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The design of cyber-physical systems (CPSs) requires methods and tools that can efficiently reason about the interaction between discrete models, e.g., representing the behaviors of ``cyber'' components, and continuous models of physical processes. Boolean methods such as satisfiability (SAT) solving are successful in tackling large combinatorial search problems for the design and verification of hardware and software components. On the other hand, problems in control, communications, signal processing, and machine learning often rely on convex programming as a powerful solution engine. However, despite their strengths, neither approach would work in isolation for CPSs. In this paper, we present a new satisfiability modulo convex programming (SMC) framework that integrates SAT solving and convex optimization to efficiently reason about Boolean and convex constraints at the same time. We exploit the properties of a class of logic formulas over Boolean and nonlinear real predicates, termed monotone satisfiability modulo convex formulas, whose satisfiability can be checked via a finite number of convex programs. Following the lazy satisfiability modulo theory (SMT) paradigm, we develop a new decision procedure for monotone SMC formulas, which coordinates SAT solving and convex programming to provide a satisfying assignment or determine that the formula is unsatisfiable. A key step in our coordination scheme is the efficient generation of succinct infeasibility proofs for inconsistent constraints that can support conflict-driven learning and accelerate the search. We demonstrate our approach on different CPS design problems, including spacecraft docking mission control, robotic motion planning, and secure state estimation. We show that SMC can handle more complex problem instances than state-of-the-art alternative techniques based on SMT solving and mixed integer convex programming.
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Applications and efficient algorithms for integer programming problems on monotone constraints
Abstract We present here classes of integer programming problems that are solvable efficiently and with combinatorial flow algorithms. The problems are characterized by constraints that have either at most two variables per inequality that appear with opposite sign coefficients, or have in addition a third variable that appears only in one constraint. Such integer programs, referred to here as monotone IP2 or IP3, are shown to be solvable in polynomial time for polynomially bounded variables. This article demonstrates the vast applicability of IP2 and IP3 as models for integer programs in multiple scenarios. Since the problems are easily recognized, the knowledge of their structure enables one to determine easily that they are efficiently solvable. The variety of applications, that previously were not known to be solved as efficiently, underlies the importance of recognizing this structure and, if appropriate, formulating problems as monotone IP2 or IP3. Additionally, if there is flexibility in the modeling of an integer programming problem, the formulation choice as monotone IP2 or IP3 leads to efficient algorithms, whereas slightly different modeling choices would lead to NP‐hard problems.
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- Award ID(s):
- 1760102
- PAR ID:
- 10455882
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Networks
- Volume:
- 77
- Issue:
- 1
- ISSN:
- 0028-3045
- Format(s):
- Medium: X Size: p. 21-49
- Size(s):
- p. 21-49
- Sponsoring Org:
- National Science Foundation
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