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New Dependencies of Hierarchies in Polynomial Optimization
[article]

2019
*
arXiv
*
pre-print

We prove a collection

arXiv:1903.04996v1
fatcat:wzjwnxn3ore3nmtmcxtgrmp5ea
*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 + ). ...##
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On the Lasserre Hierarchy of Semidefinite Programming Relaxations of Convex Polynomial Optimization Problems

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). ...

##
###
A new approximation hierarchy for polynomial conic optimization

2019
*
Computational optimization and applications
*

We propose a

doi:10.1007/s10589-019-00066-0
fatcat:td66r5i57zepjbvr4cuathxbzy
*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 ...##
###
Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator
[article]

2020
*
arXiv
*
pre-print

This is important since it signicantly simplies further step where we nd the

arXiv:2007.12550v1
fatcat:kh6bkj36fbdd7m4hfb2wlfthvm
*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] . ...##
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Response to "Counterexample to global convergence of DSOS and SDSOS hierarchies"
[article]

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 ...

##
###
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]

2015
*
arXiv
*
pre-print

We consider a

arXiv:1501.06126v2
fatcat:vjo762ovb5fajoth46czljan2e
*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*. ...##
###
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

2015
*
EURO Journal on Computational Optimization
*

We consider a

doi:10.1007/s13675-015-0050-y
fatcat:6xptlfd4hjgahe7grrrarwii5m
*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*. ...##
###
Complexity of counting the optimal solutions

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. ...

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

2021
*
arXiv
*
pre-print

TSSOS can be applied to numerous problems ranging from power networks to eigenvalue and trace

arXiv:2103.00915v1
fatcat:vdpwdttfrvbczjrmar27lxrn5i
*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*. ...##
###
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

2018
*
Scientific Reports
*

The local structure

doi:10.1038/s41598-018-29131-0
pmid:30054491
pmcid:PMC6063948
fatcat:udtkuwl3bvc5vbp2yiifexpgkm
*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 ...##
###
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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