Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-28T22:16:59.768Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  14 January 2025

Amitabh Basu
Amitabh Basu
Affiliation:
Johns Hopkins University
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Convexity and its Applications in Discrete and Continuous Optimization , pp. 291 - 304
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Fred G.Abramson, . Effective computation over the real numbers. In 12th Annual Symposium on Switching and Automata Theory (SWAT 1971), pages 33–37. IEEE Computer Society, 1971.Google Scholar
Aggarwal, Divesh, Dadush, Daniel, Regev, Oded, and Stephens-Davidowitz, Noah. Solving the shortest vector problem in 2n time using discrete Gaussian sampling. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC), pages 733–742. Association for Computing Machinery, 2015.CrossRefGoogle Scholar
Aggarwal, Divesh, Dadush, Daniel, and Stephens-Davidowitz, Noah. Solving the closest vector problem in 2n time – the discrete Gaussian strikes again! In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 563–582. IEEE, 2015.CrossRefGoogle Scholar
Aggarwal, Divesh and Stephens-Davidowitz, Noah. Just take the average! An embarrassingly simple 2n-time algorithm for SVP (and CVP). arXiv preprint arXiv:1709.01535, 2017.Google Scholar
Aho, Alfred V., Hopcroft, John E., and Ullman., Jeffrey D. The Design and Analysis of Computer Algorithms. Computer Science and Information Processing. Addison-Wesley Publishing Company, 1974.Google Scholar
Ajtai, Miklós. The shortest vector problem in L2 is NP-hard for randomized reductions. In STOC ’98: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 10–19. Association for Computing Machinery, 1998.CrossRefGoogle Scholar
Ajtai, Miklós, Kumar, Ravi, and Sivakumar, Dandapani. A sieve algorithm for the shortest lattice vector problem. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (STOC), pages 601–610. Association for Computing Machinery, 2001.CrossRefGoogle Scholar
Ajtai, Miklos, Kumar, Ravi, and Sivakumar, Dandapani. Sampling short lattice vectors and the closest lattice vector problem. In CCC ’02: Proceedings of the 17th IEEE Annual Conference on Computational Complexity. IEEE Computer Society, 2002.Google Scholar
Aliev, Iskander, Bassett, Robert, De Loera, Jesús A., and Louveaux, Quentin. A quantitative Doignon-Bell-Scarf theorem. Combinatorica, 37(3):313–332, 2017.CrossRefGoogle Scholar
Amenta, Nina, De Loera, Jesús A., and Soberón, Pablo. Helly’s theorem: new variations and applications. arXiv preprint arXiv:1508.07606, 2015.Google Scholar
Anstreicher, Kurt M.. On Vaidya’s volumetric cutting plane method for convex programming. Mathematics of Operations Research, 22(1):63–89, 1997.CrossRefGoogle Scholar
Arora, Sanjeev, Babai, László, Stern, Jacques, and Sweedyk, Z.. The hardness of approximate optima in lattices, codes, and systems of linear equations. Journal of Computer and System Sciences, 54(2):317–331, 1997.CrossRefGoogle Scholar
Arora, Sanjeev and Barak, Boaz. Computational Complexity: A Modern Approach. Cambridge University Press, 2009.CrossRefGoogle Scholar
Averkov, Gennadiy. On maximal S-free sets and the Helly number for the family of S-convex sets. SIAM Journal on Discrete Mathematics, 27(3):1610–1624, 2013.CrossRefGoogle Scholar
Averkov, Gennadiy. A proof of Lovász’s theorem on maximal lattice-free sets. Beiträge zur Algebra und Geometrie, 54(1):105–109, 2013.CrossRefGoogle Scholar
Averkov, Gennadiy, Merino, Bernardo González, Paschke, Ingo, Schymura, Matthias, and Weltge, Stefan. Tight bounds on discrete quantitative Helly numbers. Advances in Applied Mathematics, 89:76–101, 2017.CrossRefGoogle Scholar
Averkov, Gennadiy and Weismantel, Robert. Transversal numbers over subsets of linear spaces. Advances in Geometry, 12(1):19–28, 2012.CrossRefGoogle Scholar
Babai, László. On Lovász lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1–13, 1986.CrossRefGoogle Scholar
Baes, Michel, Oertel, Timm, and Weismantel, Robert. Duality for mixed-integer convex minimization. Mathematical Programming, 158(1):547–564, 2016.CrossRefGoogle Scholar
Ball, Keith. An elementary introduction to modern convex geometry. Flavors of Geometry, 31:1–58, 1997.Google Scholar
Banaszczyk, Wojciech. Inequalities for convex bodies and polar reciprocal lattices in Rn II: Application of K-convexity. Discrete & Computational Geometry, 16(3):305–311, 1996.CrossRefGoogle Scholar
Banaszczyk, Wojciech, Litvak, Alexander E., Pajor, Alain, and Szarek, Stanislaw J.. The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces. Mathematics of Operations Research, 24(3):728–750, 1999.CrossRefGoogle Scholar
Bárány, Imre. Combinatorial Convexity. American Mathematical Society, 2021.CrossRefGoogle Scholar
Barvinok, Alexander. A Course in Convexity. American Mathematical Society, 2002.CrossRefGoogle Scholar
