Modifikationen von Cutting-Plane-Methoden der ganzzahligen Optimierung

Author(s):  
H. Müller-Merbach
Keyword(s):  
4OR ◽  
2020 ◽  
Author(s):  
Michele Conforti ◽  
Marianna De Santis ◽  
Marco Di Summa ◽  
Francesco Rinaldi

AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.


2021 ◽  
Vol 3 (6) ◽  
Author(s):  
C. DiGiovanni ◽  
L. He ◽  
C. Hawkins ◽  
N. Y. Zhou ◽  
E. Biro

AbstractThe automotive industry is turning to advanced high strength steels (AHSS) to reduce vehicle weight and increase fuel efficiency. However, the zinc coating on AHSS can cause liquid metal embrittlement (LME) cracking during resistance spot welding. To understand the problem, the severity of the cracking must be measured. Typically, this is done from the weld cross-section. Currently, there is no standard procedure to determine which plane through the weld must be examined to gauge cracking severity, leading to a variety of practices for choosing a cutting plane. This work compares the magnitude and variability of LME severity measured from the plane of exhibiting the most severe surface cracking to arbitrarily chosen planes. The plane exhibiting the most severe cracks had more and longer cracks on the cross-section than the arbitrarily chosen plane, resulting in a higher crack severity measurement. This higher absolute measurement increased the relative accuracy of the examination, allowing for fewer welds to be examined to precisely determine the effect of LME mitigation methods on cracking severity, how welding parameters affect LME cracking severity and the predicted LME affected strength of a particular weld.


2000 ◽  
Vol 11 (1) ◽  
pp. 266-288 ◽  
Author(s):  
Jean-Louis Goffin ◽  
Jean-Philippe Vial

2021 ◽  
Vol 1 ◽  
pp. 30-40
Author(s):  
Natalia V. Semenova ◽  
◽  
Maria M. Lomaga ◽  
Viktor V. Semenov ◽  
◽  
...  

The lexicographic approach for solving multicriteria problems consists in the strict ordering of criteria concerning relative importance and allows to obtain optimization of more important criterion due to any losses of all another, to the criteria of less importance. Hence, a lot of problems including the ones of com­plex system optimization, of stochastic programming under risk, of dynamic character, etc. may be presented in the form of lexicographic problems of opti­mization. We have revealed conditions of existence and optimality of solutions of multicriteria problems of lexicographic optimization with an unbounded convex set of feasible solutions on the basis of applying properties of a recession cone of a convex feasible set, the cone which puts in order lexicographically a feasible set with respect to optimization criteria and local tent built at the boundary points of the feasible set. The properties of lexicographic optimal solutions are described. Received conditions and properties may be successfully used while developing algorithms for finding optimal solutions of mentioned problems of lexicographic optimization. A method of finding lexicographic of optimal solutions of convex lexicographic problems is built and grounded on the basis of ideas of method of linearization and Kelley cutting-plane method.


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