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Theorem 5.18 (Chvátal [73], Schrijver [323]). Let \( P \) be a rational polyhedron. Then there exists a positive integer \( t \) such that \( {P}^{\left( t\right) } = {P}_{I} \) . | Proof. The proof is by induction on \( d = \dim \left( P\right) \), the cases \( d = - 1, d = 0 \) being trivial. If \( \operatorname{aff}\left( P\right) \cap {\mathbb{Z}}^{n} = \varnothing \), by Theorem 1.20 there exist an integral vector \( a \) and a scalar \( d \notin \mathbb{Z} \) such that \( P \subseteq \{ x : ... | Yes |
Consider the following pure integer programming problem, which we solved in Sect. 1.2.2 using Gomory fractional cuts. | We first add slack variables \( {x}_{3} \) and \( {x}_{4} \) to turn the inequality constraints into equalities. The problem becomes:\n\n\[ z\; - {5.5}{x}_{1}\; - {2.1}{x}_{2}\; = \;0 \]\n\n\[ \begin{array}{lllll} - {x}_{1} & + {x}_{2} & + {x}_{3} & = & 2 \end{array} \]\n\n\[ 8{x}_{1}\; + 2{x}_{2}\; + {x}_{4}\; = \;{17... | Yes |
Consider the following polytope, studied by Chvátal, Cook, and Hartmann [75]\n\n\[ \nP \mathrel{\text{:=}} \left\{ {x \in {\left\lbrack 0,1\right\rbrack }^{n} : \mathop{\sum }\limits_{{j \in J}}{x}_{j} + \mathop{\sum }\limits_{{j \notin J}}\left( {1 - {x}_{j}}\right) \geq \frac{1}{2},\text{ for all }J \subseteq \{ 1,2,... | For \( j = 1,\ldots, n \), let \( {F}_{j} \) be the set of all vectors \( x \in {\mathbb{R}}^{n} \) such that \( j \) components of \( x \) are \( \frac{1}{2} \) and each of the remaining \( n - j \) components are in \( \{ 0,1\} \) . Note that \( {F}_{1} \subseteq P \) (indeed, \( P \) is the convex hull of \( {F}_{1}... | Yes |
Theorem 5.24. The specialized lift-and-project algorithm terminates after a finite number of iterations for every mixed \( 0,1 \) linear program. | Proof. The proof is in two steps.\n\n(i) We prove that, at each iteration \( k \), the \( j \) -cut computed in Step 3 of the algorithm cuts off the solution \( \bar{x} \) computed in Step 1. We first show that \( \bar{x} \) is a vertex of \( {P}^{k, j} \) . Let \( H \mathrel{\text{:=}} \left\{ {x : {x}_{t} = {\bar{x}}... | Yes |
Lemma 6.2. If the affine hull of \( P\left( B\right) \) contains a point in \( {\mathbb{Z}}^{p} \times {\mathbb{R}}^{n - p} \), then \( \operatorname{corner}\left( B\right) \) is an \( \left| N\right| \) -dimensional polyhedron. Otherwise \( \operatorname{corner}\left( B\right) \) is empty. | Proof. Since \( \operatorname{corner}\left( B\right) \) is contained in the affine hull of \( P\left( B\right) \) , \( \operatorname{corner}\left( B\right) \) is empty when the affine hull of \( P\left( B\right) \) contains no point in \( {\mathbb{Z}}^{p} \times {\mathbb{R}}^{n - p} \) .\n\nNext we assume that the affi... | Yes |
