivation Inverse network flows have been thoroughly analyzed = ⇒ Can we extend the results to tensions?? Can we find a generalization for linear programs with totally unimodular matrices? Inverse tensions might have application in many practical problems. Example: Project scheduling where the costs and time can be negociated with the customer. Cologne Twente Workshop 2008 - Gargnano Italy – p. 7/23
Inverse MCT - Rectilinear Norm (Cost) inverse minimum cost tension problem (IMCT c ): A feasible tension ˆ θ to a MCT is given = ⇒ c : ˆ � Find ˆ θ is the optimum and w ij | c − ˆ c | is minimum a ij ∈ A Cologne Twente Workshop 2008 - Gargnano Italy – p. 8/23
Inverse MCT - Rectilinear Norm (Cost) inverse minimum cost tension problem (IMCT c ): A feasible tension ˆ θ to a MCT is given = ⇒ c : ˆ � Find ˆ θ is the optimum and w ij | c − ˆ c | is minimum a ij ∈ A (Cost) inverse minimum cost flow problem (IMCF c ): [Ahuja-Orlin (2002)] inverse min cost flow under unit weight L 1 norm ≡ min cost flow problem in a unit capacity network Cologne Twente Workshop 2008 - Gargnano Italy – p. 8/23
Inverse MCT - Rectilinear Norm Definition 2. A cut ω is called residual with respect to a tension ˆ θ if ˆ ∀ a ij ∈ ω + θ ij < T ij ˆ ∀ a ij ∈ ω − θ ij > t ij The cost of a cut ω is: � � cost ( ω ) = c ij − c ij (4) a ij ∈ ω + a ij ∈ ω − Cologne Twente Workshop 2008 - Gargnano Italy – p. 9/23
Inverse MCT - Rectilinear Norm Definition 2. A cut ω is called residual with respect to a tension ˆ θ if ˆ ∀ a ij ∈ ω + θ ij < T ij ˆ ∀ a ij ∈ ω − θ ij > t ij The cost of a cut ω is: � � cost ( ω ) = c ij − c ij (4) a ij ∈ ω + a ij ∈ ω − Theorem 3. A tension ˆ θ is optimal if and only if all the residual cuts in G have nonnegative costs [Rockafellar (1984)]. Cologne Twente Workshop 2008 - Gargnano Italy – p. 9/23
Inverse MCT - Rectilinear Norm Definition 4. We call the residual cuts ω 1 and ω 2 to be arc-disjoint if ω + 1 ∩ ω + 2 = ∅ and ω − 1 ∩ ω − 2 = ∅ Cologne Twente Workshop 2008 - Gargnano Italy – p. 10/23
Inverse MCT - Rectilinear Norm Definition 4. We call the residual cuts ω 1 and ω 2 to be arc-disjoint if ω + 1 ∩ ω + 2 = ∅ and ω − 1 ∩ ω − 2 = ∅ Theorem 5. Let Ω ∗ = { ω ∗ 1 , ω ∗ 2 , . . . , ω ∗ K } be the minimum cost collection of arc-disjoint residual cuts in G and Cost (Ω ∗ ) be its cost. Then, − Cost (Ω ∗ ) is the optimal objective function value for the inverse minimum cost tension problem under unit weight rectilinear norm. Cologne Twente Workshop 2008 - Gargnano Italy – p. 10/23
Inverse MCT - Rectilinear Norm LP formulation of the inverse MCT under unit weight L 1 norm is � c ij ( π j − π i ) Minimize a ij ∈ A subject to − 1 ≤ π j − π i ≤ 1 a ij ∈ K for 0 ≤ π j − π i ≤ 1 a ij ∈ L for − 1 ≤ π j − π i ≤ 0 a ij ∈ U for π ≷ 0 where { a ij ∈ A : t ij < ˆ K := θ ij < T ij } { a ij ∈ A : ˆ L := θ ij = t ij } { a ij ∈ A : ˆ U := θ ij = T ij } Cologne Twente Workshop 2008 - Gargnano Italy – p. 11/23
Inverse MCT - Chebyshev Norm (Cost) inverse minimum cost tension problem (IMCT c ): A feasible tension ˆ θ to a MCT is given = ⇒ c : ˆ Find ˆ θ is the optimum and min max a ij ∈ A w ij | c − ˆ c | is minimum Cologne Twente Workshop 2008 - Gargnano Italy – p. 12/23
Inverse MCT - Chebyshev Norm (Cost) inverse minimum cost tension problem (IMCT c ): A feasible tension ˆ θ to a MCT is given = ⇒ c : ˆ Find ˆ θ is the optimum and min max a ij ∈ A w ij | c − ˆ c | is minimum (Cost) inverse minimum cost flow problem (IMCF c ): [Ahuja-Orlin (2002)] inverse min cost flow under unit weight L ∞ norm ≡ minimum mean cost cycle problem Cologne Twente Workshop 2008 - Gargnano Italy – p. 12/23
Inverse MCT - Chebyshev Norm ω ∗ is minimum mean residual cut in G w.r.t. ˆ θ , i.e., µ ∗ = MCost ( ω ∗ ) = cost ( ω ∗ ) / | ω ∗ | is minimum Cologne Twente Workshop 2008 - Gargnano Italy – p. 13/23
Inverse MCT - Chebyshev Norm ω ∗ is minimum mean residual cut in G w.r.t. ˆ θ , i.e., µ ∗ = MCost ( ω ∗ ) = cost ( ω ∗ ) / | ω ∗ | is minimum Theorem 6. Let µ ∗ denote the mean cost of a minimum mean residual cut in G w.r.t. ˆ θ . Then, the optimal objective function value for the inverse minimum cost tension problem under unit weight L ∞ norm is max (0 , − µ ∗ ) . Cologne Twente Workshop 2008 - Gargnano Italy – p. 13/23
Inverse MCT - Chebyshev Norm Optimal c ∗ can be defined as follows: if ˆ c ij − µ ∗ θ ij < T ij and c ij − ϕ ij < 0 if ˆ c ∗ ij = c ij + µ ∗ θ ij > t ij and c ij − ϕ ij > 0 c ij otherwise Cologne Twente Workshop 2008 - Gargnano Italy – p. 14/23
Inverse MCT - Chebyshev Norm Optimal c ∗ can be defined as follows: if ˆ c ij − µ ∗ θ ij < T ij and c ij − ϕ ij < 0 if ˆ c ∗ ij = c ij + µ ∗ θ ij > t ij and c ij − ϕ ij > 0 c ij otherwise Minimum mean cost residual cut can be found in strongly polynomial time by a Newton type algorithm [Hadjiat-Maurras (1997)]. Cologne Twente Workshop 2008 - Gargnano Italy – p. 14/23
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