| 1. |
- Strömberg, Ann-Brith, 1961, et al.
(författare)
-
Mixed-Integer Linear Optimization: Primal–Dual Relations and Dual Subgradient and Cutting-Plane Methods
- 2020
-
Ingår i: Numerical Nonsmooth Optimization: State of the Art Algorithms. - Cham : Springer International Publishing. ; , s. 499-547, s. 499-547
-
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.
|
|
| 2. |
- Kiselman, Christer, 1939-
(författare)
-
Datorskärmens geometri
- 2008
-
Ingår i: Människor och matematik. - Göteborg : Göteborgs universitet, NCM. - 9789185143085 ; , s. 211-229
-
Bokkapitel (populärvet., debatt m.m.)
|
|
| 3. |
|
|
| 4. |
- Engström, Christopher, 1987-, et al.
(författare)
-
PageRank, a Look at Small Changes in a Line of Nodes and the Complete Graph
- 2016
-
Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 223-247
-
Bokkapitel (refereegranskat)abstract
- In this article we will look at the PageRank algorithm used as part of the ranking process of different Internet pages in search engines by for example Google. This article has its main focus in the understanding of the behavior of PageRank as the system dynamically changes either by contracting or expanding such as when adding or subtracting nodes or links or groups of nodes or links. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others. We will look at PageRank as the solution of a linear system of equations and do our examination in both the ordinary normalized version of PageRank as well as the non-normalized version found by solving corresponding linear system. We will show that using two different methods we can find explicit formulas for the PageRank of some simple link structures.
|
|
| 5. |
- Engström, Christopher, 1987-, et al.
(författare)
-
PageRank, Connecting a Line of Nodes with a Complete Graph
- 2016
-
Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
-
Bokkapitel (refereegranskat)abstract
- The focus of this article is the PageRank algorithm originally defined by S. Brin and L. Page as the stationary distribution of a certain random walk on a graph used to rank homepages on the Internet. We will attempt to get a better understanding of how PageRank changes after you make some changes to the graph such as adding or removing edge between otherwise disjoint subgraphs. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others and different ways to combine the two. Both the ordinary normalized version of PageRank as well as a non-normalized version of PageRank found by solving corresponding linear system will be considered. We will see that it is possible to find explicit formulas for the PageRank in some simple link structures and using these formulas take a more in-depth look at the behavior of the ranking as the system changes.
|
|
| 6. |
|
|
| 7. |
|
|
| 8. |
- Kiselman, Christer O., 1939-
(författare)
-
Kiel rekoni rektojn kaj strekojn inter ĉiuj kurboj kaj aliaj bildoj sur la komputila ekrano?
- 2014. - 2
-
Ingår i: Internacia Simpozio "Apliko de Esperanto en la profesia agado". Prelegaro, Vol. II.. - Karlovo, BG : Akademio Internacia de la Sciencoj, AIS, San Marino. AIS-Bulgario.. ; , s. 72-92
-
Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Ni karakterizas rektecon de digitaj kurboj en la entjera ebeno per tri malsamaj metodoj: per vorta kombinatoriko; per diofantaj neegalaĵoj; kaj per diferencaj operatoroj. La laste menciita metodo ŝajnas essti nova.
|
|
| 9. |
|
|
| 10. |
- Stokes, Klara
(författare)
-
On computational anonymity
- 2012. - 7556
-
Ingår i: Privacy in Statistical Databases 2012. - : Springer. ; , s. 336-347
-
Bokkapitel (refereegranskat)
|
|
