A Formalization of Convex Polyhedra Based on the Simplex Method

Abstract

We present a formalization of convex polyhedra in the proof assistant Coq. The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination. This allows us to define the basic predicates over polyhedra in an effective way (i.e. as programs), and relate them with the corresponding usual logical counterparts. To this end, we make an extensive use of the Boolean reflection methodology. The benefit of this approach is that we can easily derive the proof of several fundamental results on polyhedra, such as Farkas' Lemma, the duality theorem of linear programming, and Minkowski's Theorem.of the effective nature of our approach is demonstrated by the fact that we easily arrive at the proof of fundamental results on polyhedra, such as several versions of Farkas' Lemma, the duality theorem of linear programming, separation from convex hulls, Minkowski's Theorem, etc.Our effective approach is made possible by implementing the simplex method inside Coq, and proving its correctness and termination. Recall that the simplex method is the first algorithm introduced to solve linear programming [10]. Two difficulties need to be overcome to formalize it. On the one hand, we need to deal with its termination. More precisely, the simplex method iterates over the so-called bases. Its termination depends on the specification of a pivoting rule, whose aim is to determine, at each iteration, the next basis. In this work, we have focused on proving that the lexicographic rule [11] ensures termination. On the other hand, the simplex method is actually composed of two parts. The part previously described, called Phase II, requires an initial basis to start with. Finding such a basis is the purpose of Phase I. It consists in building an extended problem (having a trivial initial basis), and applying to it the Phase II algorithm. Both phases need to be formalized to obtain a fully functional algorithm.We point out that our goal here is not to obtain a practically efficient implementation of the simplex method (e.g., via the code extraction facility of Coq). Rather, we use the simplex method as a tool in our proofs and, in fact, it turns out to be the cornerstone of our approach, given the intuitionistic nature of the logic in Coq. Thus, we adopt the opposite approach of most textbooks on linear programming where, firstly, theoretical results (like the ones mentioned above) are proven, and then the correctness of the simplex method is derived from them.The formalization presented in this paper can be found in a library developed by the authors called Coq-Polyhedra. This library is available through a git repository at https://github.com/nhojem/Coq-Polyhedra. The branch JAR provides the frozen version of the library corresponding to the results presented in this paper. 1 As mentioned above, our formalization is based on the Mathematical Components library [14] (MathComp for short), which is available at https://github.com/math-comp/math-comp. On top of providing a conven...

Verifying an Incremental Theory Solver for Linear Arithmetic in Isabelle/HOL

“…rationals in combination with a symbolic value δ, representing some small positive rational number. 1 In Phase 2, each constraint with exactly one variable is normalized; in all other constraints the linear polynomial is replaced by a new variable (a slack variable). Thus, Phase 2 produces a set of inequalities of the form x ≤ c or x ≥ c, where x is a variable (such constraints are called atoms).…”

“…This equality can also be used to obtain Farkas coefficients. To this end, we rewrite the equation as −x + 1 2 s − 1 2 y = 0, and use the coefficients in this equation (−1 for x, 1 2 for s, and − 1 2 for y) to form a linear combination of the corresponding atoms involving the variables: 6 Again, we here consider only the check operation, since obtaining Farkas coefficients for a conflict detected by assert is trivial, cf. footnote 3.…”

“…3 and 4. [1] formalized and verified an implementation of the simplex algorithm in Coq. Since their goal was to verify theoretical results about convex polyhedra, their formalization is considerably different from ours, as we aim at obtaining a practically efficient algorithm.…”

Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

“…Related Work. Allamigeon and Katz [ 1 ] have implemented the simplex algorithm in Coq and used it to give constructive proofs of a number of important theorems about convex polyhedra. The overlap between our work and [ 1 ] consists of formalizations of basic facts concerning cones and polyhedra, the fundamental theorem of linear inequalities, and Farkas’ lemma.…”

“…Allamigeon and Katz [ 1 ] have implemented the simplex algorithm in Coq and used it to give constructive proofs of a number of important theorems about convex polyhedra. The overlap between our work and [ 1 ] consists of formalizations of basic facts concerning cones and polyhedra, the fundamental theorem of linear inequalities, and Farkas’ lemma. However, whereas in [ 1 ] a simplex algorithm for optimization problems is implemented in order to be used in constructive mathematical proofs, we formalize theorems concerning integer programming, including bounds on the size of solutions, and use these together with the previously Isabelle-verified simplex algorithm to obtain formally verified, yet efficient, software.…”