Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

By Pablo A. Parrilo. 2000

2,361 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Numerical Analysis, Statistics and Probability

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Citations in the corpus

Verifiable Reinforcement Learning via Policy Extraction (2018, methodology): “When fπ is polynomial, we can use sum-of-squares (SOS) optimization to devise a candidate Lyapunov function, check the Lyapunov properites, and compute U [24, 32, 31]; we give a brief overview.”
Revisiting two theorems of Curto and Fialkow on moment matrices (2005, methodology): “Two software packages based on this approach have been developed for this problem: GloptiPoly, developed by Henrion and Lasserre [15], which relies on the program (3.2) involving moment matrices, and SOSTOOLS, developed by Prajna et al. [26], which relies on the sums of squares approach studied by P”
Learning dynamic polynomial proofs (2019, neutral): “Many questions in the areas of control theory [Par00], robotics [MAT13], geometry [PP04], combinatorics [Lov79], program veriﬁcation [MFK + 16] can be modeled using polynomial inequalities.”
Quantized feedback stabilization of non-linear affine systems (2004, background): “This is equivalent to a sum-of-squares problem, and can be solved efficiently using semi-definite programming (SDP); see Parrilo (2000) and Prajna et al. (2002).”
Funnel libraries for real-time robust feedback motion planning (2016, positive): “The decision problem associated with checking polynomial non-negativity is NP-hard in general (Parrilo, 2000).”
A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates (2007, methodology): “The method uses sum of squares optimization [32], [33], [38], [39]—a convex relaxation”
LMI Techniques for Optimization Over Polynomials in Control: A Survey (2010, methodology): “Global asymptotic stability of the origin can be investigated by looking for a Lyapunov function such that (see, e.g., [39] and references therein) (56) This condition can be investigated through the technique in Section III-A in the case that and are polynomial [6].”
TIGHT BOUNDS ON EXPECTED ORDER STATISTICS (2006, background): “The equivalence between the global non-negativity of a quadratic polynomial and the semidefinite representation [20] implies that Formulation (23) can be rewritten as: Z∗ n:n = min ( y′μ + Y .”
Upper bounds on algebraic connectivity via convex optimization (2006, background): “The idea of exploiting symmetry in convex optimization problems has recently found strong interest; see [24,4].”
Safety Verification of Hybrid Systems Using Barrier Certificates (2004, methodology): “The relaxation is provided by the sum of squares decomposition [18] and semidefinite programming [24], which we will describe now.”
Learning Stable Deep Dynamics Models (2020, methodology): “Although some general techniques such as sum-of-squares certification [16, 15] provide general methods for certifying stability of e.”
Challenges and opportunities in quantum optimization (2023, neutral): “Similarly, using moment or sum-of-squares techniques 134,135 , MIP can be reformulated to infinite-dimensional linear programming, with an SDP relaxation in a dimension much higher than n + d and sometimes 136 exponential in n + d .”
Predictive Control for Linear and Hybrid Systems (2017, methodology): “Such conservativeness can be reduced by constructing piecewise polynomial Lyapunov functions via semidefinite programming by means of the sum of squares (SOS) decomposition of multivariate polynomials [230].”