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Deterministic Optimization Lecture 11: IP Solution Strategies
Solution Strategies Use a linear program round to integer solution (what if not feasible?)
6.4 Use Linear Programming Solution LP solution is an upper bound on IP solution (assuming maximization) If LP is infeasible then IP is infeasible If LP solution is integral (all variables have integer values), then it is the IP solution. Round-off, sometimes it’s feasible, but not good solution, sometimes it’s infeasible.
Linear Programming Solution 1. Some LP problems will always have integer solutions transportation problem assignment problem min-cost network flow (with integer capacities) These are problems with a unimodular matrix A. (unimodular matrices have det(A) = 1). 2. Solve as linear program and round. Can violate constraints, and be non-optimal. Works OK if integer variables take on large values accuracy of constraints is questionable
6.5 Branch-and-Bound
Max z = 5x 1 + 8x 2 s.t. x 1 + x 2 6 5x 1 + 9x 2 45 x 1 , x 2 ≥ 0 integer x 1 + x 2 = 6 5x 1 + 9x 2 = 45 0 1 2 3 4 5 6 1 2 3 4 5 6 7 8 (2.25, 3.75) Z=41.25 An Example
Fact: If LP-relaxation has integral optimal solution x*, then x* is optimal for IP. In our example, (x 1 , x 2 ) = (2.25, 3.75) is the optimal solution of the LP-relaxation. The optimal value is 41.25 . Fact: z LP-relaxation z IP (for maximization problems) In our example, 41.25 is an upper bound for z IP . Utilizing Optimal Solution of LP- relaxation

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