MixedIntegerLinearProgramming
mixed Integer Linear ProgrammingCombinatorial Problem Solving (CPS)Javier Larrosa Albert Oliveras Enric Rodr guez-CarbonellMay 3, 2021Mixed Integer Linear Programs2 / 61 Amixed integer linear program(MILP, MIP) is of the formmincTxAx=bx 0xi Z i I If all variables need to be integer,it is called a(pure) integer linear program(ILP, IP) If all variables need to be0or1(binary, boolean),it is called a0 1linear programComplexity: LP vs. IP3 / 61 Including integer variables increases enourmously the modeling power,at the expense of more complexity LP s can be solved inpolynomial timewith interior-point methods(ellipsoid method, Karmarkar s algorithm) Integer Programming is anNP-hardproblem. So: There isno known polynomial-time algorithm There arelittle chancesthat one will ever be found Even small problems may be hard to solve What follows is one of the many approaches(and one of the most successful) for attacking IP sLP Relaxation of a MIP4 / 61 Given a MIP(IP)mincTxAx=bx 0xi Z i Iitslinear relaxationis the LP obtained by dropping integrality constraints:(LP)mincTxAx=bx 0 Can we solveIPby solvingLP?
A mixed integer linear program (MILP,MIP) is of the form min cTx Ax =b x ≥0 xi ∈Z ∀i ∈I If all variables need to be integer, it is called a (pure) integer linear program (ILP, IP) If all variables need to be 0or 1(binary, boolean), it is called a 0−1linear program
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