Decomposition for adjustable robust linear optimization subject to uncertainty polytope
Summary (2 min read)
1 Introduction
- Robust Optimization is now a well-developed paradigm to tackle optimization problems under uncertainty.
- In the robust counterpart of an optimization problem, the constraints involving uncertain parameters must be feasible for all values of the uncertain parameters in the convex sets.
- In these problems, the demand of the customers is usually not known with precision until the links or the facilities are constructed.
- The main advantage of affine decision rules is their tractability, since they lead to robust optimization problems with the structure of classical robust counterparts.
- The idea behind RCG has also been combined with RG heuristically by Bertsimas et al (2013).
2 Problem overview
- The authors follow a classical assumption from the robust optimization literature and suppose that T depends affinely on uncertain parameter ξ.
- One readily sees that (P ) encompasses more general problems where (i) h depends affinely on ξ and (ii) second stage variables y have fixed costs given by vector k.
- Similarly, the approaches presented in this paper can be applied (with minor modifications) to problems where set S is the intersection of Z|I1| × R|I2| and a polyhedron.
- The ideas presented in this paper rely on considering only the extreme points of Ξ, which is formalized in the result below (whose proof can be found in Ben-Tal and Nemirovski (2002), among others).
- (3) We provide next an example showing that Lemma 1 may not hold if the recourse matrix W depends on the uncertainty parameters ξ, which implies that their approach cannot handle problems with random recourse.the authors.the authors.
3 Solution approach
- Then, assuming that the separation problem can be solved adequately, the authors describe in Section 3.2 two algorithms for addressing (P ′).
- The next result is based on Farka’s lemma.
- The drawback of RG is that a single cut is added to the master problem at each iteration, facing the risk that many iterations may be required before obtaining a feasible solution for (P ′).
- For completeness, both algorithms are described formally in Algorithm 1.
4 Mixed-integer programming reformulations
- The authors propose two mixed-integer linear reformulations for (SP ).
- Notice that such a mapping always exists since the authors can represent Ξ as the set of all convex combinations of the vectors in vert(Ξ).
- This example is useless, however, because the associated polytope Ω would be the unit simplex of dimension | vert(Ξ)|.
- A different approach has been used by Mattia (2013) for a specific network design problem.
- Ut Notice that in spite of its generality, the reformulation from Theorem 3 can be very hard to solve to optimality (Mattia, 2014).
5 Application to telecommunications network design
- 1 Problem description Telecommunications networks have evolved quickly, especially in the last two decades.
- Network operators are then under constant pressure to design high speed networks with larger capacities such as to satisfy the growing need for fast connections with no interruptions and latency.
- In contrast Table 2 shows that RCG can solve only two instances with fractional values of Γ , while RG can solve roughly half of them.
- The tables also show that both algorithms spend most of their time in the solution of (SPL) apart from RCG for the instances with high numbers of iterations.
6 Conclusion
- The authors propose in this paper decomposition algorithms to solve adjustable robust linear programs under polytope uncertainty by generating the extreme points of the uncertainty polytope on the fly.
- The authors discuss algorithmic frameworks for decomposing the robust problem and solution algorithms to address the subproblem.
- The authors numerical results show that the relative efficiency of row generation and row-and-column generation algorithms depends on the type of polytope considered.
- In any case, the bottleneck of these algorithms lies in finding the extreme point of the polytope that is mostly violated by the current solution, which amounts to solve a mixed-integer linear program (MIP).
- The authors would like to thank Sara Mattia for fruitful discussions on the topic of this paper.
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References
3,729 citations
"Decomposition for adjustable robust..." refers methods in this paper
...Figure 2 shows performance profiles (Dolan and Moré, 2002) that compare the solution times of algorithms RG and RCG as well as affine routing (denoted aff)....
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...Figure 2 shows performance profiles (Dolan and Moré, 2002) that compare the solution times of algorithms RG and RCG as well as affine routing (denoted aff)....
[...]
3,364 citations
3,359 citations
"Decomposition for adjustable robust..." refers background or methods in this paper
...Whenever the uncertainty set can be obtained as an affine projection of a 0− 1 polytope, which is the case for the budgeted uncertainty polytope from Bertsimas and Sim (2004), the bilinear program can be linearized to obtain a mixed-integer linear program....