Basu, Amitabh. Complexity of optimizing over the integers. Mathematical Programming, Series B, 200:739–780, 2023.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, Cornuéjols, Gérard, Weismantel, Robert, and Weltge, Stefan. Optimality certificates for convex minimization and Helly numbers. Operations Research Letters, 45(6):671–674, 2017.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, Cornuéjols, Gérard, and Zambelli, Giacomo. Maximal lattice-free convex sets in linear subspaces. Mathematics of Operations Research, 35:704–720, 2010.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, Cornuéjols, Gérard, and Zambelli, Giacomo. Minimal inequalities for an infinite relaxation of integer programs. SIAM Journal on Discrete Mathematics, 24:158–168, February 2010.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, and Summa, Marco Di. A geometric approach to cut-generating functions. Mathematical Programming, 151(1):153–189, 2015.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, Summa, Marco Di, and Jiang, Hongyi. Split cuts in the plane. SIAM Journal on Optimization, 31(1):331–347, 2021.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, Summa, Marco Di, and Jiang, Hongyi. Complexity of branch-and-bound and cutting planes for mixed-integer optimization– II. Combinatorica, 42:971–996, 2022.CrossRefGoogle Scholar
Basu, Amitabh, Conforti, Michele, Summa, Marco Di, and Jiang, Hongyi. Complexity of branch-and-bound and cutting planes for mixed-integer optimization. Mathematical Programming, 198(1):787–810, 2023.CrossRefGoogle Scholar
Basu, Amitabh, Cornuéjols, Gérard, and Zambelli, Giacomo. Convex sets and minimal sublinear functions. Journal of Convex Analysis, 18:427–432, 2011.Google Scholar
Basu, Amitabh, Hildebrand, Robert, and Köppe, Matthias. Light on the infinite group relaxation I: Foundations and taxonomy. 4OR, 14(1):1–40, 2016.CrossRefGoogle Scholar
Basu, Amitabh, Hildebrand, Robert, and Köppe, Matthias. Light on the infinite group relaxation II: Sufficient conditions for extremality, sequences, and algorithms. 4OR, 14(2):1–25, 2016.CrossRefGoogle Scholar
Basu, Amitabh, Jiang, Hongyi, Kerger, Phillip, and Molinaro, Marco. Information complexity of mixed-integer convex optimization. In International Conference on Integer Programming and Combinatorial Optimization, pages 1–13. Springer, 2023.Google Scholar
Basu, Amitabh, Jiang, Hongyi, Kerger, Phillip, and Molinaro, Marco. Information complexity of mixed-integer convex optimization. https://arxiv.org/abs/2308.11153, 2023.Google Scholar
Basu, Amitabh and Oertel, Timm. Centerpoints: A link between optimization and convex geometry. SIAM Journal on Optimization, 27(2):866–889, 2017.CrossRefGoogle Scholar
Beame, Paul, Fleming, Noah, Impagliazzo, Russell, Kolokolova, Antonina, Pankratov, Denis, Pitassi, Toniann, and Robere, Robert. Stabbing Planes. In Karlin, Anna R., editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1–10:20. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2018.Google Scholar
Beeson., Michael J. Foundations of Constructive Mathematics: Metamathematical Studies. Springer Science & Business Media, 2012.Google Scholar
Bell, David E.. A theorem concerning the integer lattice. Studies in Applied Mathematics, 56(2):187–188, 1977.CrossRefGoogle Scholar
Ben-Tal, Aharon and Nemirovski., Arkadii Lectures on Modern Convex Optimization. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), 2001.CrossRefGoogle Scholar
Bertoni, Alberto, Mauri, Giancarlo, and Sabadini, Nicoletta. Simulations among classes of random access machines and equivalence among numbers succinctly represented. In North-Holland Mathematics Studies, pages 65–89. Elsevier, 1985.Google Scholar
Bertsekas, Dimitri P.. Convex Optimization Theory. Athena Scientific, 2009.Google Scholar
Bertsekas, Dimitri P.. Convex Optimization Algorithms. Athena Scientific, 2015.Google Scholar
Bishop, Errett. Foundations of Constructive Analysis. McGraw-Hill, 1967.Google Scholar
Bixby, Robert E.. A brief history of linear and mixed-integer programming computation. Documenta Mathematica, 107–121, 2012.CrossRefGoogle Scholar
Bixby, Robert E., Fenelon, Mary, Gu, Zonghao, Rothberg, Ed, and Wunderling, Roland. Mixed integer programming: A progress report. In The Sharpest Cut, pages 309–325. MPS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, 2004.CrossRefGoogle Scholar
Bland, Robert G., Goldfarb, Donald, and Todd, Michael J.. The ellipsoid method: A survey. Operations Research, 29(6):1039–1091, 1981.CrossRefGoogle Scholar
Blaschke, Wilhelm. Über affine Geometrie vii: Neue Extremeigenschaften von Ellipse und Ellipsoid. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl, 69:306–318, 1917.Google Scholar
Blömer, Johannes. Closest vectors, successive minima, and dual HKZ-bases of lattices. In Proceedings of 27th ICALP, pages 248–259. Lecture Notes in Computer Science. Springer, 2002.Google Scholar
Blömer, Johannes and Naewe, Stefanie. Sampling methods for shortest vectors, closest vectors and successive minima. Theoretical Computer Science, 410(18):1648–1665, 2009.CrossRefGoogle Scholar
