Persistency in 0–1 polynomial programming

An optimal solution to the continuous relaxation of a mixed-integer 0–1 linear programming problem is defined to be persistent if the set of 0–1 variables realizing binary values retains those same binary values in at least one integer optimum. A mixed-integer 0–1 linear program is said to possess the persistency property, or equivalently to be persistent, if every optimal solution to the continuous relaxation is a persistent solution. We study the issue of persistency for a special family of mixed-integer 0–1 linear programming problems which derive from the linearization strategy of Sherali and Adams. In particular, we first examine the hierarchy of linearizations which result from applying this strategy to the general class of unconstrained 0–1 polynomial programming problems, and then consider a class of specially-structured constrained problems. Our study of persistency differs from previous works in that we use the explicit algebraic representation of the convex hull of feasible binary solutions available at the highest level of this hierarchy as a tool for making inferences relative to optimal integer solutions from optimal continuous solutions. Each level of this hierarchy consists of a mixed-integer 0–1 linear program, and we provide, for any given optimal solution to the continuous relaxation of any member of this hierarchy, a set of sufficient conditions on the optimal dual multipliers for the solution to be persistent. These sufficient conditions turn out to be satisfied at every optimal solution of the quadratic level, and thus allow for the reformulation of any unconstrained 0–1 quadratic programming problem as a persistent mixed-integer linear program in a suitably-defined higher-dimensional space. We subsequently address the family of constrained 0–1 polynomial programs and use a projection operation to prove that a special family of 0–1 linear programs possesses the persistency property; included in this family is a generalization of the vertex packing problem that permits quadratic objective function coefficients. Unlike earlier published works on the vertex packing and related problems, our proofs do not rely on Balinski's extreme point characterization.