Optimization in SMT with LA(Q) Cost Functions

TL;DR

This paper introduces two methods for integrating optimization of LA(Q) cost functions into SMT solving, implemented in MathSAT, and demonstrates competitive performance against specialized optimization tools.

Contribution

It presents the first procedures to enable SMT solvers to minimize LA(Q) cost functions, filling a significant gap in the field.

Findings

The approach is implemented in MathSAT.

Experimental results show competitiveness with state-of-the-art optimization tools.

The methods often outperform existing tools on benchmark problems.

Abstract

In the contexts of automated reasoning and formal verification, important decision problems are effectively encoded into Satisfiability Modulo Theories (SMT). In the last decade efficient SMT solvers have been developed for several theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors). Surprisingly, very few work has been done to extend SMT to deal with optimization problems; in particular, we are not aware of any work on SMT solvers able to produce solutions which minimize cost functions over arithmetical variables. This is unfortunate, since some problems of interest require this functionality. In this paper we start filling this gap. We present and discuss two general procedures for leveraging SMT to handle the minimization of LA(Q) cost functions, combining SMT with standard minimization techniques. We have implemented the proposed approach within the MathSAT SMT solver. Due to the lack of competitors in AR and SMT domains, we experimentally evaluated our implementation against state-of-the-art tools for the domain of linear generalized disjunctive programming (LGDP), which is closest in spirit to our domain, on sets of problems which have been previously proposed as benchmarks for the latter tools. The results show that our tool is very competitive with, and often outperforms, these tools on these problems, clearly demonstrating the potential of the approach.