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Inverse Optimization

Author

Listed:
  • Ravindra K. Ahuja

    (Industrial and Systems Engineering Department, University of Florida, Gainesville, Florida 32611)

  • James B. Orlin

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P , let c be a specified cost vector, and x 0 be a given feasible solution. The solution x 0 may or may not be an optimal solution of P with respect to the cost vector c . The inverse optimization problem is to perturb the cost vector c to d so that x 0 is an optimal solution of P with respect to d and || d - c || p is minimum, where || d - c || p is some selected L p norm. In this paper, we consider the inverse linear programming problem under L 1 norm (where || d - c || p = (Sigma) i(in)J w j |d j -c j |, with J denoting the index set of variables x j and w j denoting the weight of the variable j ) and under L (infinity) norm (where || d - c || p = max j(in)J {w j |d j -c j |}). We prove the following results: (i) If the problem P is a linear programming problem, then its inverse problem under the L 1 as well as L (infinity) norm is also a linear programming problem. (ii) If the problem P is a shortest path, assignment or minimum cut problem, then its inverse problem under the L 1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problem P is a minimum cost flow problem, then its inverse problem under the L 1 norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problem P is a minimum cost flow problem, then its inverse problem under the L (infinity) norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problem P is polynomially solvable for linear cost functions, then inverse versions of P under the L 1 and L (infinity) norms are also polynomially solvable.

Suggested Citation

  • Ravindra K. Ahuja & James B. Orlin, 2001. "Inverse Optimization," Operations Research, INFORMS, vol. 49(5), pages 771-783, October.
    • Handle: RePEc:inm:oropre:v:49:y:2001:i:5:p:771-783
    DOI: 10.1287/opre.49.5.771.10607
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    References listed on IDEAS

    as
    1. Ravindra K. Ahuja & James B. Orlin, 2001. "A Fast Scaling Algorithm for Minimizing Separable Convex Functions Subject to Chain Constraints," Operations Research, INFORMS, vol. 49(5), pages 784-789, October.
    2. Jianzhong Zhang & Mao Cai, 1998. "Inverse problem of minimum cuts," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 47(1), pages 51-58, February.
    3. P. T. Sokkalingam & Ravindra K. Ahuja & James B. Orlin, 1999. "Solving Inverse Spanning Tree Problems Through Network Flow Techniques," Operations Research, INFORMS, vol. 47(2), pages 291-298, April.
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