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Overviewīasic LP-based branch-and-bound can be described as follows. We begin with the original MIP. Not knowing how to solve this problem directly, we remove all of the integrality restrictions. The resulting LP is called the linear-programming relaxation of the original MIP. We can then solve this LP. If the result happens to satisfy all of the integrality restrictions, even though these were not explicitly imposed, then we have been quite lucky. This solution is an optimal solution of the original MIP, and we can stop. If not, as is usually the case, then the normal procedure is to pick some variable that is restricted to be integer, but whose value in the LP relaxation is fractional. Mixed Integer Linear Programming problems are generally solved using a linear-programming based branch-and-bound algorithm. What follows is a description of the algorithm used by Gurobi to solve MILP models. The extension to MIQP and MIQCP is mostly straightforward, but we won’t describe them here. MIP models with a quadratic objective but without quadratic constraints are called Mixed Integer Quadratic Programming (MIQP) problems. MIP models with quadratic constraints are called Mixed Integer Quadratically Constrained Programming (MIQCP) problems. Models without any quadratic features are often referred to as Mixed Integer Linear Programming (MILP) problems. Some or all x must take integer values (integrality constraints) X T Q i x + q i T x ≤ b i (quadratic constraints)
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The Gurobi MIP solver can also solve models with a quadratic objective and/or quadratic constraints: Objective: For example, a variable whose values are restricted to 0 or 1, called a binary variable, can be used to decide whether or not some action is taken, such as building a warehouse or purchasing a new machine.
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The integrality constraints allow MIP models to capture the discrete nature of some decisions. Some or all xj must take integer values (integrality constraints) The problems most commonly solved by the Gurobi Parallel Mixed Integer Programming solver are of the form: Objective: