A Polynomial-Time Affine-Scaling Method for Semidefinite and Hyperbolic Programming

Renegar, James; Sondjaja, Mutiara

doi:10.48550/arxiv.1410.6734

Cited by 2 publications

(2 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As such, hyperbolic optimization problems can be solved using interior point methods as long as the polynomial p can be evaluated efficiently. More recently, other algorithmic approaches to solving hyperbolic optimization problems have been developed, including primal-dual interior point methods [MT14], affine scaling methods [RS14], firstorder methods based on applying a subgradient method to a transformation of the problem [Ren16] and accelerated modifications tailored for hyperbolic programs [Ren19].…”

Section: Hyperbolic Programmingmentioning

confidence: 99%

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Saunderson¹

2019

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems-a class of convex optimization problems that generalize semidefinite programs. We show how to produce families of nonnegative polynomials (which we call hyperbolic certificates of nonnegativity) from any hyperbolic polynomial. We investigate the pairs (n, d) for which there is a hyperbolic polynomial of degree d in n variables such that an associated hyperbolic certificate of nonnegativity is not a sum of squares. If d ≥ 4 we show that this occurs whenever n ≥ 4. In the degree three case, we find an explicit hyperbolic cubic in 43 variables that gives hyperbolic certificates that are not sums of squares. As a corollary, we obtain the first known hyperbolic cubic no power of which has a definite determinantal representation. Our approach also allows us to show that, given a cubic p, and a direction e, the decision problem "Is p hyperbolic with respect to e?" is co-NP hard.

show abstract

Section: Hyperbolic Programmingmentioning

confidence: 99%

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Saunderson¹

2019

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

show abstract

“…Renegar announced a primal-dual affine-scaling method for hyperbolic cone programming [34] (also see Renegar and Sondjaja [35]). Nesterov and Todd [25,26] present very elegant primaldual interior-point algorithms with outstanding mathematical properties in the setting of selfscaled barriers; however, beyond self-scaled barriers, there are many other primal-dual approaches that are not as symmetric, but retain some of the desired properties (see [41,18,2,3,29,21]).…”

Section: Introductionmentioning

confidence: 99%

Interior-point algorithms for convex optimization based on primal-dual metrics

Myklebust¹,

Tunçel²

2014

Preprint

View full text Add to dashboard Cite

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms we analyse are so-called shortstep algorithms and they match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming problems with given self-scaled barriers. Our results apply to any self-concordant barrier for any convex cone. We also prove that certain specializations of our algorithms to hyperbolic cone programming problems (which lie strictly between symmetric cone programming and general convex optimization problems in terms of generality) can take advantage of the favourable special structure of hyperbolic barriers. We make new connections to Riemannian geometry, integrals over operator spaces, Gaussian quadrature, and strengthen the connection of our algorithms to quasi-Newton updates and hence first-order methods in general.

show abstract

A Polynomial-Time Affine-Scaling Method for Semidefinite and Hyperbolic Programming

Cited by 2 publications

References 20 publications

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

Interior-point algorithms for convex optimization based on primal-dual metrics

Contact Info

Product

Resources

About