Blum, Lenore, Shub, Mike, and Smale, Steve. On a theory of computation and complexity over the real numbers: W-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society, 21(1):1–46, 1989.CrossRefGoogle Scholar
Bockmayr, Alexander, Eisenbrand, Friedrich, Hartmann, Mark, and Schulz, Andreas S.. On the Chvátal rank of polytopes in the 0/1 cube. Discrete Applied Mathematics, 98(1–2):21–27, 1999.CrossRefGoogle Scholar
Bonet, Maria, Pitassi, Toniann, and Raz, Ran. Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic, 62(3):708–728, 1997.CrossRefGoogle Scholar
Borodin, Allan and Munro, Ian. The Computational Complexity of Algebraic and Numeric Problems. Elsevier, 1975.Google Scholar
Borwein, Jonathan and Lewis, Adrian S.. Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer Science & Business Media, 2010.Google Scholar
Bourgain, Jean and Milman, Vitali D.. Sections Euclidiennes et volume des corps symétriques convexes dans . CR Acad. Sci. Paris, 300:435–437, 1985.Google Scholar
Boyd, Stephen and Vandenberghe, Lieven. Convex Optimization. Cambridge University Press, 2004.CrossRefGoogle Scholar
Braun, Gábor, Guzmán, Cristóbal, and Pokutta, Sebastian. Lower bounds on the oracle complexity of nonsmooth convex optimization via information theory. IEEE Transactions on Information Theory, 63(7):4709–4724, 2017.CrossRefGoogle Scholar
Bremner, David, Chen, Dan, Iacono, John, Langerman, Stefan, and Morin, Pat. Output-sensitive algorithms for Tukey depth and related problems. Statistics and Computing, 18(3):259–266, 2008.CrossRefGoogle Scholar
Busemann, Herbert. Volume in terms of concurrent cross-sections. Pacific Journal of Mathematics, 3:1–12, 1953.CrossRefGoogle Scholar
Buss, Samuel R. and Clote, Peter. Cutting planes, connectivity, and threshold logic. Archive for Mathematical Logic, 35(1):33–62, 1996.CrossRefGoogle Scholar
Caplin, Andrew and Nalebuff, Barry. On 64%-majority rule. Econometrica: Journal of the Econometric Society, 787–814, 1988.CrossRefGoogle Scholar
Carathéodory, Constantin. Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Mathematische Annalen, 64(1):95–115, 1907.CrossRefGoogle Scholar
Carmon, Yair. The Complexity of Optimization Beyond Convexity. Ph.D. thesis, Stanford University, August 2020.Google Scholar
Cassels, John William Scott. An Introduction to the Geometry of Numbers. Springer Science & Business Media, 1997.Google Scholar
Chan, Timothy M. An optimal randomized algorithm for maximum Tukey depth. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 430–436. Society for Industrial and Applied Mathematics, 2004.Google Scholar
Chvátal, Vašek. Hard knapsack problems. Operations Research, 28(6):1402–1411, 1980.CrossRefGoogle Scholar
Chvátal, Vašek. Cutting-Plane Proofs and the Stability Number of a Graph, Report Number 84326-OR. Institut für Ökonometrie und Operations Research, Universitat Bonn, Bonn, 1984.Google Scholar
Chvátal, Vašek, Cook, William J., and Hartmann, Mark. On cutting-plane proofs in combinatorial optimization. Linear Algebra and Its Applications, 114:455–499, 1989.CrossRefGoogle Scholar
Clote, Peter. Cutting planes and constant depth Frege proofs. In Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, pages 296–307. IEEE, 1992.Google Scholar
Conforti, Michele, Cornuéjols, Gérard, Daniilidis, Aris, Lemaréchal, Claude, and Malick, Jérôme. Cut-generating functions and S-free sets. Mathematics of Operations Research, 40(2): 276–301, 2015.CrossRefGoogle Scholar
Conforti, Michele, Cornuéjols, Gérard, and Zambelli, Giacomo. Integer Programming, volume 271. Berlin, Springer, 2014.CrossRefGoogle Scholar
Conforti, Michele and Summa, Marco Di. Maximal S-free convex sets and the Helly number. SIAM Journal on Discrete Mathematics, 30(4):2206–2216, 2016.CrossRefGoogle Scholar
Cook, William J., Coullard, Collette R., and Turán, Gy. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18(1):25–38, 1987.CrossRefGoogle Scholar
Cook, William J. and Dash, Sanjeeb. On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26(1):19–30, 2001.CrossRefGoogle Scholar
Cook, William J. and Hartmann, Mark. On the complexity of branch and cut methods for the traveling salesman problem. Polyhedral Combinatorics, 1:75–82, 1990.Google Scholar
Cormen, Thomas H., Leiserson, Charles E., Rivest, Ronald L., and Stein, Clifford. Introduction to Algorithms. MIT Press, 2022.Google Scholar
Dadush, Daniel. Integer Programming, Lattice Algorithms, and Deterministic Volume Estimation. Ph.D. thesis, Georgia Institute of Technology, August 2012.Google Scholar
Dadush, Daniel and Tiwari, Samarth. On the complexity of branching proofs. In Proceedings of the 35th Computational Complexity Conference. Association for Computing Machinery, 2020.Google Scholar
Danzer, Ludwig, Grünbaum, Branko, and Klee, Victor. Helly’s theorem and its relatives. In Proceedings of Symposia in Pure Mathematics VII, pages 101–180. American Mathematical Society, 1963.Google Scholar