Consider the pure integer program\n\n\\[ \max \\frac{1}{2}{x}_{2} + {x}_{3} \\]\n\n\\[ {x}_{1} + {x}_{2} + {x}_{3} \\leq 2 \\]\n\n\\[ {x}_{1} - \\frac{1}{2}{x}_{3} \\geq 0 \\]\n\n\\[ {x}_{2} - \\frac{1}{2}{x}_{3} \\geq 0 \\]\n\n(6.5)\n\n\\[ {x}_{1} + \\frac{1}{2}{x}_{3} \\leq 1 \\]\n\n\\[ - {x}_{1} + {x}_{2} + {x}_{3} ... | This problem has four feasible solutions \\( \\left( {0,0,0}\\right) ,\\left( {1,0,0}\\right) ,\\left( {0,1,0}\\right) \\), and \\( \\left( {1,1,0}\\right) \\), all satisfying \\( {x}_{3} = 0 \\) . These four points are shown in the \\( \\left( {{x}_{1},{x}_{2}}\\right) \\) - space in Fig. 6.2.\n\nWe first write the pr... | Yes |
Lemma 6.4. Assume \( \operatorname{corner}\left( B\right) \) is nonempty. Every nontrivial valid inequality for corner \( \left( B\right) \) can be written in the form \( \mathop{\sum }\limits_{{j \in N}}{\gamma }_{j}{x}_{j} \geq 1 \) where \( {\gamma }_{j} \geq 0 \) for all \( j \in N \) . | Proof. We already observed that every valid linear inequality for \( \operatorname{corner}\left( B\right) \) can be written as \( \mathop{\sum }\limits_{{j \in N}}{\gamma }_{j}{x}_{j} \geq \delta \) . We argue next that \( {\gamma }_{j} \geq 0 \) for all \( j \in N \) . Indeed, if \( {\gamma }_{k} < 0 \) for some \( k ... | Yes |
Theorem 6.5. Let \( C \subset {\mathbb{R}}^{n} \) be a closed convex set whose interior contains the point \( \bar{x} \) but no point in \( {\mathbb{Z}}^{p} \times {\mathbb{R}}^{n - p} \). The intersection cut (6.9) defined by \( C \) is a valid inequality for \( \operatorname{corner}\left( B\right) \). | Proof. The set of points of the linear relaxation \( P\left( B\right) \) of corner \( \left( B\right) \) that are cut off by (6.9) is \( S \mathrel{\text{:=}} \left\{ {x \in P\left( B\right) : \mathop{\sum }\limits_{{j \in N}}\frac{{x}_{j}}{{\alpha }_{j}} < 1}\right\} \). We will show that \( S \) is contained in the i... | Yes |
Consider the following 4-variable mixed integer linear set\n\n\[ \n{x}_{1} = {b}_{1} + {a}_{11}{y}_{1} + {a}_{12}{y}_{2} \]\n\n\[ \n{x}_{2} = {b}_{2} + {a}_{21}{y}_{1} + {a}_{22}{y}_{2} \]\n\n(6.10)\n\n\[ \nx \in {\mathbb{Z}}^{2} \]\n\n\[ \ny \geq 0 \]\n\nwhere the rays \( {r}^{1} = \left( \begin{array}{l} {a}_{11} \\ ... | Because there are two nonbasic variables in this example, the intersection cut can be represented by a line in the space of the basic variables, namely the line passing through the intersection points \( {p}^{1},{p}^{2} \) of the boundary of \( K \) with the half lines \( \left\{ {b + \alpha {r}^{1} : \alpha \geq 0}\ri... | Yes |
Example 6.10. (Gomory's Mixed Integer Cuts from the Tableau)\n\nWe already mentioned that split cuts are intersection cuts. We can interpret the formula of a Gomory mixed integer cut derived from a row of the simplex tableau (6.2) in the context of an intersection cut defined by a split set. The argument is as follows.... | Next we derive the intersection cut from the split set \( C \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n}}\right. \) : \( \left. {{\pi }_{0} \leq {\pi x} \leq {\pi }_{0} + 1}\right\} \) following Example 6.9. We will compute \( {\alpha }_{j} \) for \( j \in N \) using formula (6.11). To do this, we need to comput... | Yes |
