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Combinatorial integral approximation

Sebastian Sager, Michael Jung, Christian Kirches
2011 Mathematical Methods of Operations Research  
In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid.  ...  In Section 3 we will discuss our new method that is based on a combinatorial approximation of the integral over control deviations.  ...  Approximating the integral over the controls by MILP techniques The results of Section 2 have been used in several ways.  ... 
doi:10.1007/s00186-011-0355-4 fatcat:wfurxew4zfgivn4es36y7ywlve

Relaxed multibang regularization for the combinatorial integral approximation [article]

Paul Manns
2021 arXiv   pre-print
Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control.  ...  We extend the algorithmic framework of the combinatorial integral approximation such that a subsequence of the computed discrete-valued controls converges to the infimum of the regularized integer control  ...  integral approximation.  ... 
arXiv:2011.00205v2 fatcat:zsekio6dkrfcbh4fqsljttec7a

Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control

Clemens Zeile, Tobias Weber, Sebastian Sager
2022 Algorithms  
Here, a priori bounds were derived for a decomposition into one continuous nonlinear control problem and one mixed-integer linear program, the combinatorial integral approximation (CIA) problem.  ...  These extensions are transferable in a straightforward way, though, to recently suggested variants for certain partial differential equations, for algebraic equations, for additional combinatorial constraints  ...  Scaled Combinatorial Integral Approximation Our hypothesis is that the MILPs based on a scaled combinatorial integral approximation perform the best on instances where the binary control enters the control  ... 
doi:10.3390/a15040121 fatcat:udjkf7rbtjbyhoer3bxws2snmi

The Lagrangian relaxation for the combinatorial integral approximation problem

Michael N. Jung, Gerhard Reinelt, Sebastian Sager
2014 Optimization Methods and Software  
We decompose this MINLP into an NLP and an MILP, which is called the combinatorial integral approximation problem (CIAP).  ...  We are interested in methods to solve mixed-integer nonlinear optimal control problems (MIOCPs) constrained by ordinary differential equations and combinatorial constraints on some of the control functions  ...  The combinatorial integral approximation problem (CIAP) arises as a consequence of the following theorem from (15) . Theorem 1 (Integral Approximation Theorem).  ... 
doi:10.1080/10556788.2014.890196 fatcat:ghtafktsiraptp7gzw2x7qh62a

Compactness and convergence rates in the combinatorial integral approximation decomposition

Christian Kirches, Paul Manns, Stefan Ulbrich
2020 Mathematical programming  
AbstractThe combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and  ...  computing a discrete-valued approximation of the relaxed control.  ...  We use the results on convexification reformulations and the combinatorial integral approximation (CIA) decomposition from [12, 15, 21, 23, 24, 26, 27, 36] .  ... 
doi:10.1007/s10107-020-01598-8 fatcat:3by2lyyup5d7nc4zmebiqjxtbm

On structural similarities of combinatorial integral approximation and binary trust-region steepest descent [article]

Paul Manns and Mirko Hahn and Christian Kirches and Sven Leyffer and Sebastian Sager
2022 arXiv   pre-print
Combinatorial integral approximation and binary trust-region steepest descent are two approaches for the solution of optimal control problems with discrete (binary in the latter case) control inputs.  ...  The main difference in their analyses is an additional compactness assumption on the reduced objective functional for the combinatorial integral decomposition that implies an approximation of stationary  ...  Combinatorial integral approximation [8, 12, 23, 22] The idea that underlies CIA is to split the solution process of (P) into solving the continuous relaxation (R) and then computing a V -valued approximation  ... 
arXiv:2202.07934v1 fatcat:bqnqihma7jgn5kpcpwh6lywwsq

Sherali-Adams Integrality Gaps Matching the Log-Density Threshold

Eden Chlamtác, Pasin Manurangsi, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: that it is hard to distinguish a fully random combinatorial structure from one which contains  ...  This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest k-Subgraph, Smallest p-Edge Subgraph, their hypergraph  ...  For this problem we get integrality gaps matching the approximation guarantee of [13] : (1) . Corollary 5.  ... 
doi:10.4230/lipics.approx-random.2018.10 dblp:conf/approx/ChlamtacM18 fatcat:3l7ydhovxraqfaeka7fxtkdcsy

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

Jochen Könemann, Neil Olver, Kanstantsin Pashkovich, R. Ravi, Chaitanya Swamy, Jens Vygen, Marc Herbstritt
2017 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios  ...  We dispel this belief by showing that the integrality gap of this LP is at least 9/4. This holds even for planar graphs.  ...  This research was initiated during the Hausdorff Trimester Program "Combinatorial Optimization".  ... 
doi:10.4230/lipics.approx-random.2017.17 dblp:conf/approx/KonemannOP0SV17 fatcat:kis2fag6fndpfchfyslh7gkqau

