Lower Bounds for Polynomials Using Geometric Programming

Ghasemi, Mehdi; Marshall, Murray

doi:10.1137/110836869

Cited by 26 publications

(56 citation statements)

References 13 publications

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“…In particular, when is every nonnegative polynomial a sum of nonnegative circuit polynomials? As already mentioned in Section 7, the case of polynomials with simplex Newton polytopes is solved in [17] via geometric programming generalizing earlier work by Ghasemi in Marshall [12,13].…”

Section: Discussionmentioning

confidence: 99%

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Amoebas, nonnegative polynomials and sums of squares supported on circuits

2016

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We completely characterize sections of the cones of nonnegative polynomials, convex polynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial. Furthermore, nonnegativity of such polynomials f coincides with solidness of the amoeba of f , i.e., the Log-absolute-value image of the algebraic variety V(f ) ⊂ (C * ) n of f . These results generalize earlier works both in amoeba theory and real algebraic geometry by Fidalgo, Kovacec, Reznick, Theobald and de Wolff and solve an open problem by Reznick. They establish the first direct connection between amoeba theory and nonnegativity of real polynomials. Additionally, these statements yield a completely new class of nonnegativity certificates independent from sums of squares certificates.

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Section: Discussionmentioning

confidence: 99%

“…Since, with our choice of parameters, R − r = 1 and (r − R)/r = −t/s, it follows by (12) after replacing φ by φ = sφ that g − p is a hypotrochoid up to a rotation.…”

Section: ) the Complement Of A(f ) Has A Bounded Component If |C| > Fmentioning

confidence: 99%

Amoebas, nonnegative polynomials and sums of squares supported on circuits

2016

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“…Given the runtime comparison of the SONC and the SOS approaches in previous works [8,10,11] using GP, there is reasonable hope that our REPs are faster than SDP in several cases.…”

Section: Conclusion and Open Problemsmentioning

confidence: 96%

A Positivstellensatz for Sums of Nonnegative Circuit Polynomials

Dressler¹,

Iliman²,

Wolff³

2017

SIAM J. Appl. Algebra Geometry

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Abstract. Recently, the second and third authors developed sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity for real polynomials, which is independent of sums of squares. In this paper we show that the SONC cone is full-dimensional in the cone of nonnegative polynomials. We establish a Positivstellensatz which guarantees that every polynomial which is positive on a given compact, semialgebraic set can be represented by the constraints of the set and SONC polynomials. Based on this Positivstellensatz, we provide a hierarchy of lower bounds converging to the minimum of a polynomial on a given compact set K. Moreover, we show that these new bounds can be computed efficiently via interior point methods using results about relative entropy functions.

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“…A previous approach that has a similar spirit is the work in [40], which tackles the specific task of finding a lower bound on the minimum of a polynomial using geometric programming (GP). However, these GP-based conditions seem to be too strong and we show in [4] that this method is always outperformed by the SDSOS approach.…”

Section: Relevant Workmentioning

confidence: 99%

Funnel libraries for real-time robust feedback motion planning

Majumdar

Tedrake

2017

The International Journal of Robotics Research

332

270

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We consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we consider scenarios where these plans must be generated in real-time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Our approach is to pre-compute a library of "funnels" along different maneuvers of the system that the state is guaranteed to remain within (despite bounded disturbances) when the feedback controller corresponding to the maneuver is executed. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances.We demonstrate and validate our method using extensive hardware experiments on a small fixed-wing airplane avoiding obstacles at high speed (∼12 mph), along with thorough simulation experiments of ground vehicle and quadrotor models navigating through cluttered environments. To our knowledge, the resulting hardware demonstrations on a fixed-wing airplane constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in realtime in environments with complex geometric constraints.The key computational engine we leverage is sums-of-squares (SOS) programming. While SOS programming allows us to apply our approach to systems of relatively high dimensionality (up to approximately 10-15 dimensional state spaces), scaling our approach to higher dimensional systems such as humanoid robots requires a different set of computational tools. In this thesis, we demonstrate how DSOS and SDSOS programming, which are recently introduced alternatives to SOS programming, can be employed to achieve this improved scalability and handle control systems with as many as 30-50 state dimensions. My conversations with him on measure theory, functional analysis, and algebraic topology were extremely enriching and provided me with a set of technical tools that will no doubt be very valuable going forwards. I am grateful to Hongkai for giving me the chance to explore problems in grasping and manipulation with him. Hongkai's incredible capacity to work long hours have been an inspiration throughout my time as a graduate student and have pushed me to work that extra bit longer myself. I have lost count of the number of times a quick late night conversation with him has helped solve a problem that I was stuck on.I would also like to thank the other members of the Robot Locomotion Group for inspiration, advice, help, and entertainment during my time here. It has been an honor to work with people who will no doubt lead our field in years to come. Ian Manchester, Zack Jackowski, Michael Levashov, John Roberts,...

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Lower Bounds for Polynomials Using Geometric Programming

Cited by 26 publications

References 13 publications

Amoebas, nonnegative polynomials and sums of squares supported on circuits

Amoebas, nonnegative polynomials and sums of squares supported on circuits

A Positivstellensatz for Sums of Nonnegative Circuit Polynomials

Funnel libraries for real-time robust feedback motion planning

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