On the Complexity and Solution of Inverse Mixed Integer Linear Programs
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1 On the Complexity and Solution of Inverse Mixed Integer Linear Programs Aykut Bulut 1 Joint work with: Ted Ralphs 1 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University INFORMS Annual Meeting October 2013 Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
2 Outline 1 Inverse MILP 2 Complexity 3 Algorithm 4 Computational Results Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
3 Outline 1 Inverse MILP 2 Complexity 3 Algorithm 4 Computational Results Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
4 Outline 1 Inverse MILP 2 Complexity 3 Algorithm 4 Computational Results Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
5 Outline 1 Inverse MILP 2 Complexity 3 Algorithm 4 Computational Results Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
6 Definitions For a given d R n and P, we consider a MILP where, z IP = min x P dt x, (1) P = {x R n Ax = b, x 0} (Z r R n r ). For a given c R n, x 0 P, the inverse problem is defined as follows. min c d s.t. d T x 0 d T x x P. (2) Assumption: P is bounded. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
7 A Small Example Define forward problem feasible set P as follows and let c = ( 3, 5), x 0 = (1, 1). Figure: P, its convex hull and feasible d cone Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
8 Definitions min c d s.t. d T x 0 d T x x P. (4) Model can be linearized for l 1 and l norms. Convex hull of P is a polytope. Last constraint set can be represented with the set of extreme points of convex hull of P. Let E be the set of extreme points of convex hull of P, E is finite. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
9 Inverse MILP with l 1 norm z 1 IP = min s.t. n i=1 θ i c i d i θ i i {1, 2,..., n} (3) d i c i θ i i {1, 2,..., n} d T x 0 d T x x E. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
10 Inverse MILP with l norm z IP = min y s.t. c i d i y i {1, 2,..., n} (4) d i c i y i {1, 2,..., n} d T x 0 d T x x E. For the remainder of the presentation, we deal with the case of l norm. Let S represent feasible set of the inverse IP, defined as S = { (y, d) R R n y c d, d T (x 0 x) 0 x E }. Note that S is a polyhedron. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
11 Polynomially Solvable Forward Problems Ahuja and Orlin [1] determines the complexity of inverse problem when the forward problem is polynomially solvable. Theorem (Ahuja and Orlin [1]) If a forward problem is polynomially solvable for each linear cost function, then corresponding inverse problems under l 1 and l are polynomially solvable. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
12 Forward Problems Define the following problems related to MILP and inverse MILP. Definition MILP decision problem: Given γ Q decide whether d T x γ holds for some x P. Definition MILP optimization problem: Find solution vector x such that x argmin x P d T x or decide the problem is unbounded or decide the problem is infeasible. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
13 Inverse Problems Definition Inverse MILP decision problem: Given γ Q decide whether y γ holds for some (y, d) S. Definition Inverse MILP optimization problem: Find solution vector (y, d ), such that (y, d ) argmin (y,d) S y. Definition Inverse MILP separation problem: Given a vector (y, d) Q Q n, decide whether (y, d) is in S, and if not, [ find ] a hyperplane { [ that separates ] (y, d) } from S, i.e., find y y π Q n+1 such that π T < min π d T (y, d) S. d Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
14 Component 1 An Observation Recall inverse MILP optimization problem. z IP = min y s.t. c i d i y i {1, 2,..., n} d i c i y i {1, 2,..., n} d T x 0 d T x x E. Definition Inverse MILP separation problem: Given a vector (y, d) Q Q n, decide whether (y, d) is in S, and if not, [ find ] a hyperplane { [ that separates ] (y, d) } from S, i.e., find y y π Q n+1 such that π T < min π d T (y, d) S. d Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
15 Component 2 GLS Theorem The following theorem by Grötschel et al. indicates the relationship between separation and optimization problems. Theorem (Grötschel et al. [2]) Given an oracle for the separation problem, the optimization problem over a given polyhedron with linear objective can be solved in time, polynomial in ϕ, n and the encoding length of objective coefficient vector, where facet complexity of polyhedron is at most ϕ. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
16 Complexity of MILP optimization/decision problems Figure: Problem relations Theorem Inverse MILP optimization problem under l /l 1 norm is solvable in time polynomial of ϕ, n + 1/2n, and encoding length of (1, 0,..., 0)/(1,..., 1, 0,..., 0), given an oracle for the MILP decision problem. Corollary Inverse MILP decision problem is in P 2. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
