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Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

2020
*
International Colloquium on Automata, Languages and Programming
*

We prove that both

doi:10.4230/lipics.icalp.2020.63
dblp:conf/icalp/Hakoniemi20
fatcat:uyt7va4gwbdhxfk2krivqpnddy
*Polynomial**Calculus**and**Sums*-*of*-*Squares*proof systems admit a strong form*of**feasible**interpolation*property*for*sets*of**polynomial*equality constraints. ...*For**Sums*-*of*-*Squares*we rely on the use*of*Boolean axioms, but*for**Polynomial**Calculus*we do not assume their presence. ... A L P 2 0 2 0 63:4*Feasible**Interpolation**for**Polynomial**Calculus**and**Sums*-*Of*-*Squares*where r i , t q , u i*and*v i are arbitrary real*polynomials*. ...##
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subject index volumes 141-150

2003
*
Journal of Computational and Applied Mathematics
*

scheme
145
459
Differentiability
146
343
Differential approximants
149
457
Differential

doi:10.1016/s0377-0427(02)00844-0
fatcat:tsx7p6iw3jbxda3sdee3cdvooq
*calculus**for*p-norms 145 425 Differential*calculus**for*vector norms 145 425 Differential equation ... large numbers 142 27 Least*squares*146 179 Least*squares*147 53 Left-definite self-adjoint operator 148 213 Left-definite Sobolev space 148 213 Legendre*polynomials*148 213 Legendre ... 147 349 Toeplitz operators 142 441 Total ranking method 146 155 Transformation 143 81 Traveling wave fronts 146 309 Tree 142 61 Triangulation 145 505 Trie 142 27 Trigonometric*interpolation*...##
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Sum-of-squares optimization without semidefinite programming
[article]

2018
*
arXiv
*
pre-print

We propose a homogeneous primal-dual interior-point method to solve

arXiv:1712.01792v3
fatcat:f2rnseczuvci7cc7abjgtct5ba
*sum*-*of*-*squares*optimization problems by combining non-symmetric conic optimization techniques*and**polynomial**interpolation*. ... The approach optimizes directly over the*sum*-*of*-*squares*cone*and*its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number*of*auxiliary variables. ... The authors would like to thank the Associate Editor*and*the Referees*for*their constructive feedback. ...##
###
On the Automatizability of Resolution and Related Propositional Proof Systems
[chapter]

2002
*
Lecture Notes in Computer Science
*

This gives a better lower bound

doi:10.1007/3-540-45793-3_38
fatcat:vxg4gu2sw5aglizzinrjp6mj74
*for*the monotone*interpolation**of*Res(2)*and*a separation from Resolution as a byproduct. ... After studying the robustness*of*the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res(¤ ) is weakly automatizable*and*(iii) Res(¤ ) has*feasible**interpolation*... We are grateful to Pavel Pudlák*for*stimulating discussions on the idea*of*Theorem 11. ...##
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On the automatizability of resolution and related propositional proof systems

2004
*
Information and Computation
*

This gives a better lower bound

doi:10.1016/j.ic.2003.10.004
fatcat:sbfnlio72zdonmh2leevz7gveu
*for*the monotone*interpolation**of*Res(2)*and*a separation from Resolution as a byproduct. ... After studying the robustness*of*the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res(¤ ) is weakly automatizable*and*(iii) Res(¤ ) has*feasible**interpolation*... We are grateful to Pavel Pudlák*for*stimulating discussions on the idea*of*Theorem 11. ...##
###
The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs
[chapter]

2013
*
Lecture Notes in Computer Science
*

The inference rules are typically some more or less obvious, non-interesting,

doi:10.1007/978-3-642-39071-5_1
fatcat:4vpknc2xqrbe3fu7hw3nzi7t5m
*and**polynomially*checkable ways*of*producing some logical consequence*of*the hypotheses. ... disjunctions*and*conjunctions, general propositional formulas,*polynomial*equations over some ring,*polynomial*inequalities over some ordered ring, disjunctions*of*those, decision trees branching on variables ... Acknowledgments We thank the comments*of*Allen Van Gelder*and*an anonymous referee on the preliminary draft*of*this paper. ...##
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Proof Complexity (Dagstuhl Seminar 18051)

2018
*
Dagstuhl Reports
*

NP problem,

doi:10.4230/dagrep.8.1.124
dblp:journals/dagstuhl-reports/AtseriasNPS18
fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu
*and*in the ensuing decades many powerful techniques have been discovered*for*analyzing different proof systems. ... While many longstanding open problems from the 1980s*and*1990s still remain unsolved, recent progress gives hope that the area may be ripe*for*decisive breakthroughs. ... -Proof Complexity Russell Impagliazzo, Pavel Pudlák,*and*Jiří Sgall. Lower bounds*for*the*polynomial**calculus**and*the Gröbner basis algorithm. Computational Complexity, 8(2):127-144, 1999. ...##
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Sums of Squares, Moment Matrices and Optimization Over Polynomials
[chapter]

