[...]... multicriteria problems was first introduced in Ichida and Fujii (1990) D Scholz, Deterministic Global Optimization: Geometric Branch -and- bound Methods and Their Applications, Springer Optimization and Its Applications 63, DOI 10.1007/978-1-4614-1951-8_2, © Springer Science+Business Media, LLC 2012 15 16 2 The geometric branch -and- bound algorithm 2.1.2 Branch -and- bound methods in location theory Moreover, some... (2009) and solved using d.c decompositions Integrated scheduling and location problems have been solved in Kalsch and Drezner (2010) Finally, geometric branch -and- bound methods in multicriteria facility location problems can be found in Skriver and Anderson (2003), Fern´ ndez and T´ th a o (2009), and Scholz (2010) 2.3 The geometric branch -and- bound algorithm 17 2.2 Notations Before we present the geometric. .. published in Sch¨ bel and Scholz (2010b) o 2.1 Literature review Over the last decades, geometric branch -and- bound methods became more and more important solution algorithms for global optimization problems Although all these methods are based on the same ideas, they differ in particular in the way of calculating lower bounds 2.1.1 General branch -and- bound algorithms Some general branch -and- bound techniques... found in Horst and Tuy (1996) and Horst et al (2000) Using Lipschitzian functions, lower bounds can be constructed as suggested in Hansen and Jaumard (1995) In Horst and Thoai (1999) and Tuy and Horst (1988), branch -and- bound techniques for d.c functions are considered Global optimization using interval analysis is discussed in Ratschek and Rokne (1988), Hansen (1992), and Ratschek and Voller (1991)... X and all λ ∈ [0, 1] Definition 1.2 Let X ⊂ Rn be a convex set A function f : X → R is called convex if f (λ · x + (1 − λ ) · y) ≤ λ · f (x) + (1 − λ ) · f (y) for all x, y ∈ X and all λ ∈ [0, 1] A function f is called concave if − f is convex; that is if f (λ · x + (1 − λ ) · y) ≥ λ · f (x) + (1 − λ ) · f (y) D Scholz, Deterministic Global Optimization: Geometric Branch -and- bound Methods and Their Applications, ... these approaches and their applications to facility location problems in Section 2.1 Next, we give some notations and formally define bounding operations in Section 2.2 before the geometric branch -and- bound prototype algorithm is presented and discussed in Section 2.3 The main contribution of the present chapter can be found in Sections 2.4 and 2.5 Therein, we define the rate of convergence and results about... objective function as required throughout the geometric branch -and- bound algorithm suggested in the following chapter As an example, the natural interval extension yields an interval inclusion function due to Theorem 1.3 Chapter 2 The geometric branch -and- bound algorithm Abstract The aim of this chapter is the presentation of the fundamental geometric branch -and- bound algorithm including a general convergence... theoretical rate of convergence; see Example 2.1 2.3 The geometric branch -and- bound algorithm We consider the minimization of a continuous function f : X → R, where we assume a box X ⊂ Rn as the feasible area; L R L R X = [x1 , x1 ] × · · · × [xn , xn ] ⊂ Rn 18 2 The geometric branch -and- bound algorithm The general idea of all the geometric branch -and- bound algorithms cited in Section 2.1 is the same:... of facility location problems can be found in Love et al., Drezner (1995), or Drezner and Hamacher (2001) Although a wide range of facility location problems can be formulated as global optimization problems in small dimension, they are often hard to solve However, geometric branch -and- bound methods are convenient and commonly used solution algorithms for these problems; see Chapter 2 Therefore, all... absolute accuracy of ε > 0 To sum up, for the following geometric branch -and- bound algorithm assume an objective function f and a feasible box X Moreover, we need a bounding operation, a splitting rule, and an absolute accuracy of ε > 0 1 Let X be a list of boxes and initialize X := {X} 2 Apply the bounding operation to X and set UB := f (r(X)) and x∗ := r(X) 3 If X = 0, the algorithm stops Else set . volumes: http://www.springer.com/series/7393 Springer Optimization and Its Applications 63 Series Editor: Panos M. Pardalos Subseries: Nonconvex Optimization and Its Applications Daniel Scholz Geometric Branch -and- bound Methods and Their Applications Deterministic. Scholz, Deterministic Global Optimization: Geometric Branch -and- bound Methods and Their Applications DOI 10.1007/978-1-4614-1951-8_1, © Springer Science+Business Media, LLC 2012 1 , Springer Optimization. 9 2 The geometric branch -and- bound algorithm 15 2.1 Literature review . 15 2.1.1 General branch -and- bound algorithms 15 2.1.2 Branch -and- bound methods in location theory . . 16 2.1.3 Applications