Dash, Sanjeeb. An exponential lower bound on the length of some classes of branch-and-cut proofs. In International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 145–160. Springer, 2002.Google Scholar
Dash, Sanjeeb. Exponential lower bounds on the lengths of some classes of branch-and-cut proofs. Mathematics of Operations Research, 30(3):678–700, 2005.CrossRefGoogle Scholar
Dash, Sanjeeb. On the complexity of cutting-plane proofs using split cuts. Operations Research Letters, 38(2):109–114, 2010.CrossRefGoogle Scholar
Dash, Sanjeeb, Dobbs, Neil B., Günlük, Oktay, Nowicki, Tomasz J., and Swirszcz, Grzegorz M.. Lattice-free sets, multi-branch split disjunctions, and mixedinteger programming. Mathematical Programming, 145(1–2):483–508, 2014.CrossRefGoogle Scholar
Dash, Sanjeeb and Dubey, Yatharth. On polytopes with linear rank with respect to generalizations of the split closure. arXiv preprint arXiv:2110.04344, 2021.Google Scholar
Dash, Sanjeeb and Günlük, Oktay. On t-branch split cuts for mixed-integer programs. Mathematical Programming, 141(1-2):591–599, 2013.CrossRefGoogle Scholar
d’Aspremont, Alexandre. Smooth optimization with approximate gradient. SIAM Journal on Optimization, 19(3):1171–1183, 2008.CrossRefGoogle Scholar
Berg, Mark De, Kreveld, Marc van, Overmars, Mark, and Schwarzkopf, Otfried. Computational Geometry: Algorithms and Applications. Springer Science & Business Media, 2000.CrossRefGoogle Scholar
De Loera, Jesús A., Goaoc, Xavier, Meunier, Frederic, and Mustafa, Nabil. The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg. Bulletin of the American Mathematical Society, 56(3):415–511, 2019.CrossRefGoogle Scholar
De Loera, Jesus A. and Hogan, Thomas. Stochastic Tverberg theorems with applications in multiclass logistic regression, separability, and centerpoints of data. SIAM Journal on Mathematics of Data Science, 2(4):1151–1166, 2020.CrossRefGoogle Scholar
De Loera, Jesús A., La Haye, Reuben N., Oliveros, Déborah, and Roldán-Pensado, Edgardo. Helly numbers of algebraic subsets of Rd and an extension of Doignon’s theorem. Advances in Geometry, 17(4):473–482, 2017.CrossRefGoogle Scholar
De Loera, Jesus A., La Haye, Reuben N., Rolnick, David, and Soberón, Pablo. Quantitative Tverberg theorems over lattices and other discrete sets. Discrete & Computational Geometry, 58:435–448, 2017.CrossRefGoogle Scholar
De Loera, Jesús A., Petrović, Sonja, and Stasi, Despina. Random sampling in computational algebra: Helly numbers and violator spaces. Journal of Symbolic Computation, 77:1–15, 2016.CrossRefGoogle Scholar
Devadoss, Satyan L. and O’Rourke, Joseph. Discrete and Computational Geometry. Princeton University Press, 2011.Google Scholar
Devolder, Olivier, Glineur, François, and Nesterov, Yurii. First-order methods of smooth convex optimization with inexact oracle. Mathematical Programming, 146:37–75, 2014.CrossRefGoogle Scholar
Devroye, Luc, Györfi, László, and Lugosi, Gábor. A Probabilistic Theory of Pattern Recognition. Springer Science & Business Media, 2013.Google Scholar
Dey, Santanu S., Dubey, Yatharth, and Molinaro, Marco. Branch-and-bound solves random binary packing IPs in polytime. Mathematical Programming, 200:569–587, 2023.Google Scholar
Dey, Santanu S, Dubey, Yatharth, and Molinaro, Marco. Lower bounds on the size of general branch-and-bound trees. Mathematical Programming, 198(1):539–559, 2023.CrossRefGoogle Scholar
Dey, Santanu S., Dubey, Yatharth, Molinaro, Marco, and Shah, Prachi. A theoretical and computational analysis of full strong-branching. Mathematical Programming, 1–34, 2023.CrossRefGoogle Scholar
Dey, Santanu S. and Shah, Prachi. Lower bound on size of branch-and-bound trees for solving lot-sizing problem. Operations Research Letters, 50(5):430–433, 2022.CrossRefGoogle Scholar
Dey, Santanu S. and Wolsey, Laurence A.. Constrained infinite group relaxations of MIPs. SIAM Journal on Optimization, 20(6):2890–2912, 2010.CrossRefGoogle Scholar
Summa, Marco Di, 2019. Personal communication.Google Scholar
Dinur, Irit, Kindler, Guy, Raz, Ran, and Safra, Shmuel. Approximating CVP to within almost-polynomial factors is NP-Hard. Combinatorica, 23:205–243, April 2003.CrossRefGoogle Scholar
Doignon, J.-P.. Convexity in cristallographical lattices. Journal of Geometry, 3:71–85, 1973.CrossRefGoogle Scholar
Donoho, David L. and Gasko, Miriam. Breakdown properties of location estimates based on halfspace depth and projected outlyingness. The Annals of Statistics, 1803–1827, 1992.CrossRefGoogle Scholar
Downey, Rodney G. and Fellows, Michael R.. Parameterized Complexity. Springer Science & Business Media, 2012.Google Scholar
Downey, Rodney G. and Fellows, Michael R.. Fundamentals of Parameterized Complexity. Springer, 2013.CrossRefGoogle Scholar
Dyckerhoff, R. and Mozharovskyi, Pavlo. Exact computation of halfspace depth. arXiv preprint arXiv:1411.6927v2, 2015.Google Scholar