Example 6.11. (Intersection Cuts Can Have an Arbitrarily Large Split Rank)\n\nWe refer the reader to Sect. 5.1.3 for the definition of split rank of a valid inequality. Consider the polytope \( P \mathrel{\text{:=}} \left\{ {\left( {{x}_{1},{x}_{2}, y}\right) \in {\mathbb{R}}_{ + }^{3} : {x}_{1} \geq }\right. \)\n\n![2... | By adding slack or surplus variables, the system defining \( P \) is equivalent to\n\n\[ \n- {x}_{1} + y + {s}_{1} = 0 \]\n\n\[ \n- {x}_{2} + y + {s}_{2} = 0 \]\n\n\[ \n{x}_{1} + {x}_{2} + {2y} + {s}_{3} = 2 \]\n\n\[ \n{x}_{1},{x}_{2}, y,{s}_{1},{s}_{2},{s}_{3} \geq 0. \]\n\nThe tableau relative to the basis \( B \) de... | Yes |
Theorem 6.12. Every nontrivial facet-defining inequality for \( \operatorname{corner}\left( B\right) \) is an intersection cut. | Proof. We prove the theorem in the pure integer case, that is, when \( p = n \) (see [78] for the general case). Consider a nontrivial valid inequality for \( \operatorname{corner}\left( B\right) \) . By Lemma 6.4 it is of the form \( \mathop{\sum }\limits_{{j \in N}}{\gamma }_{j}{x}_{j} \geq 1 \) . We show that it is ... | Yes |
Lemma 6.14. Given a closed convex set \( K \) with the origin in its interior, the gauge \( {\gamma }_{K} \) is a nonnegative sublinear function. | Proof. It follows from the definition of gauge that \( {\gamma }_{K} \) is positively homogeneous and nonnegative. Since \( K \) is a closed convex set, \( {\gamma }_{K} \) is a convex function. We now show that \( {\gamma }_{K} \) is subadditive. We have that \( {\gamma }_{K}\left( {r}^{1}\right) + \) \( {\gamma }_{K}... | Yes |
Lemma 6.15. Every sublinear function \( g : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is convex, and therefore continuous. | Proof. Let \( g \) be a sublinear function. The convexity of \( g \) follows from \( \frac{1}{2}\left( {g\left( {r}^{1}\right) + g\left( {r}^{2}\right) }\right) = g\left( \frac{{r}^{1}}{2}\right) + g\left( \frac{{r}^{2}}{2}\right) \geq g\left( \frac{{r}^{1} + {r}^{2}}{2}\right) \) for every \( {r}^{1},{r}^{2} \in {\mat... | Yes |
Theorem 6.16. Let \( g : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a nonnegative sublinear function and let \( K \mathrel{\text{:=}} \left\{ {r \in {\mathbb{R}}^{n} : g\left( r\right) \leq 1}\right\} \) . Then \( K \) is a closed convex set with the origin in its interior and \( g \) is the gauge of \( K \) . | Proof. By Lemma 6.15, \( g \) is continuous and convex, therefore \( K \) is a closed convex set. Since the interior of \( K \) is \( \left\{ {r \in {\mathbb{R}}^{n} : g\left( r\right) < 1}\right\} \) and \( g\left( 0\right) = 0 \), the origin is in the interior of \( K \) .\n\nLet \( r \in {\mathbb{R}}^{n} \) . We nee... | Yes |
