By Guo L.-T.
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In space engineering, for instance, quite frequently, specific conditions on the spacecraft inertia matrix are set to address fuel consumption or attitude control concerns. 3 Additional Conditions 23 0 00 0 0 00 8β, β , β ∈ B=β < β , β, β 6¼ β X À Á Ã2 Mi wÃ2 βi þ wβ0 i ! I β00 m , ð2:30aÞ i∈I 0 00 0 0 00 8β, β , β ∈ B=β < β , β, β 6¼ β X À Á Ã2 Mi wÃ2 I β00 m , βi þ wβ0 i ð2:30bÞ i∈I where I ββ0 ðmÞ, I β00 ðmÞ and I β00 ðmÞ are (non-negative) functions of the total loaded mass m. 30b) are nonlinear constraints, giving rise to an MINLP model.
1), is usually very hard to solve. In this chapter the relevant intrinsic difficulties are examined first (Sect. 1). A heuristic philosophy is then emphasized to tackle efficiently the problem, even if just nonproven optimal solutions can, in general, be obtained. The basic concept of abstract configuration is introduced (Sect. 2). Chapter 3 reformulations, devised to solve the feasibility subproblem, are exploited to look into an initial approximate solution (Sect. 1). Two alternative heuristic procedures, thought up to improve it recursively, until a satisfactory result is reached, are discussed next (Sects.
3) of the reformulated one. 3). 2b), we shall distinguish the cases where the variables σ are zero from those where they are equal to one. Consider, for instance, σ þ βhkij ¼ 0. 5a) become Á 1XÀ wβ0hi À wβ0kj ! Lωβhi ϑωi þ Lωβkj ϑωj À Dβ . 2a). 5a) become wβ0hi À wβ0kj ! Lωβhi ϑωi þ Lωβkj ϑωj . 2a). As the same reasoning can be carried out, taking into account the cases relative to the variables σ À □ βhkij , the two models are equivalent. 1 To better understand the meaning of the general MIP model first linear reformulation, we shall make some intuitive considerations.
3-restricted connectivity of graphs with given girth by Guo L.-T.