Linear optimization on varieties and Chern-Mather classes. (English) Zbl 1539.14009
The article in question deals with the problem of linear optimization on algebraic varieties and its connection to Chern-Mather classes. It introduces the concept of linear optimization degree, which serves as an algebraic measure of the complexity of optimizing a linear objective function over an algebraic model. The authors demonstrate that the geometry of the conormal variety, expressed in terms of bidegrees, completely determines the Chern-Mather classes of the given variety. Furthermore, they establish that these bidegrees are equivalent to the linear optimization degrees of generic affine sections. The paper culminates in the proof of a theorem that shows the equality between the linear optimization bidegrees and the sectional linear optimization degrees.
The paper’s theoretical significance lies in bridging optimization theory with algebraic geometry, offering a novel linear optimization perspective akin to the maximum likelihood degree. This could influence studies in algebraic statistics and computational geometry, providing a deeper comprehension of the algebraic underpinnings of optimization.
The article is commendable for its rigorous mathematical treatment and its ability to elucidate complex ideas. The potential applicability of the results in applied mathematics and statistics underscores the paper’s relevance. In sum, the article is a noteworthy contribution to the intersection of algebraic geometry and optimization theory.
The paper’s theoretical significance lies in bridging optimization theory with algebraic geometry, offering a novel linear optimization perspective akin to the maximum likelihood degree. This could influence studies in algebraic statistics and computational geometry, providing a deeper comprehension of the algebraic underpinnings of optimization.
The article is commendable for its rigorous mathematical treatment and its ability to elucidate complex ideas. The potential applicability of the results in applied mathematics and statistics underscores the paper’s relevance. In sum, the article is a noteworthy contribution to the intersection of algebraic geometry and optimization theory.
Reviewer: Jianqiang Yang (Mengzi)
MSC:
| 14B05 | Singularities in algebraic geometry |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 57R20 | Characteristic classes and numbers in differential topology |
| 90C26 | Nonconvex programming, global optimization |
Keywords:
linear optimization degree; local Euler obstruction; Chern-Mather classes; conormal varieties; Segre classes; polar degreesReferences:
| [1] | Aluffi, P., Euler characteristics of general linear sections and polynomial Chern classes. Rend. Circ. Mat. Palermo (2), 1, 3-26 (2013) · Zbl 1312.14018 |
| [2] | Aluffi, P., Projective duality and a Chern-Mather involution. Trans. Am. Math. Soc., 3, 1803-1822 (2018) · Zbl 1422.14052 |
| [3] | Aluffi, P., Segre classes and invariants of singular varieties, 419-492 · Zbl 1524.14015 |
| [4] | Aluffi, P.; Mihalcea, L.; Schürmann, J.; Su, C., Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-Macpherson classes of Schubert cells. Duke Math. J. (2023), in press |
| [5] | Catanese, F.; Hoşten, S.; Khetan, A.; Sturmfels, B., The maximum likelihood degree. Am. J. Math., 3, 671-697 (2006) · Zbl 1123.13019 |
| [6] | Çelik, T. O.; Jamneshan, A.; Montúfar, A. G.; Sturmfels, B.; Venturello, L., Wasserstein distance to independence models. J. Symb. Comput., 855-873 (2021) · Zbl 1460.62205 |
| [7] | Clarke, P.; Cox, D. A., Moment maps, strict linear precision, and maximum likelihood degree one. Adv. Math. (2020) · Zbl 1453.14125 |