Lemma 6.17. Let \( C \) be a full-dimensional maximal \( {\mathbb{Z}}^{p} \times {\mathbb{R}}^{n - p} \) -free convex set and let \( K \) be its orthogonal projection onto \( {\mathbb{R}}^{p} \) . Then \( K \) is a maximal \( {\mathbb{Z}}^{p} \) -free convex set and \( C = K \times {\mathbb{R}}^{n - p} \) . | Proof. A classical result in convex analysis implies that the interior of \( K \) is the orthogonal projection onto \( {\mathbb{R}}^{p} \) of the interior of \( C \) (see Theorem 6.6 in Rockafellar [316]). Since \( C \) is a \( {\mathbb{Z}}^{p} \times {\mathbb{R}}^{n - p} \) -free convex set, it follows that \( K \) is... | Yes |
Theorem 6.18 (Lovász [261]). Let \( K \subset {\mathbb{R}}^{p} \) be a full-dimensional set. Then \( K \) is a maximal lattice-free convex set if and only if \( K \) is a polyhedron that does not contain any point of \( {\mathbb{Z}}^{p} \) in its interior and there is at least one point of \( {\mathbb{Z}}^{p} \) in the... | Proof of Theorem 6.18 in the bounded case. Let \( K \) be a maximal lattice-free convex set and assume that \( K \) is bounded. Then there exist vectors \( l, u \) in \( {\mathbb{Z}}^{p} \) such that \( K \) is contained in the box \( B \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{p} : l \leq x \leq u}\right\} \) .... | Yes |
Theorem 6.19. Any full-dimensional maximal lattice-free convex set \( K \subseteq {\mathbb{R}}^{p} \) has at most \( {2}^{p} \) facets. | Proof. By Theorem 6.18, each facet \( F \) contains an integral point \( {x}^{F} \) in its relative interior. If there are more than \( {2}^{p} \) facets, then there exist two distinct facets \( F,{F}^{\prime } \) such that \( {x}^{F} \) and \( {x}^{{F}^{\prime }} \) are congruent modulo 2 . Now their middle point \( \... | Yes |
Theorem 6.20. Let \( K \) be a \( {\mathbb{Z}}^{p} \) -free polyhedron containing \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{p}}\right) \) in its interior. Then \( K \) can be uniquely written in the form \( K = \left\{ {x \in {\mathbb{R}}^{p}}\right. \) : \( \left. {\mathop{\sum }\limits_{{h = 1}}^{p}{d}_{h}^{i}\left... | Proof. Every facet-defining inequality for \( K \) can be written in the form \( \mathop{\sum }\limits_{{h = 1}}^{p}{d}_{h}\left( {{x}_{h} - {\bar{x}}_{h}}\right) \leq \delta \) . Since \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{p}}\right) \) is in the interior of \( K \), it follows that \( \mathop{\sum }\limits_{{h ... | Yes |
Consider the following instance of (6.3) with no integer nonbasic variable. | \[ {x}_{1} = \frac{1}{2} + \frac{1}{4}{x}_{3} - \frac{3}{4}{x}_{4} - \frac{1}{4}{x}_{5} + {x}_{6} \]\n\n\[ \begin{matrix} {x}_{2} & = & \frac{1}{2} & + \frac{3}{4}{x}_{3} & - \frac{1}{4}{x}_{4} & + \frac{3}{4}{x}_{5} & - \frac{3}{4}{x}_{6} \end{matrix} \]\n\n\[ {x}_{1},{x}_{2} \in \mathbb{Z} \]\n\n\[ {x}_{3},{x}_{4},{x... | Yes |
Theorem 6.22 (Gomory and Johnson [179]). A function \( \pi : {\mathbb{R}}^{q} \rightarrow {\mathbb{R}}_{ + } \) is a minimal valid function for \( {G}_{f} \) if and only if \( \pi \left( 0\right) = 0,\pi \) is subadditive, periodic and satisfies the symmetry condition. | Proof. We first prove the \ | No |
