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By Ehrenborg R.

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The implies' condition is modelled thus: 2/i == 1 implies 2/2 = 1 is represented by yi — ^2 ^ 0 We now have all the machinery for modelling logical conditions which will be explained in greater depth in Hooker's chapter (see chapter 3). 5. Reformulation Techniques Despite advances in methods of solving Discrete Optimisation problems, as well as the dramatic increase in the speed and storage capacity of computers, many such problems still remain very difficult to solve. There is, however, often great flexibility in the way such problems are modelled as Integer Programmes.

For a general 0-1 problem, in order to be formulated as an equivalent continuous problem, function f{x) must possess some specific properties. 7) a^(i suppose that f is bounded on X — {x ^ W^\{) < x < e} and Lipschitz continuous on an open set A 3 {0, l}'^. 7) are equivalent. 7), equivalent to (2,2)-(2,4), has a concave objective function. For more detailed description of different relations between general integer programming and continuous optimization problems we refer the reader to [23]. 4.

A major issue is to create a systematic way of generating those cuts (ideally facet defining) which separate C from the feasible integer solutions. Before we discuss this, however, we present a major result due to Chvatal [6]. This is that all the facet defining constraints (for a PIP) can be obtained by a finite number of repeated applications of the following two procedures. (i) Add together constraints in suitable multiples (when all in the same form eg " < " or ">") and add or subtract "=" constraints in suitable multiples.

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A Bijective Answer to a Question of Zvonkin by Ehrenborg R.

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