Strong duality in Lasserre’s hierarchy for polynomial optimization

Josz, Cédric; Henrion, Didier

doi:10.1007/s11590-015-0868-5

Cited by 49 publications

(47 citation statements)

References 9 publications

(7 reference statements)

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“…Weak SDP duality implies that the optimal value of P d is always greater or equal to the optimal value of D d . In [10] the authors establish a condition which guarantees strong duality for primal and dual SDP problems in the Lasserre hierarchy. The polynomial optimisation problem POP(q 0 ; Q) is called encircled if a polynomial R 2 − i∈[n] x 2 i can be obtained as a positive linear combination of polynomials from Q of degree at most 2.…”

Section: Lasserre Hierarchymentioning

confidence: 99%

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem

Atserias

Ochremiak

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. LP, SDP or Groebner-based relaxations are now known to stay proper relaxations of isomorphism, their relative strength, besides the obvious relationships, was not fully understood. Since SDP is a proper generalization of LP, one may be tempted to guess that the Lasserre SDP hierarchy could perhaps distinguish more graphs than its LP sibling. Interestingly, we prove this not to be the case: for the isomorphism problem, the strength of the Lasserre hierarchy collapses to that of the Sherali-Adams hierarchy, up to a small loss in the level of the hierarchy.Concretely, we show that there exist a constant c ≥ 1 such that if two given graphs are distinguishable in the k-th level of the Lasserre hierarchy, then they must also be distinguishable in the ck-th level of the Sherali-Adams hierarchy. The constant c loss comes from the number of variables for expressing the SDP exact feasibility problem in the bounded-variable counting logic. It should be noted that our proof is indirect as it relies on the correspondance between indistinguishability in k-variable counting logic and the k-th level Sherali-Adams relaxation of graph isomorphism [3]. The question whether the collapse can be shown to hold directly, by lifting LP-feasible solutions into SDP-feasible ones, remains an interesting one.This collapse result has some curious consequences. For one it says that, for distinguishing graphs, the spectral methods that underlie the Lasserre hierarchy are already available in low levels of the Sherali-Adams hierarchy. This may sound surprising, but aligns well with the known fact that indistinguishability by 3-variable counting logic captures graph spectra [7], together with the abovementioned correspondance between k-variable counting logic and the k-th level of the Sherali-Adams hierarchy.By moving to the duals, our results can be read in terms of Sums-of-Squares (SOS) and Sherali-Adams (SA) proofs, and used to get consequences for Polynomial Calculus (PC) proofs as a side bonus. In terms of proofs, we show that if there is a degree-k SOS proof that two graphs are not isomorphic, then there is also a degree-ck SA proof. In turn, it was already known from before, by combining the results in [3] and [5], that if there is a degree-ck SA proof then there is also a degre...

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Section: Lasserre Hierarchymentioning

confidence: 99%

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem

Atserias

Ochremiak

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…By weak duality val (r) out er is at least the optimum value of (29). Strong duality holds for instance if the set K has a non-empty interior (since then the primal SDP is strictly feasible), or if there is a ball constraint present in the description of the set K (as shown in [28]). Then, val (r) out er is also given by the program (29), which is the case, e.g., when K is a simplex, a hypercube, or a sphere.…”

Section: Error Analysis For the Case Of Polynomial Optimizationmentioning

confidence: 99%

A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis

Klerk

Laurent

2019

Association for Women in Mathematics Series

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The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, options pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, due to J.B. Lasserre, is to approximate the convex cone of positive Borel measures by finite dimensional outer and inner conic approximations. We will review some results on these approximations, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximations.

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“…Solutions to these programs often lie on the boundary of the feasible set, and small numerical tolerances can lead to infeasibilities in subsequent programs. Adapting the work of Josz and Henrion [9], we address this issue by writing the alternations in a manner that guarantees that (1) the feasible set always has a non-empty interior and (2) that alternating solutions lie on the interior. These simple steps greatly enhance the numerical stability of bilinear alternations.…”

Section: A Bilinear Inner Approximationmentioning

confidence: 99%

“…A solution with γ < 0 is feasible for the original problem (I). As shown in [9], the feasible set of this program is guaranteed to contain a nonempty interior.…”

Section: A Bilinear Inner Approximationmentioning

confidence: 99%

Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

Posa

Koolen

Tedrake

2017

Robotics: Science and Systems XIII

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Abstract-A fundamental requirement for legged robots is to maintain balance and prevent potentially damaging falls whenever possible. As a response to outside disturbances, fall prevention can be achieved by a combination of active balancing actions, e.g. through ankle torques and upper-body motion, and through reactive step placement. While it is widely accepted that stepping is required to respond to large disturbances, the limits of active motions on balancing and step recovery are only well understood for the simplest of walking models. Recent advances in convex optimization-based verification and control techniques enable a more complete understanding of the limits and capabilities of more complex models. In this work, we present an algorithmic approach for formal analysis of the viable-capture basins of walking robots, calculating both inner and outer approximations and corresponding push recovery control strategies. Extending beyond the classic Linear Inverted Pendulum Model (LIPM), we analyze a series of centroidal momentum based planar walking models, examining the effects of center of mass height, angular momentum, and impact dynamics during stepping on capturability. This formal analysis enables an explicit calculation of the differences between these models, and assessment of whether the simplest models ultimately sacrifice capability, and thus stability, when designing push recovery control policies.

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Strong duality in Lasserre’s hierarchy for polynomial optimization

Cited by 49 publications

References 9 publications

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem

A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis

Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

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