17 Inverse MILP decision is P 2 complete Inverse Decision problem is P 2 complete by a reduction from DSAT problem. Definition Deterministic formula: Let Y 1, Y 2,..., Y k, {x 1, x 2,..., x k 1 } be sets of Boolean variables. A formula F involving these variables is said to be deterministic if it consists of the conjuction of the following clauses. (a) For each y Y 1 Y 2 either (y) or (y) is a clase of F. (b) For each j = 1, 2,..., k 1 and each y Y j+1, there are two sets of conjuctions of literals from Y j {x j }, say {C i } and {D i }, such that (C i y) and (D i y) are all clauses of F, and, furthermore, for any truth assignment for Y j {x j } exactly one of the conjuctions in {C i } {D i } is true. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
18 Definition Deterministic Satisfiability (DSAT) is the following computational problem. Given k formulas F 0 (x 1,..., x k 1, Y 1,..., Y k ), F 1 (Y 1, Z 1 ),..., F k 1 (Y k 1, Z k 1 ), where {x 1,..., x k 1 }, Y 1,..., Y k, Z 1,..., Z k 1 are disjoint sets of variables and F 0 is a deterministic formula, is there a truth assignment ˆx 1,..., ˆx k 1, Ŷ 1,..., Ŷ k for x and y variables such that (a) F 0 is satisfied, and (b) F j (Ŷ j, Z j )) is satisfiable iff ˆx j = true. Theorem (Papadimitriou [3]) DSAT is P 2 complete. Proof follows by reducing a Deterministic Truing Machine with an NP oracle to DSAT. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
19 Inverse MILP decision is P 2 complete DSAT is canonical P 2 complete problem, like SAT for NP complete. Papadimitriou defines DSAT to reduce it to unique optimum problem and shows that it is P 2 complete [3]. Using similar techniques DSAT can be reduced to inverse TSP decision version. Inverse TSP decision version is P 2 complete. Since it is a special case of inverse MILP decision, inverse MILP decision is also P 2 complete. Theorem Inverse MILP decision problem is P 2 complete. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
20 Proposed Algorithm First, we define two parametric problems named P k and InvP k as follows min x P dkt x (P k ) min y s.t. c i d i y i {1, 2,..., n} (InvP k ) d i c i y i {1, 2,..., n} d T x 0 d T x where E k 1 is the set of extreme points found so far. E k 1 Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
21 Proposed Algorithm Algorithm 1 k 1 d k c Solve P k, x k x while d kt (x 0 x k ) > 0 do k k + 1 Solve InvP k, d k d Solve P k, x k x end while x k is an optimal solution to P k, whereas d k is an optimal solution to invp k. The algorithm stops when a generated cut is not violated by the current solution. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
22 Table: MIPLIB Iteration number l 1 p. name pert. 0.1 pert. 0.2 pert. 0.3 pert. 0.4 pert. 0.5 pert. 0.6 bell3a blend dcmulti egout enigma flugpl gen gt l152lav lseu mas misc misc mod mod modglob p p pk pp08acuts qiu rgn rout stein vpm Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
23 Table: MIPLIB Iteration number l inf p. name pert. 0.1 pert. 0.2 pert. 0.3 pert. 0.4 pert. 0.5 pert. 0.6 air blend cap dcmulti egout enigma flugpl gen gt harp l152lav lseu mas misc misc mod mod modglob p p pk pp08acuts qnet rgn rout stein vpm Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
24 Table: TSPLIB Iteration number l 1 p. name pert. 0.1 pert. 0.2 pert. 0.3 pert. 0.4 pert. 0.5 pert. 0.6 att berlin bier burma ch ch eil eil eil kroa kroc krod kroe lin pr pr pr rat rd st u ulysses ulysses Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
25 Table: TSPLIB Iteration number l inf p. name pert. 0.1 pert. 0.2 pert. 0.3 pert. 0.4 pert. 0.5 pert. 0.6 att berlin bier burma ch ch eil eil eil kroa kroa krob kroc krod kroe lin pr pr pr pr pr pr rat rd st u ulysses ulysses Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
26 References Ravindra K. Ahuja and James B. Orlin. Inverse optimization. Operations Research, 49(5): , September/October Martin Grötschel, Lászlo Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, second corrected edition edition, Christos H. Papadimitriou. On the complexity of unique solutions. J. ACM, 31(2): , March Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
27 End of presentation This is end of presentation! Thank you for listening! Bulut, Ralphs Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
28 A Small Example: Iteration 1 Let P and convex hull of P given as in Figure. Let c = ( 3, 5) and x 0 = (1, 1). We know d 1 = c. When objective coefficient of forward problem is d 1, optimal solution is x 1 = (8, 2). Figure: Iteration 1, d 1 and x 1 Using x 1, we generate cut d (x 0 x 1 ) 0, i.e. 7d 1 d 2 0. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
29 A Small Example: Iteration 2 Following figure shows feasible cone for d. d 2 is the inverse optimal with the current cut. x 2 is the forward optimal solution when d 2 is objective coefficient vector. Figure: Iteration 2, feasible d cone ((3, 0) is apex), d 2 and x 2 Using x 2, we generate cut d (x 0 x 2 ) 0, i.e. 2d 1 + d 2 0. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
30 A Small Example: Iteration 3 Figure: Iteration 3, feasible d cone, d 3 and x 3 Using x 3, we generate cut d (x 0 x 3 ) 0, i.e. d 1 + d 2 0. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
31 A Small Example: Iteration 4 Figure: Iteration 4, feasible d cone, d 4 and x 4 Using x 4, we generate cut d (x 0 x 4 ) 0, i.e. d 1 + d 2 0. Note that current d 4 does not violate this cut, then d 4 is the optimal solution of inverse problem. Bulut, Ralphs (COR@L Lab) On the Complexity and Solution of Inverse MILP INFORMS, October / 28
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