Consider a continuous nonnegative periodic function \( \pi : \mathbb{R} \rightarrow {\mathbb{R}}_{ + } \) that is piecewise-linear in the interval \( \left\lbrack {0,1}\right\rbrack \) and satisfies \( \pi \left( 0\right) = 0 \) . By Theorem 6.22, such a function \( \pi \) is minimal if it is subadditive and satisfies ... | Using this, the reader can verify that all three functions given above are minimal. | No |
The third function is extreme when \( f \geq t + 1/2 \) | The proof is left as an exercise (Exercise 6.18). | No |
Lemma 6.25. Let \( \pi \) be a minimal valid function. Assume \( \pi = \frac{1}{2}{\pi }_{1} + \frac{1}{2}{\pi }_{2} \) , where \( {\pi }_{1} \) and \( {\pi }_{2} \) are valid functions. Then \( {\pi }_{1} \) and \( {\pi }_{2} \) are minimal functions and \( E\left( \pi \right) \subseteq E\left( {\pi }_{1}\right) \cap ... | Proof. Suppose \( {\pi }_{1} \) is not minimal. Let \( {\pi }_{1}^{\prime } \neq \pi \) be a valid function, such that \( {\pi }_{1}^{\prime } \leq {\pi }_{1} \) . Then \( {\pi }^{\prime } = \frac{1}{2}{\pi }_{1}^{\prime } + \frac{1}{2}{\pi }_{2} \) is a valid function, distinct from \( \pi \), and \( {\pi }^{\prime } ... | Yes |
Lemma 6.28. Let \( B \) be a \( {\mathbb{Z}}^{q} \) -free closed convex set with \( f \) in its interior. Let \( \psi \) be the gauge of \( B - f \) . Then \( \psi \) is a valid function. | Proof. By Lemma 6.14, \( \psi \) is sublinear. Consider \( \bar{y} \in {R}_{f} \) . Then \( \sum r{\bar{y}}_{r} = \) \( \bar{x} - f \), for some \( \bar{x} \in {\mathbb{Z}}^{q} \) . \[ \sum \psi \left( r\right) {\bar{y}}_{r} = \sum \psi \left( {r{\bar{y}}_{r}}\right) \geq \psi \left( {\sum r{\bar{y}}_{r}}\right) = \psi... | Yes |
A function \( \psi \) is a minimal valid function for \( {R}_{f} \) if and only if there exists some maximal \( {\mathbb{Z}}^{q} \) -free convex set \( B \) such that \( \psi \) is the gauge of \( B - f \) . | For the \ | No |
Consider the maximal \( {\mathbb{Z}}^{2} \) -free set \( B \) defined in Example 6.21, and let \( f = \left( \frac{\frac{1}{2}}{\frac{1}{2}}\right) \) . The corresponding function \( \psi \) has three pieces, corresponding to the three polyhedral cones \( {P}_{1},{P}_{2},{P}_{3} \) shown in Fig. 6.10, and we have | \[ \psi \left( r\right) = \left\{ {\begin{array}{ll} - 2{r}_{1} & \text{ for }r \in {P}_{1} \\ - 2{r}_{2} & \text{ for }r \in {P}_{2} \\ {r}_{1} + {r}_{2} & \text{ for }r \in {P}_{3} \end{array}\;r \in {\mathbb{R}}^{2}.}\right. \] | Yes |
Lemma 6.33. Let \( \left( {\pi ,\psi }\right) \) be a minimal valid function for \( {M}_{f} \) . Then \( \pi \leq \psi \) and \( \psi \) is a nonnegative sublinear function. | Proof. The same proof as that in Lemma 6.29 shows that \( \psi \) is nonnegative and sublinear. We next show that \( \pi \leq \psi \) . Suppose not, and let \( \widetilde{r} \in {\mathbb{R}}^{q} \) such that \( \pi \left( \widetilde{r}\right) > \psi \left( \widetilde{r}\right) \) . Let \( {\pi }^{\prime } \) be the fun... | Yes |
