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New Dependencies of Hierarchies in Polynomial Optimization [article]

Adam Kurpisz, Timo de Wolff
2019 arXiv   pre-print
We prove a collection of dependencies among these hierarchies both for general CPOPs and for optimization problems on the Boolean hypercube.  ...  We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC)  ...  In order to relax (CPOP) to a finite size optimization problem we introduce polynomial hierarchies. Definition 2.1. Let G be a collection of polynomials and let GEN be a subset of K(G + ).  ... 
arXiv:1903.04996v1 fatcat:wzjwnxn3ore3nmtmcxtgrmp5ea

On the Lasserre Hierarchy of Semidefinite Programming Relaxations of Convex Polynomial Optimization Problems

Etienne de Klerk, Monique Laurent
2011 SIAM Journal on Optimization  
In particular, the size of the semidefinite program that gives the exact reformulation of the convex polynomial optimization problem may be exponential in the input size.  ...  The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre.  ...  We have given a new proof of the finite convergence of the Lasserre hierarchy for convex polynomial optimization problems, under weaker assumptions than were known before (Theorem 3.2).  ... 
doi:10.1137/100814147 fatcat:xyvcr2pn2nhzxf7mqcrdehqwku

A new approximation hierarchy for polynomial conic optimization

Peter J. C. Dickinson, Janez Povh
2019 Computational optimization and applications  
We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Pólya's Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic  ...  The hierarchy of relaxations, which contain SOS polynomials and matrix SOS polynomials, is recaptured in Subsection 4.2.  ...  The second author started this research when he was affiliated to the Faculty of Information Studies in Novo Mesto, Slovenia, and continued the work after moving to the University of Ljubljana, Slovenia  ... 
doi:10.1007/s10589-019-00066-0 fatcat:td66r5i57zepjbvr4cuathxbzy

Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator [article]

Pavel Trutman
2020 arXiv   pre-print
This is important since it signicantly simplies further step where we nd the optimal solution by Lasserre relaxations of non-convex polynomial systems.  ...  We demonstrate the method on the 7DOF KUKA LBR IIWA manipulator and show that we are able to compute the optimal IK or certify in-feasibility in 99 % tested poses.  ...  Semi-algebraic optimization problems are in general non-convex but can be solved with certified global optimality [18] using the Lasserre hierarchy of convex optimization problems [17] .  ... 
arXiv:2007.12550v1 fatcat:kh6bkj36fbdd7m4hfb2wlfthvm

Response to "Counterexample to global convergence of DSOS and SDSOS hierarchies" [article]

Amir Ali Ahmadi, Anirudha Majumdar
2017 arXiv   pre-print
In a recent note [8], the author provides a counterexample to the global convergence of what his work refers to as "the DSOS and SDSOS hierarchies" for polynomial optimization problems (POPs) and purports  ...  DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization. In proceedings of the 48th annual IEEE Conference on Information Sciences and Systems, 2014. [5] A. A.  ...  Indeed, we believe that the true power of the (S)DSOS framework comes from the fact that it can provide solutions in situations where even the first level of the SOS relaxation hierarchy is simply too  ... 
arXiv:1710.02901v1 fatcat:adeo54g2szaqrixrakcfouttiq

Page 1746 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews  
The existence of the average time hierarchy was known, but depended on languages defined by complicated diagonalization.  ...  in combinatorial optimization.  ... 

Page 564 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews  
First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy.  ...  Summary: “We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH3(k), where BH;(k) is the kth level of the Boolean hierarchy over Lf.  ... 

A bounded degree SOS hierarchy for polynomial optimization [article]

Jean-Bernard Lasserre, Toh Kim-Chuan, Yang Shouguang
2015 arXiv   pre-print
We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem (P): f^∗={ f(x):x∈ K } on a compact basic semi-algebraic set K⊂^n.  ...  (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems.  ...  Conclusion We have described and tested a new hierarchy of semideifinite relaxations for global polynomial optimization.  ... 
arXiv:1501.06126v2 fatcat:vjo762ovb5fajoth46czljan2e

Page 1489 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
In this hierarchy, the best ‘solutions’ p are optimal points of the optimization problem.  ...  We develop polynomial schemes for special cases of the MCNDP. As a result, we obtain polynomial schemes for new, non-trivial cases of the TSP.  ... 

A bounded degree SOS hierarchy for polynomial optimization

Jean B. Lasserre, Kim-Chuan Toh, Shouguang Yang
2015 EURO Journal on Computational Optimization  
We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem (P ) : f * = min{ f (x) : x ∈ K } on a compact basic semi-algebraic set K ⊂ R n .  ...  (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems.  ...  Conclusion We have described and tested a new hierarchy of semideifinite relaxations for global polynomial optimization.  ... 
doi:10.1007/s13675-015-0050-y fatcat:6xptlfd4hjgahe7grrrarwii5m

Complexity of counting the optimal solutions

Miki Hermann, Reinhard Pichler
2009 Theoretical Computer Science  
In particular, the classes #·Π k P with k ≥ 1 corresponding to all levels of the polynomial hierarchy have thus been studied.  ...  In order to remedy this unsatisfactory situation, we introduce a hierarchy of new counting complexity classes #·Opt k P and #·Opt k P[log n] with k ≥ 1.  ...  Acknowledgment: We thank Arnaud Durand and Yann Strozecki for their remarks and for the discussion on the proof of Theorem 9.  ... 
doi:10.1016/j.tcs.2009.05.025 fatcat:64qcmnafzjdt7pgq2vf4umtgsu

TSSOS: a Julia library to exploit sparsity for large-scale polynomial optimization [article]

Victor Magron, Jie Wang
2021 arXiv   pre-print
TSSOS can be applied to numerous problems ranging from power networks to eigenvalue and trace optimization of noncommutative polynomials, involving up to tens of thousands of variables and constraints.  ...  The Julia library TSSOS aims at helping polynomial optimizers to solve large-scale problems with sparse input data.  ...  A commonly present structure in large-scale polynomial optimization is sparsity. In view of this, TSSOS implements the sparsity-adapted moment-SOS hierarchies.  ... 
arXiv:2103.00915v1 fatcat:vdpwdttfrvbczjrmar27lxrn5i

Page 467 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP-complete sets.” 96a:68033 68Q15 05C05  ...  circuits (in symbols NP C P/poly) implies that the polynomial-time hierarchy collapses to level two (in sym- bols PH = 2} = I$).  ... 

The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization

Franz-Benjamin Mocnik
2018 Scientific Reports  
The local structure of space and global optimization can both be found in transport, brain, and communication networks, which suggests the polynomial volume law, often in combination with hierarchies or  ...  A model of hierarchical spatial networks is introduced to examine the effect of global structures, in particular of hierarchies, on the polynomial volume law.  ...  The author has been funded by Deutsche Forschungsgemeinschaft as part of the project A framework for measuring the fitness for purpose of OpenStreetMap data based on intrinsic quality indicators (FA 1189  ... 
doi:10.1038/s41598-018-29131-0 pmid:30054491 pmcid:PMC6063948 fatcat:udtkuwl3bvc5vbp2yiifexpgkm

Page 5110 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews  
Summary: “The main result in this paper is that P/T! c P/+'T'+*, that is, the PT-hierarchy is a proper hierarchy (i.e., it does not collapse).  ...  Depending on the conditions on the monoid, the problem can be in AC’, be P-complete, be NP- complete or be PSPACE-complete.  ... 
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