Dual Growth with Variable Rates: An Improved Integrality Gap for Steiner Tree [article]

Ali Çivril
2022 arXiv   pre-print
In this paper, we extend the approach provided by these authors and show that the integrality gap of BCR is at most 7/6 on quasi-bipartite graphs via a fast combinatorial algorithm.  ...  Using this, they gave a √(2)-approximation algorithm for quasi-bipartite graphs and showed that the integrality gap of the relaxation is at most 4/3 for this class of graphs.  ...  combinatorial optimization problems.  ... 
arXiv:1704.08680v6 fatcat:55x76hwamndj3ej64o6s3hvzem

A Combinatorial Optimized Knapsack Linear Space for Information Retrieval

Varghese S Chooralil, Vinodh P Vijayan, Biju Paul, Anishin Raj M.M, Karthikeyan B, G.Manikandan
2021 Computers Materials & Continua  
In this work, we consider the strength of a term and the influence of a term as a combinatorial optimization, called Combinatorial Optimized Linear Space Knapsack for Information Retrieval (COLSK-IR).  ...  However, the influence of the terms that coexist with each other can be part of the frequency of the terms of their semantic dependence, as they are non-integrative and their individual meaning cannot  ...  Moreover, the approximation factor with perturbation sets in the COLSK-IR method considers both the non-integrative and integrative snippets to reduce the convergent time on semantic data analysis by 45%  ... 
doi:10.32604/cmc.2021.012796 fatcat:zunc3g6a6fbg5by4qodciifuiq

Preface

A. Ridha Mahjoub, Giovanni Rinaldi, Gerhard Woeginger
2015 Mathematical programming  
In the past years, combinatorial optimization has undergone rapid developments, major advances being obtained in different areas such as computational complexity, approximation algorithms, cutting-plane  ...  edition of ISCO was the second of a series of biennial conferences on combinatorial optimization with its first venue held in Hammamet, Tunisia in March 2010.  ...  They give a 3/2-approximation algorithm for this class improving a 2-approximation algorithm given before by Goemans and Williamson.  ... 
doi:10.1007/s10107-015-0869-2 fatcat:epqoua7uqbcupmakq7kurfegbm

Computational identities for extensions of some families of special numbers and polynomials

2021 Turkish Journal of Mathematics  
Furthermore, we provide an approximation for the combinatorial-type 11 numbers with the aid of the Stirling's approximation for factorials.  ...  By the implementation of the p -adic integral approach 9 to the combinatorial-type polynomials with multi-variables, we also obtain formulas for the Volkenborn integral and the 10 fermionic p -adic integral  ...  the p -adic integrals approach to the combinatorial-type polynomials with 12 the Volkenborn and fermionic p -adic integrals, we obtain some formulas for the Volkenborn and fermionic p-13 adic integrals  ... 
doi:10.3906/mat-2101-83 fatcat:y7olaz2kw5es5fzvnzp44s6pxq

Page 14 of Annals of Mathematics Vol. 38, Issue 1 [page]

1937 Annals of Mathematics  
The combinatory invariants of complexes are properties in the large, and as such they have been extended to sets by means of approximation.  ...  The possibility of applying combinatory methods in general topology is based entirely on approximation of sets by complexes and cycles, and this also explains the character of the results, which express  ... 

On the relationship between algebra and analysis

Jon M. Beck
1980 Journal of Pure and Applied Algebra  
The functions and the integral operator are interpreted combinatorially. i=O linearly interpolated, provide a piecewise linear approximation to the solution, y.  ...  Note that y(x) is also an exact fixed point for the combinatorial integral operator T. us model the method of successive approximations in the system C(21'tk) (O~jsm).  ... 
doi:10.1016/0022-4049(80)90092-4 fatcat:ruatrtpr4feodj5t2dflt2f3qq

Front Matter, Table of Contents, Preface, Program Committees, External Reviewers, List of Authors

Klaus Jansen, Claire Mathieu, José D. P. Rolim, Chris Umans, Marc Herbstritt
2016 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Cohen, Cameron Musco, and Jakub Pachocki . . . . . . . . . . . . . . . . . . . . . . . . 7:1-7:18 Oblivious Rounding and the Integrality Gap Uriel Feige, Michal Feldman, and Inbal Talgam-Cohen . . . .  ...  into Constant-Dimensional p Spaces Artūrs Bačkurs and Anastasios Sidiropoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1:1-1:15 Computing Approximate PSD Factorizations  ...  Preface This volume contains the papers presented at the 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2016) and the 20th International Workshop  ... 
doi:10.4230/lipics.approx-random.2016.0 dblp:conf/approx/X16 fatcat:5aka2x4msbawlcwc55jleb3htm
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