Semidefinite programming and arithmetic circuit evaluation

We address the exact semidefinite programming feasibility problem (SDFP) consisting in checking that intersection of the cone of positive semidefinite matrices and some affine subspace of matrices with rational entries is not empty. SDFP is a convex programming problem and is often considered as tractable since some of its approximate versions can be efficiently solved, e.g. by the ellipsoid algorithm. We prove that SDFP can decide comparison of numbers represented by the arithmetic circuits, i.e. circuits that use standard arithmetical operations as gates. Our reduction may give evidence to the intrinsic difficulty of SDFP (contrary to the common expectations) and clarify the complexity status of the exact SDP-an old open problem in the field of mathematical programming.

[1]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[2]  Andrew Chiu,et al.  Division in logspace-uniform NC1 , 2001, RAIRO Theor. Informatics Appl..

[3]  T. Apostol.  Introduction to analytic number theory , 1976 .

[4]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[5]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[6]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[7]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[8]  Pascal Koiran.  Valiant’s model and the cost of computing integers , 2004, computational complexity.

[9]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[10]  Leslie G. Valiant,et al.  Completeness classes in algebra , 1979, STOC.

[11]  Arnold Schönhage,et al.  On the Power of Random Access Machines , 1979, ICALP.

[12]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[13]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[14]  Leonid Khachiyan,et al.  On the Complexity of Semidefinite Programs , 1997, J. Glob. Optim..

[15]  Klaus W. Wagner,et al.  The complexity of combinatorial problems with succinct input representation , 1986, Acta Informatica.

[16]  Bruce E. Litow,et al.  Fast Parallel Arithmetic via Modular Representation , 1991, SIAM J. Comput..

[17]  R. Freund.  Review of A mathematical view of interior-point methods in convex optimization, by James Renegar, SIAM, Philadelphia, PA , 2004 .

[18]  Motakuri V. Ramana,et al.  An exact duality theory for semidefinite programming and its complexity implications , 1997, Math. Program..

[19]  M. Shub,et al.  On The Intractability Of Hilbert's Nullstellensatz And An Algebraic Version Of . . , 1995 .

[20]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[21]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[22]  Dima Grigoriev,et al.  On the Power of Real Turing Machines Over Binary Inputs , 1997, SIAM J. Comput..

[23]  Komaravolu Chandrasekharan,et al.  Introduction to Analytic Number Theory , 1969 .

[24]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..