A Cutting-plane Method for Semidefinite Programming with Potential Applications on Noisy Quantum Devices

There is an increasing interest in quantum algorithms for optimization problems. Within convex optimization, interior-point methods and other recently proposed quantum algorithms are non-trivial to implement on noisy quantum devices. Here, we discuss how to utilize an alternative approach to convex optimization, in general, and semidefinite programming (SDP), in particular. This approach is based on a randomized variant of the cutting-plane method. We show how to leverage quantum speed-up of an eigensolver in speeding up an SDP solver utilizing the cutting-plane method. For the first time, we demonstrate a practical implementation of a randomized variant of the cuttingplane method for semidefinite programming on instances from SDPLIB, a well-known benchmark. Furthermore, we show that the RCP method is very robust to noise in the boundary oracle, which may make RCP suitable for use even on noisy quantum devices.

[1]  Santosh S. Vempala,et al.  Solving convex programs by random walks , 2004, JACM.

[2]  Nicholas J. Higham,et al.  The Conditioning of Linearizations of Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[3]  Zhi-hao Cao.  On a deflation method for the symmetric generalized eigenvalue problem , 1987 .

[4]  M. Berhanu.  The Polynomial Eigenvalue Problem , 2005 .

[5]  Santosh S. Vempala,et al.  Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[6]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I: theory and algorithms , 1993, TOMS.

[7]  G. Calafiore.  Random walks for probabilistic robustness , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[8]  Amás,et al.  Efficient Use of Quantum Linear System Algorithms in Interior Point Methods for Linear Optimization , 2022, 2205.01220.

[9]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[10]  Luis Rademacher,et al.  Approximating the centroid is hard , 2007, SCG '07.

[11]  Carlos Beltrán,et al.  The Polynomial Eigenvalue Problem is Well Conditioned for Random Inputs , 2019, SIAM J. Matrix Anal. Appl..

[12]  Shouvanik Chakrabarti,et al.  Quantum algorithms and lower bounds for convex optimization , 2018, Quantum.

[13]  Sylvain Pion,et al.  Hamiltonian Monte Carlo with boundary reflections, and application to polytope volume calculations , 2018 .

[14]  Alexei Y. Kitaev,et al.  Quantum measurements and the Abelian Stabilizer Problem , 1995, Electron. Colloquium Comput. Complex..

[15]  N. Z. Shor.  Cut-off method with space extension in convex programming problems , 1977, Cybernetics.

[16]  Pravin M. Vaidya,et al.  A cutting plane algorithm for convex programming that uses analytic centers , 1995, Math. Program..

[17]  Stephen P. Boyd,et al.  Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding , 2013, Journal of Optimization Theory and Applications.

[18]  Fabrizio Dabbene,et al.  A Randomized Cutting Plane Method with Probabilistic Geometric Convergence , 2010, SIAM J. Optim..

[19]  D. J. Newman,et al.  Location of the Maximum on Unimodal Surfaces , 1965, JACM.

[20]  Jakub Marecek,et al.  Warm-starting quantum optimization , 2021, Quantum.

[21]  Boris Polyak,et al.  Billiard walk - a new sampling algorithm for control and optimization , 2014 .

[22]  B. Grünbaum.  Partitions of mass-distributions and of convex bodies by hyperplanes. , 1960 .

[23]  B. Parlett.  Analysis of Algorithms for Reflections in Bisectors , 1971 .

[24]  S. Duane,et al.  Hybrid Monte Carlo , 1987 .

[25]  Fabrizio Dabbene,et al.  A randomized cutting plane scheme with geometric convergence: Probabilistic analysis and SDP applications , 2008, 2008 47th IEEE Conference on Decision and Control.

[26]  M. Ruiz Espejo.  Sampling , 2013, Encyclopedic Dictionary of Archaeology.

[27]  Michael J. Todd,et al.  Polynomial Algorithms for Linear Programming , 1988 .

[28]  Claude Lemaréchal,et al.  An Algorithm for Minimizing Convex Functions , 1974, IFIP Congress.

[29]  Ronald de Wolf,et al.  Convex optimization using quantum oracles , 2018, Quantum.

[30]  Boris T. Polyak,et al.  The D-decomposition technique for linear matrix inequalities , 2006 .

[31]  Vissarion Fisikopoulos,et al.  Efficient Sampling from Feasible Sets of SDPs and Volume Approximation , 2020, ArXiv.

[32]  Robert L. Smith,et al.  Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions , 1984, Oper. Res..

[33]  Yin Tat Lee,et al.  An improved cutting plane method for convex optimization, convex-concave games, and its applications , 2020, STOC.

[34]  Ronald de Wolf,et al.  Quantum SDP-Solvers: Better Upper and Lower Bounds , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[35]  Brian Borchers,et al.  SDPLIB 1.1, A Library of Semidefinite Programming Test Problems , 1998 .

[36]  Sergio Boixo,et al.  Spectral Gap Amplification , 2011, SIAM J. Comput..

[37]  Ewin Tang,et al.  Quantum-inspired classical algorithms for principal component analysis and supervised clustering , 2018, ArXiv.

[38]  G. Fix,et al.  An Algorithm for the Ill-Conditioned Generalized Eigenvalue Problem , 1972 .

[39]  S. Vempala.  Geometric Random Walks: a Survey , 2007 .

[40]  Volker Mehrmann,et al.  Vector Spaces of Linearizations for Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[41]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[42]  László Lovász,et al.  Hit-and-run mixes fast , 1999, Math. Program..

[43]  Yin Tat Lee,et al.  A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[44]  Khazhgali Kozhasov,et al.  The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average , 2018, Found. Comput. Math..

[45]  Jean-Daniel Boissonnat,et al.  Effective computational geometry for curves and surfaces , 2006 .

[46]  Anmer Daskin.  Quantum Principal Component Analysis , 2015 .

[47]  Vlatko Vedral,et al.  NISQ Algorithm for Semidefinite Programming , 2021, ArXiv.

[48]  Hans D. Mittelmann,et al.  An independent benchmarking of SDP and SOCP solvers , 2003, Math. Program..

[49]  Justin Domke,et al.  Reflection, Refraction, and Hamiltonian Monte Carlo , 2015, NIPS.

[50]  J. Parker,et al.  Quantum phase estimation for a class of generalized eigenvalue problems , 2020, Physical Review A.

[51]  Sébastien Bubeck,et al.  Convex Optimization: Algorithms and Complexity , 2014, Found. Trends Mach. Learn..

[52]  Frann Coise Tisseur.  Backward Error and Condition of Polynomial Eigenvalue Problems , 1999 .

[53]  Jan van den Brand.  A Deterministic Linear Program Solver in Current Matrix Multiplication Time , 2020, SODA.

[54]  Giuseppe Carlo Calafiore,et al.  A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs , 2007, Autom..

[55]  Angelika Bunse-Gerstner,et al.  An algorithm for the symmetric generalized eigenvalue problem , 1984 .

[56]  J. Radon.  Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[57]  Peter Lindqvist,et al.  A NONLINEAR EIGENVALUE PROBLEM , 2004 .