Dual-Feasible Functions for Integer Programming and by Cláudio Alves, Francois Clautiaux, José Valério de Carvalho,

By Cláudio Alves, Francois Clautiaux, José Valério de Carvalho, Jürgen Rietz

This ebook offers a postgraduate viewers the keys they should comprehend and additional boost a collection of instruments for the effective computation of decrease bounds and legitimate inequalities in integer courses and combinatorial optimization difficulties. After discussing the classical techniques defined within the literature, the booklet addresses how one can expand those instruments to different non-standard formulations that could be utilized to a wide set of functions. Examples are supplied to demonstrate the underlying ideas and to pave the way in which for destiny contributions.

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These functions are in fact discrete DFF. A dual-feasible function is defined formally as follows. 1 A function f W Œ0; 1 ! 0; 1. 1 shows a parameter dependent staircase function f W Œ0; 1 ! x/ WD bCxc=bCc; © Springer International Publishing Switzerland 2016 C. 1) 21 22 2 Classical Dual-Feasible Functions y 1 y 1 y 1 1 x y 1 1 x 1 x Fig. 1) for parameter values C 2 ˚ 12 11 ; 12 36 60 ; 7 ; 11 7 1 x « for several parameter values C 1. Black filled circles mean that point belongs to the graph of the function, while white circles exclude that point.

X/ holds for all x 2 Œ0; ˇ. x/, respectively. The function uA is a simple staircase function. The dependence of uB and uD on the parameter a is shown in Fig. 10 for p D 3. 4) (p. 25) for C WD 1=ˇ. Checking this observation is left as an exercise. The function fFS;1 W Œ0; 1 ! Œ0; 1 depending on the parameter k 2 N n f0g discussed in Sect. k C 1/ xc=k; otherwise; is a maximal dual-feasible function. 5. k C1/ remain unchanged, while the other ones are submitted to a rounding function. 4 Examples 43 y 1 y 1 uB (·;3, 15 ) y 1 uB (·;3, 13 72 ) 1 x y 1 1 x y 1 uD (·;3, 15 ) 5 uB (·;3, 24 ) 1 x y 1 uD (·;3, 13 72 ) 1 x 5 uD (·;3, 24 ) 1 x Fig.

3 Symmetry Maximal dual-feasible functions can be built from superadditive functions by keeping the images of the values smaller than 1=2, and by computing the images of the values larger than 1=2 by symmetry. 4 Let f W Œ0; 1 ! Œ0; 1 be a superadditive function. The following function fO W Œ0; 1 ! 1 x/; if 1=2 < x Ä 1; is a maximal dual-feasible function dominating f . 4. An example is provided and discussed in Sect. 4. 4 Using the Limiting Behaviour of a Function A maximal dual-feasible function can sometimes be built from a superadditive function f W Œ0; 1 !

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