| [8] | Di Rocco, S.; Eklund, D.; Weinstein, M., The bottleneck degree of algebraic varieties. SIAM J. Appl. Algebra Geom., 1, 227-253 (2020) · Zbl 1505.14122 |
| [9] | Draisma, J.; Horobeţ, E.; Ottaviani, G.; Sturmfels, B.; Thomas, R. R., The Euclidean distance degree of an algebraic variety. Found. Comput. Math., 1, 99-149 (2016) · Zbl 1370.51020 |
| [10] | Fulton, W., Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0885.14002 |
| [11] | Ginsburg, V., Characteristic varieties and vanishing cycles. Invent. Math., 2, 327-402 (1986) · Zbl 0598.32013 |
| [12] | Graf von Bothmer, H.-C.; Ranestad, K., A general formula for the algebraic degree in semidefinite programming. Bull. Lond. Math. Soc., 2, 193-197 (2009) · Zbl 1185.14047 |
| [13] | Gross, E.; Drton, M.; Petrović, S., Maximum likelihood degree of variance component models. Electron. J. Stat., 993-1016 (2012) · Zbl 1281.62159 |
| [14] | Hartley, R. I.; Sturm, P., Triangulation. Comput. Vis. Image Underst., 2, 146-157 (1997) |
| [15] | Hoşten, S.; Khetan, A.; Sturmfels, B., Solving the likelihood equations. Found. Comput. Math., 4, 389-407 (2005) · Zbl 1097.13035 |
| [16] | Huh, J., The maximum likelihood degree of a very affine variety. Compos. Math., 8, 1245-1266 (2013) · Zbl 1282.14007 |
| [17] | Huh, J.; Sturmfels, B., Likelihood geometry, 63-117 · Zbl 1328.14004 |
| [18] | Kubjas, K.; Kuznetsova, O.; Sodomaco, L., Algebraic degree of optimization over a variety with an application to \(p\)-norm distance degree (2021), arXiv preprint |
| [19] | MacPherson, R. D., Chern classes for singular algebraic varieties. Ann. Math. (2), 423-432 (1974) · Zbl 0311.14001 |
| [20] | Maxim, L.; Rodriguez, J.; Wang, B.; Wu, L., Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels involution conjecture, Comm. Pure Appl. Math. (published online) · Zbl 07793230 |
| [21] | Maxim, L. G.; Rodriguez, J. I.; Wang, B., Euclidean distance degree of the multiview variety. SIAM J. Appl. Algebra Geom., 1, 28-48 (2020) · Zbl 1437.13049 |
| [22] | Nie, J.; Ranestad, K., Algebraic degree of polynomial optimization. SIAM J. Optim., 1, 485-502 (2009) · Zbl 1190.14051 |
| [23] | Nie, J.; Ranestad, K.; Sturmfels, B., The algebraic degree of semidefinite programming. Math. Program., 2, Ser. A, 379-405 (2010) · Zbl 1184.90119 |
| [24] | Parusiński, A.; Pragacz, P., Characteristic classes of hypersurfaces and characteristic cycles. J. Algebraic Geom., 1, 63-79 (2001) · Zbl 1072.14505 |
| [25] | Rodriguez, J. I.; Wang, B., The maximum likelihood degree of mixtures of independence models. SIAM J. Appl. Algebra Geom., 1, 484-506 (2017) · Zbl 1377.62082 |
| [26] | Rostalski, P.; Sturmfels, B., Dualities, 203-249 · Zbl 1360.90195 |
| [27] | Sabbah, C., Quelques remarques sur la géométrie des espaces conormaux, 161-192 · Zbl 0598.32011 |
| [28] | Schürmann, J.; Tibăr, M., Index formula for MacPherson cycles of affine algebraic varieties. Tohoku Math. J. (2), 1, 29-44 (2010) · Zbl 1187.14013 |
| [29] | Seade, J.; Tibăr, M.; Verjovsky, A., Global Euler obstruction and polar invariants. Math. Ann., 2, 393-403 (2005) · Zbl 1089.32020 |
| [30] | Serre, J.-P., Local Algebra. Springer Monographs in Mathematics (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0959.13010 |
| [31] | Sturmfels, B., Beyond linear algebra (2021) |
| [32] | Sturmfels, B.; Telen, S., Likelihood equations and scattering amplitudes. Algebraic Stat., 2, 167-186 (2021) · Zbl 1515.62137 |
| [33] | Tráng, L. D.; Teissier, B., Variétés polaires locales et classes de Chern des variétés singulières. Ann. Math. (2), 3, 457-491 (1981) · Zbl 0488.32004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