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...Finally, the big-M coefficients can be set to 1, because 0 ≤ πm ≤ 1, which yields constraints (15) and (16). ut An interesting example of non-bijective affine mapping arises with the budgeted uncertainty set introduced in Bertsimas and Sim (2004): ΞΓ ≡ { ξ ∈ [0, 1]|K| s.t. ∑ k∈K ξk ≤ Γ } ....
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...Our uncertainty set is based on the budgeted uncertainty polytope from Bertsimas and Sim (2004)....
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2,501 citations
"Decomposition for adjustable robust..." refers background in this paper
...The framework has experienced its revival in the late nineties independently by Ben-Tal and Nemirovski (1998); El Ghaoui et al (1998); Kouvelis and Yu (1997), and has witnessed an increasing attention in J. Ayoub CDS Consultant at Murex, Lebanon E-mail: josetteayoub@hotmail.com M. Poss UMR CNRS…...
[...]
1,463 citations
"Decomposition for adjustable robust..." refers background in this paper
...The framework has experienced its revival in the late nineties independently by Ben-Tal and Nemirovski (1998); El Ghaoui et al (1998); Kouvelis and Yu (1997), and has witnessed an increasing attention in J. Ayoub CDS Consultant at Murex, Lebanon E-mail: josetteayoub@hotmail.com M. Poss UMR CNRS…...
[...]
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Frequently Asked Questions (10)
Q2. What is the first mixed-integer linear formulation for (SP )?
Their first mixed-integer linear formulation for (SP ) is based on reformulating Ξ as the affine mapping of a 0 − 1 polytope Ω ⊂ R|K′| ofreasonable dimension |K ′|.
Q3. What is the main advantage of affine decision rules?
The main advantage of affine decision rules is their tractability, since they lead to robust optimization problems with the structure of classical robust counterparts.
Q4. What are the main advantages of affine decision rules?
Chen et al (2008) introduce deflected linear and segregated linear decision rules, Chen and Zhang (2009) propose to define affine decision rules on extended descriptions of the uncertainty set, Goh and Sim (2010) introduce complex piece-wise linear decision rules defined through liftings and projections, and Bertsimas and Georghiou (2015) propose piece-wise affine decision rules having fixed number of affine pieces.
Q5. What are the advantages of affine decision rules?
Authors have also studied more complex decision rules that offer more flexibility than affine decision rules while providing tractable optimization problems.
Q6. How can one assess the quality of the aforementioned approximations?
To assess numerically the quality of the aforementioned approximations, one needs to compute the (exact) optimal solutions, at least on small instances.
Q7. What is the time limit of the CPLEX call?
CPLEX is called with its default parameters and the authors have set a time limit of T seconds of CPU time for every individual run and their computing times are presented in seconds.
Q8. what is the bottleneck of the polytope generation algorithm?
In any case, the bottleneck of these algorithms lies in finding the extreme point of the polytope that is mostly violated by the current solution, which amounts to solve a mixed-integer linear program (MIP).
Q9. What is the way to solve the optimization problem?
Vector x∗ belongs to K if and only if the optimal solution of the following optimization problem is non-positivemax (h− T̃ 0x∗) · π − ∑ k∈K (T̃ 1kx∗) · vk(SPL) s.t. ω ∈ Ω WTπ = 01 · π = 1 vkm ≥ πm − (1− ωk) k ∈ K ′,m ∈M (15) vkm ≤ ωk k ∈ K ′,m ∈M (16) π, vkm ≥ 0, ω ∈ {0, 1}|K ′|where T̃ 1k = ∑ h∈K T 1hAhk for each k ∈ K and T̃ 0 = T 0 + ∑ k∈K T 1k ∑ h∈K Akhb k.Proof
Q10. What is the example of a non-bijective affine mapping?
An interesting example of non-bijective affine mapping arises with the budgeted uncertainty set introduced in Bertsimas and Sim (2004):ΞΓ ≡ { ξ ∈ [0, 1]|K| s.t.∑ k∈K ξk ≤