Eisenbrand, Friedrich. Integer programming and algorithmic geometry of numbers. In Jünger, M., Liebling, T., Naddef, D., Pulleyblank, W., Reinelt, G., Rinaldi, G., and Wolsey, L., editors, 50 Years of Integer Programming 1958–2008. Springer-Verlag, 2010.Google Scholar
Eisenbrand, Friedrich and Schulz, Andreas S.. Bounds on the Chvatal rank of polytopes in the 0/1-cube. Combinatorica, 23(2):245–261, 2003.CrossRefGoogle Scholar
Eisenbrand, Friedrich and Venzin, Moritz. Approximate CVPp in time 20.802n. Journal of Computer and System Sciences, 124:129–139, 2022.CrossRefGoogle Scholar
Ekeland, Ivar and Temam, Roger. Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics, 1999.CrossRefGoogle Scholar
Erickson, Jeff, Hoog, Ivor Van Der, and Miltzow, Tillmann. Smoothing the gap between NP and ER. SIAM Journal on Computing, 2022.CrossRefGoogle Scholar
Ewald, Günter. Combinatorial Convexity and Algebraic Geometry. Springer Science & Business Media, 2012.Google Scholar
Fleming, Noah, Göös, Mika, Impagliazzo, Russell, Pitassi, Toniann, Robere, Robert, Tan, Li-Yang, and Wigderson, Avi. On the power and limitations of branch and cut. arXiv preprint arXiv:2102.05019, 2021.Google Scholar
Friedman, Harvey. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. In Logic Colloquium ’69, pages 361–389. Elsevier, 1971.CrossRefGoogle Scholar
Furstenberg, Harry and Tzkoni, Isaac. Spherical functions and integral geometry. Israel Journal of Mathematics, 10:327–338, 1971.CrossRefGoogle Scholar
Goerdt, Andreas. Cutting plane versus Frege proof systems. In International Workshop on Computer Science Logic, pages 174–194. Springer, 1990.Google Scholar
Goerdt, Andreas. The cutting plane proof system with bounded degree of falsity. In International Workshop on Computer Science Logic, pages 119–133. Springer, 1991.Google Scholar
Grigoriev, Dima, Hirsch, Edward A., and V, Dmitrii. Pasechnik. Complexity of semi-algebraic proofs. In Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 419–430. Springer, 2002.Google Scholar
Grötschel, Martin, Lovász, László, and Schrijver, Alexander. Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics: Study and Research Texts. Springer-Verlag, 1988.CrossRefGoogle Scholar
Gruber, Peter M.. Convex and Discrete Geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, 2007.Google Scholar
Gruber, Peter M. and Lekkerkerker, Cornelis G.. Geometry of Numbers, Second edition, North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1987.Google Scholar
Grünbaum, Branko. Partitions of mass-distributions and of convex bodies by hyperplanes. Pacific Journal of Mathematics, 10:1257–1261, 1960.CrossRefGoogle Scholar
Grünbaum, Branko. Measures of symmetry for convex sets. In Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, page 233. American Mathematical Society, 1963.CrossRefGoogle Scholar
Grünbaum, Branko. Convex Polytopes. Springer Science & Business Media, 2013.Google Scholar
Grzegorczyk, Andrzej. Computable functionals. Fundamenta Mathematicae, 42:168–202, 1955.CrossRefGoogle Scholar
Guggenheimer, Heinrich W.. A formula of Furstenberg-Tzkoni type. Israel Journal of Mathematics, 14(3):281–282, 1973.CrossRefGoogle Scholar
Güler, Osman. Foundations of Optimization. Springer Science & Business Media, 2010.CrossRefGoogle Scholar
Halmos, Paul R.. Finite-Dimensional Vector Spaces. Courier Dover Publications, 2017.Google Scholar
Hartmanis, Juris and Simon, Janos. On the power of multiplication in random access machines. In 15th Annual Symposium on Switching and Automata Theory (swat 1974), pages 13–23. IEEE, 1974.CrossRefGoogle Scholar
Heinz, Sebastian. Complexity of integer quasiconvex polynomial optimization. Journal of Complexity, 21(4):543–556, 2005.CrossRefGoogle Scholar
Helly, Eduard. Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175–176, 1923.Google Scholar
Helly, Eduard. Uber Systeme abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatshefte für Mathematik, 37:281–302, 1930.CrossRefGoogle Scholar
Henk, Martin. Löwner-John ellipsoids. Documenta Mathematica, Extra volume ISMP:95–106, 2012.CrossRefGoogle Scholar
Hildebrand, Robert and Köppe, Matthias. A new Lenstra-type algorithm for quasiconvex polynomial integer minimization with complexity 2O(n log n). Discrete Optimization, 10(1):69–84, 2013.CrossRefGoogle Scholar
Hintermüller, Michael. A proximal bundle method based on approximate subgradients. Computational Optimization and Applications, 20:245–266, 2001.CrossRefGoogle Scholar
Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. Convex Analysis and Minimization Algorithms I, Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1993.Google Scholar
Hiriart-Urruty, Jean-Baptiste and Lemaréchal, Claude. Convex Analysis and Minimization Algorithms II, Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, 1993.Google Scholar
Hoffman, Alan J.. Binding constraints and Helly numbers. Annals of the New York Academy of Sciences, 319:284–288, 1979.CrossRefGoogle Scholar
Holmes, Richard B.. Geometric Functional Analysis and Its Applications. Springer-Verlag, 1975.CrossRefGoogle Scholar
