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Towards a Practical Volumetric Cutting Plane Method for Convex Programming

Kurt M. Anstreicher
1998 SIAM Journal on Optimization  
We consider the volumetric cutting plane method for nding a point in a convex set C < n that is characterized by a separation oracle.  ...  We prove polynomiality of the algorithm with each added cut placed directly through the current point, and show that this \central cut" version of the method can be implemented using no more than 25n constraints  ...  In 3], Anstreicher describes a strengthened version of Vaidya's volumetric cutting plane algorithm for the convex feasibility problem.  ... 
doi:10.1137/s1052623497318013 fatcat:xuliqbfj6rayhm2akepueskq2e

Page 1313 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
Pierre Loridan (Monchy Saint-Eloi) 98b:90103 90C25 Anstreicher, Kurt M. (1-IA-MG; lowa City, IA) On Vaidya’s volumetric cutting plane method for convex programming. (English summary) Math. Oper.  ...  This paper concerns a particular type of cutting plane method for convex feasibility problems, based on the notion of volumetric centers.  ... 

An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applications [article]

Haotian Jiang, Yin Tat Lee, Zhao Song, Sam Chiu-wai Wong
2020 arXiv   pre-print
A bottleneck of previous cutting plane methods is to compute leverage scores, a measure of the relative importance of past constraints.  ...  We propose a new cutting plane algorithm that uses an optimal O(n log (κ)) evaluations of the oracle and an additional O(n^2) time per evaluation, where κ = nR/ϵ. ∙ This improves upon Vaidya's O( SO· n  ...  This project was supported in part by Special Year on Optimization, Statistics, and Theoretical Machine Learning (being led by Sanjeev Arora) at Institute for Advanced Study.  ... 
arXiv:2004.04250v1 fatcat:imxhan7abfbbhbhygdkd6nbhsu

A unifying geometric solution framework and complexity analysis for variational inequalities

Thomas L. Magnanti, Georgia Perakis
1995 Mathematical programming  
To establish this result we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids,  ...  and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems.  ...  Acknowledgments: We are grateful to Robert Freund for several suggestions that improved the technical content of this paper and to a referee whose probing questions helped us to improve and clarify several  ... 
doi:10.1007/bf01590959 fatcat:qclcpupuvrftjdjzp253b4myn4

Ellipsoidal Approximations of Convex Sets Based on the Volumetric Barrier

K. M. Anstreicher
1999 Mathematics of Operations Research  
We show that a modi cation of the volumetric cutting plane method obtains an O(n 3=2 )-rounding of C in O(n 2 ln(nR)) oracle calls.  ...  Let C R n be a convex set. We assume that kxk 1 1 for all x 2 C, and that C contains a ball of radius 1=R.  ...  Acknowlegement I would like to thank Martin Gr otschel and Yuri Nesterov for several conversations on the topic of this paper, and Laci L ovasz for drawing my attention to Kochol (1994) .  ... 
doi:10.1287/moor.24.1.193 fatcat:r57qpyefh5e7lgw73icxs6ft3u

Algorithm for Constrained Markov Decision Process with Linear Convergence [article]

Egor Gladin, Maksim Lavrik-Karmazin, Karina Zainullina, Varvara Rudenko, Alexander Gasnikov, Martin Takáč
2022 arXiv   pre-print
An agent aims to maximize the expected accumulated discounted reward subject to multiple constraints on its costs (the number of constraints is relatively small).  ...  A new dual approach is proposed with the integration of two ingredients: entropy regularized policy optimizer and Vaidya's dual optimizer, both of which are critical to achieve faster convergence.  ...  agreement (agreement identifier 000000D730321P5Q0002 ) and the agreement with the Ivannikov Institute for System Programming of the Russian Academy of Sciences dated November 2, 2021 No. 70-2021-00142  ... 
arXiv:2206.01666v1 fatcat:oieriptfbrbf3cdp3ephedfdpi

A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization [article]

Yin Tat Lee, Aaron Sidford, Sam Chiu-wai Wong
2015 arXiv   pre-print
We improve upon the running time for finding a point in a convex set given a separation oracle.  ...  Semidefinite Programming: Our runtime is Õ(n(n^2+m^ω+S)), improving upon the previous best of Õ(n(n^ω+m^ω+S)) for the regime where the number of nonzeros S is small.  ...  Lastly, we would like to thank Vaidya for his beautiful work on his cutting plane method.  ... 
arXiv:1508.04874v2 fatcat:74i57p5lardbnmstsdml66ovta

Using Selective Orthonormalization to Update the Analytic Center after Addition of Multiple Cuts

