| 11. |
- Strömberg, Ann-Brith, 1961, et al.
(författare)
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Mixed-Integer Linear Optimization: Primal–Dual Relations and Dual Subgradient and Cutting-Plane Methods
- 2020
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Ingår i: Numerical Nonsmooth Optimization: State of the Art Algorithms. - Cham : Springer International Publishing. ; , s. 499-547, s. 499-547
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- This chapter presents several solution methodologies for mixed-integer linear optimization, stated as mixed-binary optimization problems, by means of Lagrangian duals, subgradient optimization, cutting-planes, and recovery of primal solutions. It covers Lagrangian duality theory for mixed-binary linear optimization, a problem framework for which ultimate success—in most cases—is hard to accomplish, since strong duality cannot be inferred. First, a simple conditional subgradient optimization method for solving the dual problem is presented. Then, we show how ergodic sequences of Lagrangian subproblem solutions can be computed and used to recover mixed-binary primal solutions. We establish that the ergodic sequences accumulate at solutions to a convexified version of the original mixed-binary optimization problem. We also present a cutting-plane approach to the Lagrangian dual, which amounts to solving the convexified problem by Dantzig–Wolfe decomposition, as well as a two-phase method that benefits from the advantages of both subgradient optimization and Dantzig–Wolfe decomposition. Finally, we describe how the Lagrangian dual approach can be used to find near optimal solutions to mixed-binary optimization problems by utilizing the ergodic sequences in a Lagrangian heuristic, to construct a core problem, as well as to guide the branching in a branch-and-bound method. The chapter is concluded with a section comprising notes, references, historical downturns, and reading tips.
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| 12. |
- Berman, Robert, 1976
(författare)
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On the strict convexity of the K-energy
- 2019
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Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 15:4, s. 983-999
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Tidskriftsartikel (refereegranskat)abstract
- Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.
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| 13. |
- Fotedar, Sunney, 1989, et al.
(författare)
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A criterion space decomposition approach to generalized tri-objective tactical resource allocation
- 2023
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Ingår i: Computational Management Science. - : Springer Science and Business Media LLC. - 1619-697X .- 1619-6988. ; 20, s. 1-28
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Tidskriftsartikel (refereegranskat)abstract
- We present a tri-objective mixed-integer linear programming model of the tactical resource allocation problem with inventories, called the generalized tactical resource allocation problem (GTRAP). We propose a specialized criterion space decomposition strategy, in which the projected two-dimensional criterion space is partitioned and the corresponding sub-problems are solved in parallel by application of the quadrant shrinking method (QSM) (Boland in Eur J Oper Res 260(3):873–885, 2017) for identifying non-dominated points. To obtain an efficient implementation of the parallel variant of the QSM we suggest some modifications to reduce redundancies. Our approach is tailored for the GTRAP and is shown to have superior computational performance as compared to using the QSM without parallelization when applied to industrial instances.
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| 14. |
- Arkeryd, Leif, 1940, et al.
(författare)
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On stationary solutions to normal, coplanar discrete Boltzmann equation models
- 2020
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Ingår i: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 18:8, s. 2215-2234
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Tidskriftsartikel (refereegranskat)abstract
- The paper proves existence of renormalized solutions for a class of velocity-discrete coplanar stationary Boltzmann equations with given indata. The proof is based on the construction of a sequence of approximations with L-1- compactness for the integrated collision frequency and gain term. L-1-compactness of a sequence of approximations is obtained using the Kolmogorov-Riesz theorem and replaces the L-1-compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete.
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| 15. |
- Brandes, Julia, 1986
(författare)
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The density of rational lines on hypersurfaces: a bihomogeneous perspective
- 2021
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Ingår i: Monatshefte für Mathematik. - : Springer Science and Business Media LLC. - 1436-5081 .- 0026-9255. ; 195:2, s. 191-231
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Tidskriftsartikel (refereegranskat)abstract
- Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points (x, y) with | x| ⩽ X and | y| ⩽ Y which generate a line lying in the hypersurface defined by F, provided that n> 2 d-1d4(d+ 1) (d+ 2). In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y.
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| 16. |
- Samieinia, Shiva, 1978-
(författare)
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Digital Geometry, Combinatorics, and Discrete Optimization
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of two parts: digital geometry and discrete optimization. In the first part we study the structure of digital straight line segments. We also study digital curves from a combinatorial point of view. In Paper I we study the straightness in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments. We show that we can investigate the straightness of Khalimsky arcs by using our knowledge from the 8-connected plane. In Paper II we determine the number of Khalimsky-continuous functions with 2, 3 and 4 points in their codomain. These enumerations yield examples of known sequences as well as new ones. We also study the asymptotic behavior of each of them. In Paper III we study the number of Khalimsky-continuous functions with codomain Z and N. This gives us examples of Schröder and Delannoy numbers. As a byproduct we get some relations between these numbers. In Paper IV we study the number of Khalimsky-continuous functions between two points in a rectangle. Using a generating function we get a recurrence formula yielding this numbers. In the second part we study an analogue of discrete convexity, namely lateral convexity. In Paper V we define by means of difference operators the class of lateral convexity. The functions have plus infinity in their codomain. For the real-valued functions we need to check the difference operators for a smaller number of points. We study the relation between this class and integral convexity. In Paper VI we study the marginal function of real-valued functions in this class and its generalization. We show that for two points with a certain distance we have a Lipschitz property for the points where the infimum is attained. We show that if a function is in this class, the marginal function is also in the same class.
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| 17. |
- Balkanova, O., et al.
(författare)
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NON-VANISHING OF MAASS FORM L-FUNCTIONS AT THE CENTRAL POINT
- 2021
-
Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 149:2, s. 509-523
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider the family {L-j(s)}(j=1)(infinity) of L-functions associated to an orthonormal basis {u(j)}(j=1)(infinity) of even Hecke-Maass forms for the modular group SL(2, Z) with eigenvalues {lambda(j) = kappa(2)(j) + 1/4}(j=1)(infinity). We prove the following effective non-vanishing result: At least 50% of the central values L-j(1/2) with kappa(j) <= T do not vanish as T -> infinity. Furthermore, we establish effective non-vanishing results in short intervals.
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| 18. |
- Mertens, M. H., et al.
(författare)
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Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms
- 2021
-
Ingår i: Mathematical Research Letters. - 1073-2780 .- 1945-001X. ; 28:2, s. 511-561
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Tidskriftsartikel (refereegranskat)abstract
- The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of SL2(Z)-representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms.
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| 19. |
- Raum, Martin
(författare)
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Relations among Ramanujan-type congruences I
- 2022
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 409
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Tidskriftsartikel (refereegranskat)abstract
- We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations.(c) 2022 The Author(s). Published by Elsevier Inc. This is an (http://creativecommons.org/licenses/by/4.0/).
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| 20. |
- Westerholt-Raum, Martin, 1985, et al.
(författare)
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All modular forms of weight 2 can be expressed by Eisenstein series
- 2020
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Ingår i: Research in Number Theory. - : Springer Science and Business Media LLC. - 2522-0160 .- 2363-9555. ; 6:3
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Tidskriftsartikel (refereegranskat)abstract
- We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central L-values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.
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