By M. L. Balinski

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**Additional info for Approaches to Integer Programming**

**Sample text**

Let A be a matrix and let Let be a vector of multipliers. Starting from the linear system the Chvátal-Gomory procedure produces a valid inequality for of the type and adds it to It is a well known result [36] that any valid inequality for can be derived by applying the Chvátal-Gomory procedure a finite number of times. Now let us turn to our problem, where is a set of Metric Inequalities. Let and let be the inequality derived by the ChvátalGomory procedure. It can be easily proved that It follows that at any stage of the procedure the system A includes only inequalities of the form with Let an extreme ray of Met(G) and let be valid for Since cannot be obtained as a conic combination of two or more other distinct metrics, it follows that can be derived applying the Chvátal-Gomory procedure with a vector of multipliers where the multiplier associated with the itself is 1 and the multipliers associated with the other inequalities are null and then Let S and T be a partition of V and let be a cut on G.

Lemma 1. 2 If (and The quantity in this case increases, the degree of asymmetry of G then the degree of asymmetry of G is is a positive semidefinite matrix then The Bound Theorem 1. For an affine variational inequality problem with problem function with for all we have: Proof: We first observe that since solves VI(F, K) and inequality formulation implies that then the variational 50 G. Perakis (multiplying with S and (from Cauchy’s inequality, and (from the norm inequality) (Definition 1) For every we have leading to This in turn implies that if we choose We thus obtain that for all If we further select then and and since If we further impose the condition Given that we have freedom to select by solving we obtain we obtain the bound: and we find the best upper bound The optimal solution to Problem (1) is given as follows: If the optimal solution is leading to If the optimal solution is leading to Remark: Notice that the results in this section apply to a feasible region that is non-convex and as a result includes integers.

36 3 S. Dash and O. Günlük A Simple Polyhedral Set In this section we study simple three variable mixed-integer sets. We first look at the set when and satisfy and Note that variable is required to be non-negative. In a recent study Agra and Constantino [2] study when is an arbitrary positive number and give a polynomial-time algorithm that enumerates all of its facets. Under our restrictions on and it is possible to describe the convex hull of explicitly. Lemma 2. The following inequalities are facet defining for and together with the non-negativity requirements, give a complete description of Proof, (validity) Note that inequality (4) can be obtained by treating as a continuous variable and writing the MIR inequality based on To obtain inequality (5), we start with inequality (4), divide it by and relax the resulting inequality as follows: Writing the MIR inequality where is treated as a continuous variable and is treated as an integer variable gives inequality (5).