Howell, Rodney R.. On asymptotic notation with multiple variables. Tech. Rep., 2008.Google Scholar
Hungerford, Thomas W.. Algebra. Springer Science & Business Media, 2012.Google Scholar
Impagliazzo, Russell, Pitassi, Toniann, and Urquhart, Alasdair. Upper and lower bounds for tree-like cutting planes proofs. In Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science, pages 220–228. IEEE, 1994.CrossRefGoogle Scholar
Jeroslow, Robert G.. Trivial integer programs unsolvable by branch-and-bound. Mathematical Programming, 6(1):105–109, December 1974.CrossRefGoogle Scholar
Jiang, Haotian, Lee, Yin Tat, Song, Zhao, and Wong, Sam Chiu-wai. An improved cutting plane method for convex optimization, convex-concave games, and its applications. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 944–953. Association for Computing Machinery, 2020.CrossRefGoogle Scholar
John, Fritz. Extremum problems with inequalities as subsidiary conditions. Studies and Essays, Courant Anniversary Volume:187–204, 1948.Google Scholar
Kannan, Ravindran. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):415–440, 1987.CrossRefGoogle Scholar
Kannan, Ravindran and Lovász, László. Covering minima and lattice-point-free convex bodies. Annals of Mathematics, 577–602, 1988.CrossRefGoogle Scholar
Khachiyan, Leonid and Porkolab, Lorant. Integer optimization on convex semial-gebraic sets. Discrete & Computational Geometry, 23(2):207–224, 2000.CrossRefGoogle Scholar
Kirkpatrick, David and Reisch, Stefan. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28(3):263–276, 1983.CrossRefGoogle Scholar
Kiwiel, Krzysztof C.. A proximal bundle method with approximate subgradient linearizations. SIAM Journal on Optimization, 16(4):1007–1023, 2006.CrossRefGoogle Scholar
Ko, Ker-I. Complexity Theory of Real Functions. Birkhauser Boston Inc., 1991.CrossRefGoogle Scholar
Koshevoy, Gleb A.. The Tukey depth characterizes the atomic measure. Journal of Multivariate Analysis, 83(2):360–364, 2002.CrossRefGoogle Scholar
Krajíček, Jan. Discretely ordered modules as a first-order extension of the cutting planes proof system. The Journal of Symbolic Logic, 63(4):1582–1596, 1998.CrossRefGoogle Scholar
Lacombe, Daniel. Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. Comptes Rendus Académie des Sciences Paris, 241:151–153, 1955.Google Scholar
Lagarias, Jeffrey C., Jr.Lenstra, Hendrik W., and Schnorr, Claus-Peter. Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, 10(4):333–348, 1990.CrossRefGoogle Scholar
Lan, Guanghui. Convex Optimization Under Inexact First-Order Information. Georgia Institute of Technology, 2009.Google Scholar
Land, Ailsa H. and G, Alison. Doig. An automatic method of solving discrete programming problems. Econometrica, 28(3):497–520, 1960.CrossRefGoogle Scholar
Lau, Lap Chi. Convexity and optimization. https://cs.uwaterloo.ca/~lapchi/cs798/notes.html, 2017.Google Scholar
Lee, Yin Tat, Sidford, Aaron, and Wong, Sam Chiu-wai. A faster cutting plane method and its implications for combinatorial and convex optimization. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1049–1065. IEEE, 2015.CrossRefGoogle Scholar
Lenstra, Arjen K., Jr.Lenstra, Hendrik W., and Lovász, László. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, 1982.CrossRefGoogle Scholar
JrLenstra, Hendrik W.. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538–548, 1983.CrossRefGoogle Scholar
Li, Yanjun and Richard, Jean-Philippe P.. Cook, Kannan and Schrijver’s example revisited. Discrete Optimization, 5(4):724–734, 2008.CrossRefGoogle Scholar
Lodi, Andrea. Mixed integer programming computation. In 50 Years of Integer Programming 1958–2008, pages 619–645. Springer, 2010.CrossRefGoogle Scholar
Lovász, László. An Algorithmic Theory of Numbers, Graphs, and Convexity, volume 50. Society for Industrial and Applied Mathematics, 1986.CrossRefGoogle Scholar
Lovász, László. Geometry of numbers and integer programming. In Iri, M. and Tanabe, K., editors, Mathematical Programming: State of the Art, pages 177–201. Mathematical Programming Society, 1989.Google Scholar
Luenberger, David G.. Optimization by Vector Space Methods. Wiley-Interscience, 1996.Google Scholar
Lutwak, Erwin. On some ellipsoid formulas of Busemann, Furstenberg and Tzkoni, Guggenheimer, and Petty. Journal of Mathematical Analysis and Applications, 159(1):18–26, 1991.CrossRefGoogle Scholar
Marsden, Annie, Sharan, Vatsal, Sidford, Aaron, and Valiant, Gregory. Efficient convex optimization requires superlinear memory. In Conference on Learning Theory, pages 2390–2430. Proceedings of Machine Learning Research., 2022.Google Scholar
Matousek, Jiri. Lectures on Discrete Geometry. Springer Science & Business Media, 2002.CrossRefGoogle Scholar
Micciancio, Daniele and Voulgaris, Panagiotis. A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. In Proceedings of the 42nd ACM Symposium on Theory of Computing, pages 351–358. Association for Computing Machinery, 2010.Google Scholar
