12,641 research outputs found
Delta-Complete Decision Procedures for Satisfiability over the Reals
We introduce the notion of "\delta-complete decision procedures" for solving
SMT problems over the real numbers, with the aim of handling a wide range of
nonlinear functions including transcendental functions and solutions of
Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational
number \delta, a \delta-complete decision procedure determines either that
\varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is
satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that
allows \delta-bounded numerical perturbations on \varphi. We prove the
existence of \delta-complete decision procedures for bounded SMT over reals
with functions mentioned above. For functions in Type 2 complexity class C,
under mild assumptions, the bounded \delta-SMT problem is in NP^C.
\delta-Complete decision procedures can exploit scalable numerical methods for
handling nonlinearity, and we propose to use this notion as an ideal
requirement for numerically-driven decision procedures. As a concrete example,
we formally analyze the DPLL framework, which integrates Interval
Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient
conditions for its \delta-completeness. We discuss practical applications of
\delta-complete decision procedures for correctness-critical applications
including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201
A General Large Neighborhood Search Framework for Solving Integer Programs
This paper studies how to design abstractions of large-scale combinatorial optimization problems that can leverage existing state-of-the-art solvers in general purpose ways, and that are amenable to data-driven design. The goal is to arrive at new approaches that can reliably outperform existing solvers in wall-clock time. We focus on solving integer programs, and ground our approach in the large neighborhood search (LNS) paradigm, which iteratively chooses a subset of variables to optimize while leaving the remainder fixed. The appeal of LNS is that it can easily use any existing solver as a subroutine, and thus can inherit the benefits of carefully engineered heuristic approaches and their software implementations. We also show that one can learn a good neighborhood selector from training data. Through an extensive empirical validation, we demonstrate that our LNS framework can significantly outperform, in wall-clock time, compared to state-of-the-art commercial solvers such as Gurobi
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Convex Relaxations and Approximations of Chance-Constrained AC-OPF Problems
This paper deals with the impact of linear approximations for the unknown
nonconvex confidence region of chance-constrained AC optimal power flow
problems. Such approximations are required for the formulation of tractable
chance constraints. In this context, we introduce the first formulation of a
chance-constrained second-order cone (SOC) OPF. The proposed formulation
provides convergence guarantees due to its convexity, while it demonstrates
high computational efficiency. Combined with an AC feasibility recovery, it is
able to identify better solutions than chance-constrained nonconvex AC-OPF
formulations. To the best of our knowledge, this paper is the first to perform
a rigorous analysis of the AC feasibility recovery procedures for robust
SOC-OPF problems. We identify the issues that arise from the linear
approximations, and by using a reformulation of the quadratic chance
constraints, we introduce new parameters able to reshape the approximation of
the confidence region. We demonstrate our method on the IEEE 118-bus system
Disjunctive Answer Set Solvers via Templates
Answer set programming is a declarative programming paradigm oriented towards
difficult combinatorial search problems. A fundamental task in answer set
programming is to compute stable models, i.e., solutions of logic programs.
Answer set solvers are the programs that perform this task. The problem of
deciding whether a disjunctive program has a stable model is
-complete. The high complexity of reasoning within disjunctive
logic programming is responsible for few solvers capable of dealing with such
programs, namely DLV, GnT, Cmodels, CLASP and WASP. In this paper we show that
transition systems introduced by Nieuwenhuis, Oliveras, and Tinelli to model
and analyze satisfiability solvers can be adapted for disjunctive answer set
solvers. Transition systems give a unifying perspective and bring clarity in
the description and comparison of solvers. They can be effectively used for
analyzing, comparing and proving correctness of search algorithms as well as
inspiring new ideas in the design of disjunctive answer set solvers. In this
light, we introduce a general template, which accounts for major techniques
implemented in disjunctive solvers. We then illustrate how this general
template captures solvers DLV, GnT and Cmodels. We also show how this framework
provides a convenient tool for designing new solving algorithms by means of
combinations of techniques employed in different solvers.Comment: To appear in Theory and Practice of Logic Programming (TPLP
Effective Lower Bounding Techniques for Pseudo-Boolean Optimization
Linear Pseudo-Boolean Optimization (PBO) is a widely used modeling framework in Electronic Design Automation (EDA). Due to significant advances in Boolean Satisfiability (SAT), new algorithms for PBO have emerged, which are effective on highly constrained instances. However, these algorithms fail to handle effectively the information provided by the cost function of PBO. This paper addresses the integration of lower bound estimation methods with SAT-related techniques in PBO solvers. Moreover, the paper shows that the utilization of lower bound estimates can dramatically improve the overall performance of PBO solvers for most existing benchmarks from EDA. 1
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