MIPMinIP

MIPMinIP, short for Minimization of Inner Product in Mixed-Integer Programming, refers to a family of optimization problems and solution approaches that handle objectives or constraints involving inner products of decision variable vectors, often mixing binary and continuous variables. In practice, this yields mixed-integer bilinear programs, where the product terms x_i y_j appear in the objective or constraints.

Typical formulation takes the form: minimize sum_{i in B} sum_{j in C} w_{ij} x_i y_j subject to standard MIP constraints, where x_i are binary, y_j are continuous within bounds. The bilinear terms are nonconvex, so they are linearized by introducing auxiliary variables z_{ij} to represent x_i y_j, along with constraints to enforce z_{ij} = x_i y_j for all i, j. Linearization can use big-M constraints or convex hull formulations, depending on variable bounds and desired tightness.

Solving MIPMinIP relies on mixed-integer programming techniques, typically branch-and-bound with cutting planes, supported by modern MILP solvers. The quality of the linearization, tightness of bounds, and symmetry breaking influence solve times. Some problem classes admit special reformulations that yield tighter relaxations or even exact convex representations in particular cases.

Applications appear in supply chain design with interaction costs, network design with bilateral effects, facility location with pairwise penalties, scheduling with setup or compatibility costs, and portfolio optimization where discrete decisions interact with continuous allocations.

While related to mixed-integer linear programming and mixed-integer bilinear programming, MIPMinIP is not a single standardized method but a label used for various formulations and linearization strategies that aim to optimize inner-product terms within a MIP framework.