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Generating feasible points for mixed-integer convex optimization problems by inner parallel cuts

Christoph Neumann (christoph.neumann***at***kit.edu)
Oliver Stein (stein***at***kit.edu)

Abstract: In this article we introduce an inner parallel cutting plane method (IPCP) to compute good feasible points for mixed-integer convex optimization problems. The method iteratively generates polyhedral outer approximations of an enlarged inner parallel set (EIPS) of the continuously relaxed feasible set. This EIPS possesses the crucial property that any rounding of any of its elements is feasible for the original problem. The outer approximations are refined in each iteration by using modified Kelley cutting planes, which are defined via rounded optimal points of linear optimization problems (LPs). We show that the method either computes a feasible point or certifies that the EIPS is empty. Moreover, we provide bounds on the objective value of the generated feasible point. As there exist consistent problems which possess an empty EIPS, the IPCP is not guaranteed to find a feasible point for the latter. Yet, the crucial advantage of the method lies in the complexity of each iteration: While other approaches need to solve a mixed-integer linear optimization problem, the IPCP only needs to solve an LP, which can be carried out efficiently. Our computational study indicates that the IPCP is able to quickly find feasible points for many practical applications. It further demonstrates that the objective values of the computed feasible points are generally of good quality and sometimes not easily obtainable by other methods.

Keywords: Mixed-integer programming, feasibility problem, cutting plane method, inner parallel set

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 2: Integer Programming (Cutting Plane Approaches )

Citation: Optimization Online, Preprint ID 2018-11-6947, 2018

Download: [PDF]

Entry Submitted: 11/27/2018
Entry Accepted: 11/27/2018
Entry Last Modified: 11/27/2018

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