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Homework answers / question archive / Let P: = {x € Rn : Ax £ b} be a polyhedron, let J Ì {1,… ,n}, and let S: ={x ? P: xj € Z, j € I} be a mixed integer set, where I indexes the integer variables

Let P: = {x € R^{n} : Ax £ b} be a polyhedron, let J Ì {1,… ,n}, and let S: ={x ? P: x_{j} € Z, j € I} be a mixed integer set, where I indexes the integer variables. C: = {1,…, n}\ I to be the index set of the continuous variables.

The Theory of Valid inequalities

Problem 1

Given a vector π € Z^{n} such that π_{j} = 0 for all j € C, πx is integer for all x € S. Thus, for any π_{0} € Z, it follows that every x € S satisfies exactly one of πx £ π_{0} or πx ³ π_{0} + 1

P_{1}: = PÇ{x : πx £ π_{0}}

P_{2}: = PÇ{x : πx ³ π_{0} + 1}

Clearly, conv(S) Ì conv(P_{1}È P_{2}). For u € R^{m}, let u^{+} be defined by u^{+}_{i} := max{0,u_{i}} and let u^{-} := (—u)^{+}. A_{I} and A_{C} are the matrices comprising the columns of A with indices I and C respectively. Suppose uA_{I} is integral, uAc = 0, and ub Ï Z.

(a) Show the following inequality is valid for S

U^{+}(b— Ax)/¦ + u^{-}(b— Ax)/1-¦ ³1

Where f := ub — |ub|.

(b) Suppose now P := {x € R^{n} : Ax =b,x __>__ 0}. Let a := uA, b := ub, f := ub— [ub], and ¦_{j} = a_{j} —|a_{j}], j € I. Use Part (a) to prove the following valid inequality for S

S_{j}_{?}_{I,}_{¦j}_{£}_{¦}¦_{j}/¦.x_{j} + S_{ j}_{?}_{I,}_{¦j>}_{¦}1-¦_{j}/1-¦.x_{j} + S_{j}_{?C,}_{aj}_{³0}a_{j}/¦.x_{j} - S_{j}_{?C,}_{aj<0}a_{j}/1-¦.x _{j}³1.

This is Gomory’s mixed integer inequality.

(c) For the two-dimensional mixed integer set {(e, v) € Z x R_{+} e - v £ b} recall the rounding inequality

e - 1/1-¦.v £ [b],

Where f := b — |b|. Suppose that we are given a valid inequality for P written in the form

πx – (c - cx) £ b

Such that π_{I} is integral, πc = 0, and cx £ c is a valid inequality for P. Then πx is an integer and c — cx ³ 0 for all x € S. Using the simple mixed integer rounding with variable substitution gives the inequality

πx – 1/1-¦. (c - cx) £ b

Show that any inequality of the form in part (a) can be derived using the mixed integer rounding procedure above.

(d) Let T= {x € Z^{n}, y ? R^{p}_{+} : S a_{j}x_{j} + S J_{j}y_{j }£ b}, where a_{j} ? Z for all j, gcd(a_{1},--- ,a_{n}) = 1, and b Ï Z. Show that the following mixed integer rounding inequality defines a facet of conv(T).

S a_{j}x_{j} + 1/1-¦ S_{j}_{?J-} J_{j}y_{j }£ b,

Where J^{-} = {j : g_{j} < O} and f = b- |b}.

Problem 2

For mixed integer set S := {x € P : x_{j} € Z,j € I} where P:= {x € R^{n} : Ax __<__ b}. The Chvatal inequality is u __>__ 0

(uA)x £ ub.

The Chvatal closure P^{ch} of P is the set of points in P satisfying all the Chvatal inequalities.

P^{Ch} := {x € P: (uA)x __<__ |ub| for all u __>__ 0 s.t. uA_{I} € Z^{I},uAc = 0}.

(a) Suppose S := PÇZ^{n} is a pure integer set. Show that P^{Ch} is the set of all points in P satisfying the Chvatal inequalities (uA) __<__ [ub] for all u such that uA € Z^{n} and 0 __<__ u __<__ 1.

(b) Show that for pure integer linear sets P^{Ch} is a polyhedron.

(c) Let P := {(x,y) ? R^{2} : 2x __>__ y, 2x + y __<__ 2,y __>__ 0} and S := PÇ(Z x R). Show that conv(S) ¹ P and that the Chvatal closure of P is P itself.

Valid Inequalities for Structured Integer Programs

Consider the 0, 1 knapsack set

K := {x € {0,1}" : S^{n}_{j=1} a_{j}x_{j} £ b}

Where b > 0 and a_{j} > 0. We further assume that a_{j} __<__ b for all j € N.

Problem 3

Recall that a cover is a subset C Ì N such that S_{j}_{?C}a_{j} > b and it is minimal if S_{j}_{?C}\_{{k}}a_{j} £ b for all k € C. For any cover C, the cover inequality associated with C is

S_{j}_{?C}x_{j} £ ?C? - 1,

(a) Let C be a cover for K. Show that the cover inequality associated with C is facet-defining for

P_{C} := conv(K)Ç{x € R^{n} : x_{j} =0, j € N\C} if and only if C is a minimal cover.

(b) Let C be a minimal cover, let h € C such that a_{h} = max_{j}_{?C} a_{j}. Show that the inequality

S_{j}_{?C}x_{j} + S_{j}_{?N}\_{C} [a_{j}/a_{h}]x_{j} £ ?C? - 1

is a Chvatal inequality for P := {x ? R^{n} : S^{n}_{j=1}a_{j}x_{j} £ b, 0 £ x £ 1}.

(c) Consider a binary set S := {x € {0,1}" : Ax £ b} of dimension n, where A is a nonnegative matrix. Suppose we start with a facet-defining inequality S_{j}_{?C}a_{j}x_{j} £ b of conv(S)Ç{x : x_{j} = 0, j € N\C}. Consider the following lifting procedure

Choose an ordering j_{1},… , j_{I} of the indices in N\C. Let C_{0} = C and C_{h} = C_{h-1} {j_{h}}

For h = 1 up to h = i, compute

a_{jh} := b — max{ S_{j}_{?Ch-1}a_{j}x_{j} : x ? S, x_{j} = 0, j ? N\C_{h}, x_{jh} = 1).

Show that the inequality S^{n}_{j=1}a_{j}x_{j} £ b obtained this way is facet-defining for conv(S)