Extended Formulations via Decision Diagrams

Abstract

We propose a general algorithm of constructing an extended formulation for any given set of linear constraints with integer coefficients. Our algorithm consists of two phases: first construct a decision diagram (V,E)(V,E) that somehow represents a given m×nm \times n constraint matrix, and then build an equivalent set of ∣E∣|E| linear constraints over n+∣V∣n+|V| variables. That is, the size of the resultant extended formulation depends not explicitly on the number mm of the original constraints, but on its decision diagram representation. Therefore, we may significantly reduce the computation time for optimization problems with integer constraint matrices by solving them under the extended formulations, especially when we obtain concise decision diagram representations for the matrices. We can apply our method to 11-norm regularized hard margin optimization over the binary instance space {0,1}n\{0,1\}^n, which can be formulated as a linear programming problem with mm constraints with {−1,0,1}\{-1,0,1\}-valued coefficients over nn variables, where mm is the size of the given sample. Furthermore, introducing slack variables over the edges of the decision diagram, we establish a variant formulation of soft margin optimization. We demonstrate the effectiveness of our extended formulations for integer programming and the 11-norm regularized soft margin optimization tasks over synthetic and real datasets

    Similar works

    Full text

    thumbnail-image

    Available Versions