Interior-point methods for unconstrained geometric programming and scaling problems

We provide a condition-based analysis of two interior-point methods for unconstrained geometric programs, a class of convex programs that arise naturally in applications including matrix scaling, matrix balancing, and entropy maximization. Our condition numbers are natural geometric quantities associated with the Newton polytope of the geometric program, and lead to diameter bounds on approximate minimizers. We also provide effective bounds on the condition numbers both in general and under combinatorial assumptions on the Newton polytope. In this way, we generalize the iteration complexity of recent interior-point methods for matrix scaling and matrix balancing. Recently, there has been much work on algorithms for certain optimization problems on Lie groups, known as capacity and scaling problems. For commutative groups, these problems reduce to unconstrained geometric programs, which serves as a particular source of motivation for our work.

[1]  Nisheeth K. Vishnoi,et al.  Maximum Entropy Distributions: Bit Complexity and Stability , 2017, COLT.

[2]  Cole Franks.  Operator scaling with specified marginals , 2018, STOC.

[3]  James Bruce Lee,et al.  Theory and Application , 2019, Wearable Sensors in Sport.

[4]  Levent Tunçel,et al.  Primal-Dual Interior-Point Methods for Domain-Driven Formulations , 2018, Math. Oper. Res..

[5]  Ankur Moitra,et al.  Rigorous Guarantees for Tyler's M-estimator via quantum expansion , 2020, COLT 2020.

[6]  Avi Wigderson,et al.  Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling , 2016, Geometric and Functional Analysis.

[7]  M. Walter,et al.  Minimal length in an orbit closure as a semiclassical limit , 2020, 2004.14872.

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

[9]  Yin Tat Lee,et al.  The Paulsen problem, continuous operator scaling, and smoothed analysis , 2017, STOC.

[10]  Yinyu Ye,et al.  A Computational Study of the Homogeneous Algorithm for Large-scale Convex Optimization , 1998, Comput. Optim. Appl..

[11]  B. Kostant.  On convexity, the Weyl group and the Iwasawa decomposition , 1973 .

[12]  Shlomo Sternberg,et al.  Convexity properties of the moment mapping. II , 1982 .

[13]  Avi Wigderson,et al.  Much Faster Algorithms for Matrix Scaling , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[15]  Aleksander Madry,et al.  Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[16]  Avi Wigderson,et al.  Operator Scaling: Theory and Applications , 2015, Found. Comput. Math..

[17]  Felipe Cucker,et al.  Condition - The Geometry of Numerical Algorithms , 2013, Grundlehren der mathematischen Wissenschaften.

[18]  Linda Ness,et al.  A Stratification of the Null Cone Via the Moment Map , 1984 .

[19]  Peter Bürgisser,et al.  Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[20]  S. Sternberg,et al.  Convexity properties of the moment mapping , 1982 .

[21]  Avi Wigderson,et al.  Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing , 2018, STOC.

[22]  Yinyu Ye,et al.  An infeasible interior-point algorithm for solving primal and dual geometric programs , 1997, Math. Program..

[23]  Clarence Zener,et al.  Geometric Programming : Theory and Application , 1967 .

[24]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[25]  Stephen P. Boyd,et al.  A tutorial on geometric programming , 2007, Optimization and Engineering.

[26]  Peter Bürgisser,et al.  Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory , 2017, ITCS.

[27]  Alex Samorodnitsky,et al.  A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents , 1998, STOC '98.

[28]  Nisheeth K. Vishnoi,et al.  Fair Distributions from Biased Samples: A Maximum Entropy Optimization Framework , 2019, ArXiv.

[29]  F. Kirwan.  Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31 , 1984 .

[30]  Lap Chi Lau,et al.  Spectral Analysis of Matrix Scaling and Operator Scaling , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[31]  Mohit Singh,et al.  Entropy, optimization and counting , 2013, STOC.

[32]  Harm Derksen,et al.  Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations , 2020, SIAM J. Appl. Algebra Geom..

[33]  Michael Atiyah,et al.  Convexity and Commuting Hamiltonians , 1982 .

[34]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[35]  Leonid Gurvits.  Combinatorial and algorithmic aspects of hyperbolic polynomials , 2004, Electron. Colloquium Comput. Complex..

[36]  Jean-Louis Goffin,et al.  The Relaxation Method for Solving Systems of Linear Inequalities , 1980, Math. Oper. Res..

[37]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[38]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[39]  Levent Tunçel,et al.  “Cone-free” primal-dual path-following and potential-reduction polynomial time interior-point methods , 2005, Math. Program..

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

[41]  László A. Végh,et al.  Rescaling Algorithms for Linear Conic Feasibility , 2016, Math. Oper. Res..