Computation with Polynomial Equations and Inequalities Arising in Combinatorial Optimization

Loera, Jesús A. De; Malkin, Peter N.; Parrilo, Pablo A.

doi:10.1007/978-1-4614-1927-3_16

Cited by 12 publications

(16 citation statements)

References 49 publications

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“…As pathological convex optimization problems do arise in practice [51,33,36,74,77], there is practical value in studying how well-behaved and robust an algorithm is in such setups However, the there had been surprisingly little work investigating the behavior of the popular methods DRS and ADMM under pathologies. The understanding is still incomplete, but there has been some recent progress: [10,15,16,50] analyze DRS under specific pathological setups, [12,13,16] analyze DRS under general setups, and [65,72,6] analyze ADMM under specific pathological setups for conic programs.…”

Section: Prior Workmentioning

confidence: 99%

Douglas–Rachford splitting and ADMM for pathological convex optimization

2019

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Despite the vast literature on DRS and ADMM, there has been very little work analyzing their behavior under pathologies. Most analyses assume a primal solution exists, a dual solution exists, and strong duality holds. When these assumptions are not met, i.e., under pathologies, the theory often breaks down and the empirical performance may degrade significantly. In this paper, we establish that DRS only requires strong duality to work, in the sense that asymptotically iterates are approximately feasible and approximately optimal.Keywords Douglas-Rachford splitting · Strong Duality · Pathological convex programs Mathematics Subject Classification (2010) 90C46 · 49N15 · 90C25 1 Introduction Douglas-Rachford splitting (DRS) and alternating directions method of multipliers (ADMM) are classical methods originally presented in [61,35,49,47] and [44,46], respectively. DRS and ADMM are closely related. Over the last decade, these methods have enjoyed a resurgence of popularity, as the demand to solve ever larger problems grew.DRS and ADMM have strong theoretical guarantees and empirical performance, but such results are often limited to non-pathological problems; in Ernest K. Ryu UCLA

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Section: Prior Workmentioning

confidence: 99%

Douglas–Rachford splitting and ADMM for pathological convex optimization

2019

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“…If a system is known to have a Nullstellensatz certificate of small constant degree (over a finite field), one can simply find this certificate by a series of linear algebra computations [14,15,16]. There are well-known upper bounds for the degrees of the coefficients αi in the Nullstellensatz certificate for general systems of polynomials that grow with the number of variables [28].…”

Section: Introductionmentioning

confidence: 98%

Graph-Coloring Ideals

Loera

Margulies

Pernpeintner

et al. 2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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We consider a well-known family of polynomial ideals encoding the problem of graph-k-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gröb-ner bases and Nullstellensatz certificates for the coloring ideals of general graphs. We revisit the fact that computing a Gröbner basis is NP-hard and prove a robust notion of hardness derived from the inapproximability of coloring problems. For chordal graphs, however, we explicitly describe a Gröbner basis for the coloring ideal and provide a polynomial-time algorithm to construct it.

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“…Since f is a polynomial then the maximum value of f (x) if exists in its polyhedra is not singular, that there is a local maximum and there is a global maximum (optimal value), so there must be a guarantee that the maximum value local is a global maximum value, namely if the function f : S → X on S ⊆ X, assuming a convex subset S and f is also convex function of the local maximum is a global maximum. At [3] have discussed the issue of feasible area or a polynomial combinatorial polyhedron, specifically with linear objective function and constraints of any polynomial with integer coefficients. It also defines the convexity problem a function at integer, as an extension of the definition of convexity on a continuous function in R, also examines the properties of the polyhedron obtained.…”

Section: Introductionmentioning

confidence: 99%