Hybrid Optimization: The Ten Years of CPAIOR by Pascal van Hentenryck, Michela Milano

By Pascal van Hentenryck, Michela Milano

This quantity specializes in the combination of man-made intelligence and constraint programming to unravel difficulties in operations learn and combinatorial optimization. This quantity collects the contributions of specialists from quite a few examine parts together with selection concept, structures engineering, propositional satisfiability, mathematical optimization, and synthetic intelligence. those invited students describe and reveal probably the most vital issues and effects from the final ten years of analysis with regards to hybrid optimization. Key positive aspects: - comprises either good verified learn effects, and instructions for destiny examine. - offers numerous resolution tools for universal combinatorial optimization and choice difficulties. - provides either theoretical recommendations and real-world purposes in man made intelligence and operations study. Hybrid Optimization can function a worthwhile source for graduate scholars, researchers and practitioners learning man made intelligence or operations learn who're attracted to investigating or making use of constraint programming techniques.

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The set packing problem can be formulated with 0–1 knapsack inequalities. Let 0 otherwise. Let variable xj D1 Aij D1 when item i belongs to set Sj , and Aij D P when set j is selected. The knapsack inequality njD1 Aij xij Ä 1 prevents the Hybrid Modeling 27 selection of any two sets containing item i . Thus, the system Ax Ä e of knapsack inequalities, where e is a vector of ones, P prevents the selection of any two overlapping sets. The objective is to maximize njD1 xj subject to Ax Ä e and x 2 f0; 1gn, which is an MILP problem.

To equalize recession cones, the modeling system can automatically add user-supplied bounds L Ä x Ä U to the disjunction, which becomes ! L Ä x Ä U / (39) LÄxÄU ı The system now generates a convex-hull MILP model for (39). N. Ai x for i 2 I should be written _ Ai x b i ı i 2I bi / i The disjunct corresponding to ıi D 0 for all i 2 I does not appear because it is empty. 4 Network Flow Problems An MILP model is the preferred choice for a network flow problem, not only because it is a natural and intuitive formulation, but also because it has the substantial advantage of total unimodularity.

7 assigns m tasks to n workers (m Ä n). It is a special case of a network flow problem, which means that the MILP model is totally unimodular. This provides a strong incentive to use an MILP formulation at some stage of the solution process. The flow network corresponding to an assignment problem is bipartite, as illustrated in Fig. 7. If m < n, then dummy task nodes are created so that supply balances demand. The cost cij is set to zero for a dummy task i . A unit flow yij D 1 indicates that worker i is assigned task j .

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