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On the Integrality Gap of Binary Integer Programs with Gaussian Data [article]

Sander Borst, Daniel Dadush, Sophie Huiberts, Samarth Tiwari
2021 arXiv   pre-print
Building upon recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), we show that the integrality gap implies that branch-and-bound requires n^poly(m) time on random Gaussian IPs with good probability  ...  For a binary integer program (IP) max c^𝖳 x, Ax ≤ b, x ∈{0,1}^n, where A ∈ℝ^m × n and c ∈ℝ^n have independent Gaussian entries and the right-hand side b ∈ℝ^m satisfies that its negative coordinates have  ...  On the Integrality Gap of Binary Integer Programs with Gaussian Data 37 We now sketch the high level ideas of the proof of Lemma 16.  ... 
arXiv:2012.08346v2 fatcat:clry6bhkofg7pbnx5q6jn6fp6u

MAP Clustering under the Gaussian Mixture Model via Mixed Integer Nonlinear Optimization [article]

Patrick Flaherty, Pitchaya Wiratchotisatian, Ji Ah Lee, Zhou Tang, Andrew C. Trapp
2020 arXiv   pre-print
An important benefit of our approach is the explicit quantification of the degree of suboptimality, via the optimality gap, en route to finding the globally optimal MAP clustering.  ...  We approximate the MINLP through a mixed-integer quadratic program (MIQP) transformation that improves computational aspects while guaranteeing ϵ-global optimality.  ...  Maximum a posteriori clustering under the Gaussian mixture model (1) with known covariance is a biconvex mixed-integer nonlinear programming optimization problem. Proof.  ... 
arXiv:1911.04285v2 fatcat:svl7kad6qnc2xgch3i6aipbyjq

Support Vector Machines with the Ramp Loss and the Hard Margin Loss

J. Paul Brooks
2011 Operations Research  
Based on results on simulated and real-world data, we conclude that SVM with the ramp loss is preferred to SVM with the hard margin loss.  ...  In the interest of deriving classifiers that are robust to outlier observations, we present integer programming formulations of Vapnik's support vector machine (SVM) with the ramp loss and hard margin  ...  , the integrality gap is at least 94.1%.  ... 
doi:10.1287/opre.1100.0854 fatcat:hvderjcljbat7m6zvu7y7s767i

Polynomial-time algorithms for the integer minimal principle for centrosymmetric structures

Anastasia Vaia, Nikolaos V. Sahinidis
2005 Acta Crystallographica Section A Foundations of Crystallography  
However, integer programming techniques may require an exponential number of iterations to exhaust the search space.  ...  Two specialized Gaussian elimination algorithms are then developed to solve this system of equations in polynomial time in the number of atoms.  ...  Yet computational experience with the integer minimal principle model indicates that the gap between its optimal solution and its linear programming relaxation is zero.  ... 
doi:10.1107/s010876730501648x pmid:15972998 fatcat:g5yfjhczlzdz7ldz3l5aw3rdda

Second-Order Conic and Polyhedral Approximations of the Exponential Cone: Application to Mixed-Integer Exponential Conic Programs [article]

Qing Ye, Weijun Xie
2021 arXiv   pre-print
Since the exponential cone can be viewed as the epigraph of perspective of the natural exponential function or the hypograph of perspective of the natural logarithm function, many mixed-integer nonlinear  ...  To harvest the past efforts on MILPs and MISOCPs, this paper presents second-order conic (SOC) and poly-hedral approximation schemes for the exponential cone with application to MIECPs.  ...  The Gaussian quadrature is quite accurate to approximate one-dimensional integration. In fact, according to Chapter 19 [24] , the error of Gaussian quadrature decays geometrically.  ... 
arXiv:2106.09123v2 fatcat:gtxq3tru7jcfrebdgstg56c4x4

Submodularization for Binary Pairwise Energies

Lena Gorelick, Yuri Boykov, Olga Veksler, Ismail Ben Ayed, Andrew Delong
2014 2014 IEEE Conference on Computer Vision and Pattern Recognition  
Many computer vision problems require optimization of binary non-submodular energies. We propose a general optimization framework based on local submodular approximations (LSA).  ...  These methods obtain state-of-the-art results on a wide range of applications outperforming many standard techniques such as LBP, QPBO, and TRWS.  ...  Figures 3 (top-left) shows a binary image after convolution with a uniform 3 × 3 and adding Gaussian noise (σ = 0.05). The goal of binary deconvolution is to recover Figure 5 .  ... 
doi:10.1109/cvpr.2014.151 dblp:conf/cvpr/GorelickBVAD14 fatcat:4qjxhd3blvhitchofkco4dcafq

Cyclic coded integer-forcing equalization

Or Ordentlich, Uri Erez
2010 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
Explicit bounds on the performance of the proposed scheme are derived.  ...  A discrete-time intersymbol interference (ISI) channel with additive Gaussian noise is considered, where only the receiver has knowledge of the channel impulse response.  ...  ACKNOWLEDGMENT The authors wish to thank J. Zhan, B. Nazer, and M. Gastpar for discussions on IF equalization. They also thank Y. Kochman for helpful comments.  ... 
doi:10.1109/allerton.2010.5706944 fatcat:inqiuwfstvde5kmp5tvbnexhrq