J. E. Mitchell, S. Ramaswamy
2005 Journal of Optimization Theory and Applications  
This is an important issue that arises at every 'stage' in a cutting plane algorithm.  ...  If q ≤ n cuts are to be added, we show that we can use a 'Selective Orthonormalization' procedure to modify the cuts before adding them -it is then easy to identify a direction for an affine step into  ...  Acknowledgments We would like to thank the two anonymous referees for their careful reading of the paper and useful suggestions.  ... 
doi:10.1007/s10957-004-1858-4 fatcat:lybkbtmsabcohii3fn4b2ma4hi

Minimizing Convex Functions with Rational Minimizers [article]

Haotian Jiang
2022 arXiv   pre-print
Given a separation oracle 𝖲𝖮 for a convex function f defined on ℝ^n that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most (a) O(n (n loglog  ...  This improves upon the previously best oracle complexity of O(n^2 (n + log(R))) for polynomial time algorithms and O(n^2log(nR)) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver  ...  I also thank Thomas Rothvoss for other useful comments and his wonderful lecture notes on integer optimization and lattice theory.  ... 
arXiv:2007.01445v5 fatcat:h56qsjo24ff67fnb44oxmxypku

Convex Optimization: Algorithms and Complexity [article]

Sébastien Bubeck
2015 arXiv   pre-print
Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient  ...  We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.  ...  insightful discussions about cutting-plane methods.  ... 
arXiv:1405.4980v2 fatcat:lnzpsn24hbganftjbaps6h2lmq

Fast Core Pricing for Rich Advertising Auctions [article]

Rad Niazadeh, Jason Hartline, Nicole Immorlica, Mohammad Reza Khani, Brendan Lucier
2020 arXiv   pre-print
We conclude that core pricing is implementable even for very time sensitive practical use cases such as realtime auctions for online advertising and can yield more revenue.  ...  We justify this claim experimentally using the Microsoft Bing Ad Auction data, through which we show our core pricing algorithm generates almost 26% more revenue than VCG on average, about 9% more revenue  ...  We would also like to thank the anonymous referees and the associate editor for extraordinarily helpful comments during the revision process.  ... 
arXiv:1610.03564v4 fatcat:yr7ob5whafewzciijne4qmlqzu

A Utility Theory Based Interactive Approach to Robustness in Linear Optimization

Mehdi Karimi, Somayeh Moazeni, Levent Tunçel
2017 Journal of Global Optimization  
After introducing our approach, we develop interactive cutting-plane algorithms for robust optimization, based on concave and quasi-concave utility functions.  ...  We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making.  ...  More sophisticated algorithms have been developed based on Vaidya's volumetric cutting plane method [43, 1] . 5.6. Solutions for Practical Concerns.  ... 
doi:10.1007/s10898-017-0581-2 fatcat:hzt3gskmjjfqndlr44ocfqbpqq

Models and algorithms for distributionally robust least squares problems

Sanjay Mehrotra, He Zhang
2013 Mathematical programming  
For these three cases we derive equivalent formulations and show that the resulting optimization problem can be solved efficiently.  ...  The three probability ambiguity descriptions are given by: (1) confidence interval over the first two moments; (2) bounds on the probability measure with moments constraints; (3) confidence interval over  ...  For the first case, we show that the equivalent semi-infinite programming formulation can be solved efficiently by using ellipsoid method or Vaidya's volumetric cutting plane method with oracle determined  ... 
doi:10.1007/s10107-013-0681-9 fatcat:kpbtuiaop5getnujpn37obd3ga

Semidefinite Programming [chapter]

2017 Convex Optimization for Signal Processing and Communications  
Most interior-point methods for linear programming have been generalized to semide nite programs.  ...  Such a constraint is nonlinear and nonsmooth, but convex, so semide nite programs are convex optimization problems.  ...  Nesterov and Nemirovsky NN94, x5.5] have generalized Vaidya's volumetric and combined volumetric barriers to the cone of positive semide nite matrices.  ... 
doi:10.1201/9781315366920-9 fatcat:nwgwjg7btnfpvmjjeds7uypj7u

4. Semidefinite Programming [chapter]

2001 Lectures on Modern Convex Optimization  
Most interior-point methods for linear programming have been generalized to semide nite programs.  ...  Such a constraint is nonlinear and nonsmooth, but convex, so semide nite programs are convex optimization problems.  ...  Nesterov and Nemirovsky NN94, x5.5] have generalized Vaidya's volumetric and combined volumetric barriers to the cone of positive semide nite matrices.  ... 
doi:10.1137/1.9780898718829.ch4 fatcat:54pwgbfqezd4rnmqewhsccilmq
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