Micciancio, Daniele and Voulgaris, Panagiotis. A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM Journal on Computing, 42(3):1364–1391, 2013.CrossRefGoogle Scholar
Diego, A. Morán, R. and Dey, Santanu S.. On maximal S-free convex sets. SIAM Journal on Discrete Mathematics, 25(1):379, 2011.Google Scholar
Diego, A. Morán, R., Dey, Santanu S., and Pablo Vielma, Juan. A strong dual for conic mixed-integer programs. SIAM Journal on Optimization, 22(3):1136–1150, 2012.Google Scholar
Mosler, Karl. Depth statistics. In Robustness and Complex Data Structures, pages 17–34. Springer, 2013.CrossRefGoogle Scholar
Naderi, Mohammad Javad, Buchanan, Austin, and L, Jose. Walteros. Worst-case analysis of clique MIPs. Mathematical Programming, 195(1):517–551, 2022.CrossRefGoogle Scholar
Nemhauser, George L. and Wolsey, Laurence A.. Integer and Combinatorial Optimization. Wiley, 1988.CrossRefGoogle Scholar
Nemirovski, Arkadii. Efficient methods in convex programming. Lecture Notes. www2.isye.gatech.edu/~nemirovs/LecEEMCO.pdf, 1994.Google Scholar
Nemirovski, Arkadii S. and Yudin, David B.. Problem Complexity and Method Efficiency in Optimization. John Wiley, 1983.Google Scholar
Nesterov, Yuri. Rounding of convex sets and efficient gradient methods for linear programming problems. Optimization Methods Software, 23(1):109–128, 2008.CrossRefGoogle Scholar
Nesterov, Yurii E.. Introductory Lectures on Convex Optimization, Applied Optimization. Kluwer Academic Publishers, 2004.CrossRefGoogle Scholar
Nocedal, Jorge and Wright, Stephen. Numerical Optimization. Springer Science & Business Media, 2006.Google Scholar
Oertel, Timm. Integer Convex Minimization in Low Dimensions. Ph.D. thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 22288, 2014.Google Scholar
Pisier, Gilles. The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, 1999.Google Scholar
Polyak, Boris T.. Introduction to Optimization. Translations Series in Mathematics and Engineering. Optimization Software Inc. Publications Division, 1987.Google Scholar
Pour-El, Marian Boykan and Richards, Ian. Computability and noncomputability in classical analysis. Transactions of the American Mathematical Society, 275(2):539–560, 1983.CrossRefGoogle Scholar
Pratt, Vaughan R., Rabin, Michael O., and Stockmeyer, Larry J.. A characterization of the power of vector machines. In Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pages 122–134. Association for Computing Machinery, 1974.Google Scholar
Preparata, Franco P. and Shamos., Michael I. Computational Geometry: An Introduction. Springer Science & Business Media, 2012.Google Scholar
Pudlák., Pavel Lower bounds for resolution and cutting plane proofs and monotone computations. The Journal of Symbolic Logic, 62(3):981–998, 1997.CrossRefGoogle Scholar
Pudlák, Pavel. On the complexity of the propositional calculus. In On the Complexity of the Propositional Calculus, pages 197–218, London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.Google Scholar
Radon, Johann. Uber eine Erweiterung des Begriffs der konvexen Functionen mit einer Anwendung auf die Theorei der Konvexen korper. Sitzungsberichte der Wiener Akademie, 125:241–258, 1916.Google Scholar
Radon, Johann. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Mathematische Annalen, 83(1–2):113–115, 1921.CrossRefGoogle Scholar
Razborov, Alexander A.. On the width of semialgebraic proofs and algorithms. Mathematics of Operations Research, 42(4):1106–1134, 2017.CrossRefGoogle Scholar
Reis, Victor and Rothvoss, Thomas. The subspace flatness conjecture and faster integer programming. arXiv preprint arXiv:2303.14605, 2023.Google Scholar
Richard, Jean-Philippe P. and Dey, Santanu S.. The group-theoretic approach in mixed integer programming. In Jünger, Michael, Liebling, Thomas M., Naddef, Denis, Nemhauser, George L., Pulleyblank,, William R. Reinelt, Gerhard, Rinaldi, Giovanni, and Wolsey, Laurence A., editors, 50 Years of Integer Programming 1958–2008, pages 727–801. Springer, 2010.Google Scholar
Rockafellar, R. Tyrell. Convex Analysis. Princeton Mathematical Series. Princeton University Press, 1970.CrossRefGoogle Scholar
Rothvoss, Thomas. Asymptotic convex geometry. https://sites.math.washington.edu/~rothvoss7lecturenotes7AsymptoticConvexGeometry.pdf, 2021.Google Scholar
Rothvoß, Thomas and Sanitá, Laura. 0/1 polytopes with quadratic Chvátal rank. In International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 349–361. Springer, 2013.Google Scholar
Rothvoss, Thomas and Venzin, Moritz. Approximate CVP in time 20.802n – now in any norm! In International Conference on Integer Programming and Combinatorial Optimization, pages 440–453. Springer, 2022.Google Scholar
Rousseeuw, Peter J. and Ruts, Ida. The depth function of a population distribution. Metrika, 49(3):213–244, 1999.CrossRefGoogle Scholar
Royden, Halsey L. and Fitzpatrick, Patrick M.. Real Analysis, Fourth edition. Prentice Hall, 2010.Google Scholar