Cyclic-Coded Integer-Forcing Equalization

Or Ordentlich, Uri Erez
2012 IEEE Transactions on Information Theory  
Explicit bounds on the performance of the proposed scheme are derived.  ...  A discrete-time intersymbol interference (ISI) channel with additive Gaussian noise is considered, where only the receiver has knowledge of the channel impulse response.  ...  ACKNOWLEDGMENT The authors wish to thank J. Zhan, B. Nazer, and M. Gastpar for discussions on IF equalization. They also thank Y. Kochman for helpful comments.  ... 
doi:10.1109/tit.2012.2202633 fatcat:bxwo5xvukngrha2hsbppdodw7y

Submodularization for Quadratic Pseudo-Boolean Optimization [article]

Lena Gorelick, Yuri Boykov, Olga Veksler, Ismail Ben Ayed, Andrew Delong
2014 arXiv   pre-print
Many computer vision problems require optimization of binary non-submodular energies. We propose a general optimization framework based on local submodular approximations (LSA).  ...  These methods obtain state-of-the-art results on a wide range of applications outperforming many standard techniques such as LBP, QPBO, and TRWS.  ...  Overview of submodularization Linearization has been a popular approximation approach to integer quadratic problem (1)-(2), but it often requires relaxation leading to the integrality gap problem.  ... 
arXiv:1311.1856v2 fatcat:xq5e5f7x6jgadp2zizpzh5uuai

Phase identification in smart grids

V. Arya, D. Seetharam, S. Kalyanaraman, K. Dontas, C. Pavlovski, S. Hoy, J. R. Kalagnanam
2011 2011 IEEE International Conference on Smart Grid Communications (SmartGridComm)  
Furthermore, as the number of measurements increase, continuous relaxations of integer programs may also be used to retrieve household phase efficiently.  ...  Our methods involve the use of integer programming and solutions can be retrieved using branch and bound search algorithms implemented by MIP solvers such as CPLEX.  ...  Then one could find all solutions X with a relative optimality gap of α = {g(X)−g(X * )}/g(X). α can be set close to 0, for e.g 5%.  ... 
doi:10.1109/smartgridcomm.2011.6102329 dblp:conf/smartgridcomm/AryaSKDPHK11 fatcat:6yjitb4slvbfjieh447qubjhke

Integrated optimization of stope boundary selection and scheduling for sublevel stoping operations

T. Copland
2016 Journal of the Southern African Institute of Mining and Metallurgy  
Randomly generated data-sets for gold-copper mineralization are used to investigate the model performance, describing solution time as a function of data complexity. integer programming, operations research  ...  the typical processing power of onsite technical services systems, on standard size data-sets (Little, 2012).  ...  These problems are a subset of IP and are termed binary integer programming (BIP).  ... 
doi:10.17159/2411-9717/2016/v116n12a7 fatcat:ijmzi2xspngfnjgkb7acscw6mq

Polynomial algorithms for LP over a subring of the algebraic integers with applications to LP with circulant matrices

Ilan Adler, Peter A. Beling
1992 Mathematical programming  
We show that a modified variant of the interior point method can solve linear programs (LPs) whose coefficients are real numbers from a subring of the algebraic integers.  ...  We then extend the Tardos scheme to our case, obtaining a running time which is independent of the objective and right-hand side data.  ...  Acknowledgement This research was funded by the National Science Foundation DMS8810192. We greatly appreciate this financial support. under grant  ... 
doi:10.1007/bf01581077 fatcat:5qj4mv6bdnguzaykwvnymw5oua

Near-optimal quantization and linear network coding for relay networks [article]

Anand Muralidhar, P. R. Kumar
2012 arXiv   pre-print
We prove that this digital interface is near-optimal for Gaussian relay networks and the capacities of the Gaussian and the discrete networks are within a bounded gap of O(M^2) bits, where M is the number  ...  the network within a bounded gap of the network's capacity.  ...  In this section, we start the program of approximating Gaussian networks with the discrete model by proving that the cut-set bounds of the two networks are within a bounded gap.  ... 
arXiv:1203.2316v1 fatcat:t7nxrqxn7vdpbk7p4p2pxuulvy

TI-Stan: Model Comparison Using Thermodynamic Integration and HMC

R. Wesley Henderson, Paul M. Goggans
2019 Entropy  
Existing successful implementations use binary slice sampling on the Hilbert curve to accomplish this task.  ...  MC-Stan uses automatic differentiation to compute the gradients of the likelihood that HMC requires in about the same amount of time as it computes the likelihood function itself, easing the programming  ...  Within the TI-based methods proposed here, parameters are always represented as either integer values on [0, 2 B ] (for TI with Binary Slice Sampling) or floating point values on the interval [0, 1] (for  ... 
doi:10.3390/e21121161 fatcat:blg5ypvhxbhl3d644w7vtvwofa

An algorithm for generating stochastic cloud fields from radar profile statistics

K. Franklin Evans, Warren J. Wiscombe
2004 Atmospheric research  
The stochastic cloud generation algorithm is tested with 3 months of boundary layer cumulus cloud data from an 8.6-mm wavelength radar on the island of Nauru.  ...  The major assumptions of the algorithm are that the statistics of the fields are translationally invariant in the horizontal and independent of horizontal direction; however, the statistics do depend on  ...  The rest of the Nauru data were obtained from the Atmospheric Radiation Measurement (ARM) Program sponsored by the U.S.  ... 
doi:10.1016/j.atmosres.2004.03.016 fatcat:x56cwreq4beqxpyqutfmxuem44
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