2008
*
IMA Volumes in Mathematics and its Applications
*

Hierarchies

doi:10.1007/978-0-387-09686-5_7
fatcat:buio7ftpcbfdbdobnaqopag5nu
*of*semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices*and*the dual theory*of**sums**of**squares**of**polynomials*. ... We review the mathematical tools underlying these properties, in particular, some*sums**of**squares*representation results*for*positive*polynomials*, some results about moment matrices (in particular,*of*... I am very grateful to Markus Schweighofer*for*discussion*and*his help with some proofs in the survey,*and*to Hayato Waki*for*his careful reading*and**for*his useful questions*and*suggestions. ...##
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Behavior Formula Extraction for Object Trajectory using Curve Fitting Method

2014
*
International Journal of Computer Applications
*

Curve fitting, also known as regression analysis, is used to find the "best fit" line or curve

doi:10.5120/18177-9068
fatcat:cv723gdqzfcwdgjb73p6ljjqoa
*for*a series*of*data points. ... One*of*the main goals*of*computer vision is to enable computers to replicate the basic functions*of*human vision such as motion perception*and*scene understanding. ... the minimum*sums**of**squares**for*the linear equation do not give minimum*sums**of**squares**for*the un linear, or original form*of*the equation. ...##
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Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF
[chapter]

2016
*
Lecture Notes in Computer Science
*

An algorithm

doi:10.1007/978-3-319-40229-1_14
fatcat:bf2mwiy5tvdz3o7jjmdxhatwyi
*for*generating*interpolants**for*formulas which are conjunctions*of*quadratic*polynomial*inequalities (both strict*and*nonstrict) is proposed. ... theory*of*concave quadratic*polynomial*inequalities*and*the equality theory over uninterpreted functions (EUF). ... Under a technical condition that if no nonpositive constant combination*of*nonstrict inequalities is a*sum**of**squares**polynomials*, then such an*interpolant*can be generated essentially using the linearization ...##
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Complexity of Semialgebraic Proofs

2002
*
Moscow Mathematical Journal
*

*calculus*proofs),

*and*Tseitin's tautologies (which are hard

*for*many known proof systems). ... It is a known approach to translate propositional formulas into systems

*of*

*polynomial*inequalities

*and*consider proof systems

*for*the latter. ... RM1-2409-ST-02

*of*the U. S. Civilian Research & Development Foundation

*for*the Independent States

*of*the Former Soviet Union (CRDF). ...

##
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subject index 10 volumes

2005
*
Journal of Computational and Applied Mathematics
*

180 345
Mesh equidistribution 176 131
Method

doi:10.1016/s0377-0427(05)00274-8
fatcat:lnl66qx5lngarjw7i3ducv7qta
*of**feasible*direction 175 447 Method*of*lines 176 293 Method*of*moments 175 77 m-Function 171 93, 411 Mittag-Leffler functions 171 39 Mixed boundary ... 180 461 Indefinite weight 171 93 Indeterminate moment problems 178 91, 235 Index*of*determinacy 178 169 Inequalities 172 289 Inequalities*of*Hardy, Sobolev*and*Laptev/Weidl 171 59 Inequality constrained ...##
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Reviews and Descriptions of Tables and Books

1966
*
Mathematics of Computation
*

the Leverrier-Faddeev method), least

doi:10.1090/s0025-5718-66-99938-8
fatcat:qoyz4x4e2rdndlhtwi3hse4squ
*squares*, difference tables,*polynomial**interpolation**and*numerical differentiation,*and*finally a brief chapter on ordinary differential equations presenting some ... Walsh on simultaneous approximation*and**interpolation*by*polynomials*,*and*the Stone-Weierstrass theorem. ... This is a self-instructional manual*for*FORTRAN II as designed*for*the IBM 700 series*of*computers. The material is presented at a very elementary level (high- ...##
###
Galerkin Optimal Control

2016
*
Journal of Optimization Theory and Applications
*

Send comments regarding this burden estimate or any other aspect

doi:10.1007/s10957-016-0918-x
fatcat:ppa5viqtn5gwrkebxh5aseetkm
*of*this collection*of*information, including suggestions*for*reducing this burden, to Washington headquarters Services, Directorate*for*... is estimated to average 1 hour per response, including the time*for*reviewing instruction, searching existing data sources, gathering*and*maintaining the data needed,*and*completing*and*reviewing the ... As with the*calculus**of*variations both necessary*and*sufficient conditions*for*optimality may be established. ...##
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Least-squares image resizing using finite differences

2001
*
IEEE Transactions on Image Processing
*

*For*a given choice

*of*basis functions, the results

*of*our method are consistently better than those

*of*the standard

*interpolation*procedure; the present scheme achieves a reduction

*of*artifacts such as ... The method can be generalized to include other classes

*of*piecewise

*polynomial*functions, expressed as linear combinations

*of*B-splines

*and*their derivatives. ... Thévenaz,

*of*the Swiss Federal Institute

*of*Technology,

*for*his careful proofreading

*of*this paper. The authors thank also the reviewers

*for*their insightful comments. ...

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