Rudelson, Mark. Distances between non-symmetric convex bodies and the MM*-estimate. Positivity, 4(2):161–178, 2000.CrossRefGoogle Scholar
Rudin, Walter. Principles of Mathematical Analysis, Third edition. McGraw-Hill, 1976.Google Scholar
Santaló, Luis A.. Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugaliae Mathematica, 8:155–161, 1949.Google Scholar
Scarf, Herbert E.. An observation on the structure of production sets with indivisibilities. Proceedings of the National Academy of Sciences, 74(9):3637–3641, 1977.CrossRefGoogle ScholarPubMed
Schmidt, Mark, Roux, Nicolas, and Bach, Francis. Convergence rates of inexact proximal-gradient methods for convex optimization. Advances in Neural Information Processing Systems, 24, 2011.Google Scholar
Schneider, Rolf. Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 1993.CrossRefGoogle Scholar
Schönhage, Arnold. On the power of random access machines. In International Colloquium on Automata, Languages, and Programming, pages 520–529. Springer, 1979.CrossRefGoogle Scholar
Schrijver, Alexander. Theory of Linear and Integer Programming. John Wiley & Sons, 1986.Google Scholar
Shor, Naum Z.. Convergence rate of the gradient descent method with dilatation of the space. Cybernetics, 6(2):102–108, 1970.CrossRefGoogle Scholar
Shor, Naum Z.. Utilization of the operation of space dilatation in the minimization of convex functions. Cybernetics, 6(1):7–15, 1972.CrossRefGoogle Scholar
Shor, Naum Z.. Cut-off method with space extension in convex programming problems. Cybernetics, 13(1):94–96, 1977.CrossRefGoogle Scholar
Shor, Naum Z.. Minimization Methods for Non-Differentiable Functions. Springer Series in Computational Mathematics. Springer, 1985.CrossRefGoogle Scholar
Siegel, Carl Ludwig. Lectures on the Geometry of Numbers. Springer Science & Business Media, 1989.CrossRefGoogle Scholar
Stoer, Josef and Witzgall, Christoph. Convexity and Optimization in Finite Dimensions I. Springer Science & Business Media, 1970.CrossRefGoogle Scholar
Strang, Gilbert. Linear Algebra and Its Applications. Thomson, Brooks/Cole, 2006.Google Scholar
Tiel, Jan van. Convex Analysis. John Wiley, 1984.Google Scholar
Toth, Gabor. Measures of Symmetry for Convex Sets and Stability. Springer, 2015.CrossRefGoogle Scholar
Traub, Joseph Frederick, Wasilkowski, Grzegorz Wlodzimierz, and Woźniakowski, Henryk. Information, Uncertainty, Complexity. Addison-Wesley, 1983.Google Scholar
Traub, Joseph Frederick, Wasilkowski, Grzegorz Wlodzimierz, and Woźniakowski, Henryk. Information-Based Complexity. Academic Press, 1988.Google Scholar
Turing, Alan Mathison. On computable numbers, with an application to the “Entscheidungsproblem.” Proceedings of the London Mathematical Society, 42(2):230–265, 1936.Google Scholar
Turing, Alan Mathison. On computable numbers, with an application to the “Entscheidungsproblem.” a correction. Proceedings of the London Mathematical Society, 43(2):544–546, 1937.Google Scholar
Vaidya, Pravin M.. A new algorithm for minimizing convex functions over convex sets. Mathematical Programming, 73(3):291–341, 1996.CrossRefGoogle Scholar
Boas, Peter van Emde. Another NP-complete problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Instituut, Mathematisch, University of Amsterdam, 1981.Google Scholar
Vapnik, Vladimir N. and Y, Alexey. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications, 16(2):264–280, 1971.CrossRefGoogle Scholar
Villani, Cédric. Optimal Transport: Old and New. Springer, 2009.CrossRefGoogle Scholar
Weihrauch, Klaus. Computable Analysis: An Introduction. Springer Science & Business Media, 2000.CrossRefGoogle Scholar
Wright, Stephen J.. Coordinate descent algorithms. Mathematical Programming, 151(1):3–34, 2015.CrossRefGoogle Scholar
Yudin, David B. and S, Arkadii. Nemirovskii. Evaluation of the informational complexity of mathematical programming problems. Matekon, 13(2):3–25, 1976.Google Scholar
Yudin, David B. and Nemirovskii, Arkadii S.. Informational complexity and efficient methods for the solution of convex extremal problems. Matekon, 13(3):25–45, 1977.Google Scholar
Ziegler, Günter M.. Lectures on Polytopes. Springer, 1995.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Amitabh Basu, Johns Hopkins University
  • Book: Convexity and its Applications in Discrete and Continuous Optimization
  • Online publication: 14 January 2025
  • Chapter DOI: https://doi.org/10.1017/9781108946650.012
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Amitabh Basu, Johns Hopkins University
  • Book: Convexity and its Applications in Discrete and Continuous Optimization
  • Online publication: 14 January 2025
  • Chapter DOI: https://doi.org/10.1017/9781108946650.012
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Amitabh Basu, Johns Hopkins University
  • Book: Convexity and its Applications in Discrete and Continuous Optimization
  • Online publication: 14 January 2025
  • Chapter DOI: https://doi.org/10.1017/9781108946650.012
Available formats
×