Certification of real inequalities: templates and sums of squares

Magron, Victor; Allamigeon, Xavier; Gaubert, Stéphane; Werner, Benjamin

doi:10.1007/s10107-014-0834-5

Cited by 3 publications

(5 citation statements)

References 35 publications

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“…involves univariate non-semialgebraic components such as exp or sin. We use the method presented in [Magron et al 2015a], based on maxplus approximation of semiconvex transcendental functions by quadratic functions. This idea comes from optimal control [McEneaney 2006] and was developed further to represent the value function by a "maxplus linear combination", which is a supremum of quadratic polynomials at given points x i .…”

Section: Semialgebraic Programsmentioning

confidence: 99%

“…upper) approximation l − (resp. l + ) of l over K. For more details in the context of global optimization, we refer the reader to [Magron et al 2015a]. Then, calling the procedure sdp_sa allows us to get interval enclosures of l − as well as l + .…”

Section: Semialgebraic Programsmentioning

confidence: 99%

“…The procedure relies on SDP to provide sparse sumof-squares decompositions of nonnegative polynomials. Our framework handles polynomial program analysis (involving the operations +, ×, −) as well as extensions to the more general class of semialgebraic and transcendental programs (involving √ ,/, min, max, arctan, exp), following the approximation scheme described in [Magron et al 2015a].…”

Section: Introductionmentioning

confidence: 99%

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Certified Roundoff Error Bounds Using Semidefinite Programming

Magron

Constantinides

Donaldson

2017

ACM Trans. Math. Softw.

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Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementations. This problem becomes challenging when the program does not employ solely linear operations, and non-linearities are inherent to many interesting computational problems in real-world applications.Existing solutions to reasoning possibly lead to either inaccurate bounds or high analysis time in the presence of nonlinear correlations between variables. Furthermore, while it is easy to implement a straightforward method such as interval arithmetic, sophisticated techniques are less straightforward to implement in a formal setting. Thus there is a need for methods which output certificates that can be formally validated inside a proof assistant.We present a framework to provide upper bounds on absolute roundoff errors of floating-point nonlinear programs. This framework is based on optimization techniques employing semidefinite programming and sums of squares certificates, which can be checked inside the Coq theorem prover to provide formal roundoff error bounds for polynomial programs. Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization and space control. Our tool produces more accurate error bounds for 23 % of all programs and yields better performance in 66 % of all programs.

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Section: Semialgebraic Programsmentioning

confidence: 99%

Section: Semialgebraic Programsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Certified Roundoff Error Bounds Using Semidefinite Programming

Magron

Constantinides

Donaldson

2017

ACM Trans. Math. Softw.

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“…For the case of boundedness property, it would allow to decrease the value of the bounds on the variables. Finally, our method could be generalized to a larger class of programs, involving semialgebraic or transcendental assignments, while applying the same polynomial reduction techniques as in [MAGW14].…”

Section: Example 8 (Adapted From Example 3 In [Aj13])mentioning

confidence: 99%

Property-based Polynomial Invariant Generation Using Sums-of-Squares Optimization

2015

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While abstract interpretation is not theoretically restricted to specific kinds of properties, it is, in practice, mainly developed to compute linear over-approximations of reachable sets, aka. the collecting semantics of the program. The verification of user-provided properties is not easily compatible with the usual forward fixpoint computation using numerical abstract domains. We propose here to rely on sums-of-squares programming to characterize a property-driven polynomial invariant. This invariant generation can be guided by either boundedness, or in contrary, a given zone of the state space to avoid. While the target property is not necessarily inductive with respect to the program semantics, our method identifies a stronger inductive polynomial invariant using numerical optimization. Our method applies to a wide set of programs: a main while loop composed of a disjunction (ifthen-else) of polynomial updates e.g. piecewise polynomial controllers. It has been evaluated on various programs.

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“…One motivation of the present work is to reduce this total verification time using alternative formal methods. In a previous work, the authors developed an informal framework, built on top of the certified -template based -global optimization method [MAGW,AGMW13a,AGMW13b]. The nonlinear template method is a certification framework, aiming at handling the approximation of transcendental functions and increasing the size of certifiable instances.…”

Section: Contextmentioning

confidence: 99%

Formal Proofs for Nonlinear Optimization

Magron,

Allamigeon,

Gaubert

et al. 2014

Preprint

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We present a formally verified global optimization framework. Given a semialgebraic or transcendental function f and a compact semialgebraic domain K, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of f over K. This method allows to bound in a modular way some of the constituents of f 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 under-approximations 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.The research leading to these results has received funding from the European Union's 7 th Framework Programme under grant agreement no. 243847 (ForMath).

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Certification of real inequalities: templates and sums of squares

Cited by 3 publications

References 35 publications

Certified Roundoff Error Bounds Using Semidefinite Programming

Certified Roundoff Error Bounds Using Semidefinite Programming

Property-based Polynomial Invariant Generation Using Sums-of-Squares Optimization

Formal Proofs for Nonlinear Optimization

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