Consider the three functions \( \pi ,{\pi }_{1},{\pi }_{2} \) of Fig. 6.8, where \( t > 0 \) and \( t + 1/2 < f < 1 \) . As discussed in Example 6.24, these functions are extreme for \( {G}_{f} \) . For ease of notation, let \( {\pi }_{0} \mathrel{\text{:=}} \pi \) . For \( i = 0,1,2 \), let \( {s}_{i}^{ + } \) be the ... | \[ {\psi }_{i}\left( r\right) \mathrel{\text{:=}} \left\{ {\begin{array}{ll} {s}_{i}^{ + }r & \text{ if }r \geq 0 \\ {s}_{i}^{ - }r & \text{ if }r < 0 \end{array}\;r \in \mathbb{R}.}\right. \] The positive slopes are identical \( \left( {{s}_{i}^{ + } = {\left( 1 - f\right) }^{-1}}\right. \) for \( \left. {i = 0,1,2}\r... | Yes |
Let us consider the case \( q = 1 \) . Assume that \( 0 < f < 1 \) , and Let \( B = \left\lbrack {0,1}\right\rbrack \) . Let \( \psi \) be the gauge of \( B - f \) . As one can easily check, | \[ \psi \left( r\right) = \max \left\{ {\frac{r}{1 - f}, - \frac{r}{f}}\right\} . \] One can verify that the trivial lifting \( \bar{\pi } \) for \( \psi \) is the following \[ \bar{\pi }\left( r\right) = \left\{ \begin{array}{ll} \frac{r-\lfloor r\rfloor }{1 - f} & \text{ if }r - \lfloor r\rfloor \leq 1 - f \\ \frac{\... | Yes |
Lemma 6.38. Let \( \left( {\pi ,\psi }\right) \) be a minimal valid function for \( {M}_{f} \) . Given \( {r}^{ * } \in {\mathbb{R}}^{q} \), if\n\n\[ \psi \left( {r}^{ * }\right) + \psi \left( {z - f - {r}^{ * }}\right) = \psi \left( {z - f}\right) = 1\;\text{ for some }z \in {\mathbb{Z}}^{q}, \]\n\n(6.30)\n\nthen \( \... | Proof. Given \( z \in {\mathbb{Z}}^{q} \), define\n\n\[ {x}_{r} \mathrel{\text{:=}} \left\{ {\begin{array}{ll} 1 & \text{ for }r = {r}^{ * } \\ 0 & \text{ for }r \neq {r}^{ * } \end{array}\;{y}_{r} \mathrel{\text{:=}} \left\{ \begin{array}{ll} 1 & \text{ for }r = z - f - {r}^{ * } \\ 0 & \text{ for }r \neq z - f - {r}^... | Yes |
Example 6.39. (Dey and Wolsey [118]) Let \( q = 2 \) . Consider the maximal lattice-free triangle \( B = \operatorname{conv}\left( {\left( \begin{array}{l} 0 \\ 0 \end{array}\right) ,\left( \begin{array}{l} 2 \\ 0 \end{array}\right) ,\left( \begin{array}{l} 0 \\ 2 \end{array}\right) }\right) \), and let \( f \) be a po... | For each of the three points \( {z}_{1} = \left( \begin{array}{l} 1 \\ 0 \end{array}\right) ,{z}_{2} = \left( \begin{array}{l} 0 \\ 1 \end{array}\right) ,{z}_{3} = \left( \begin{array}{l} 1 \\ 1 \end{array}\right) \) on the boundary of \( B \), we have that \( \psi \left( {{z}_{i} - f}\right) = 1 \) . For \( i = 1,2,3 ... | Yes |
Proposition 7.1. Let \( C \) be a cover for \( K \) . The cover inequality associated with \( C \) is facet-defining for \( {P}_{C} \mathrel{\text{:=}} \operatorname{conv}\left( K\right) \cap \left\{ {x \in {\mathbb{R}}^{n} : {x}_{j} = 0, j \in N \smallsetminus C}\right\} \) if and only if \( C \) is a minimal cover. | Proof. Note that \( \dim \left( {P}_{C}\right) = \left| C\right| \) . Assume \( C \) is a minimal cover. For all \( j \in C \), let \( {x}^{j} \) be the point defined by \( {x}_{i}^{j} = 1 \) for all \( i \in C \smallsetminus \{ j\} \) and \( {x}_{i}^{j} = 0 \) for all \( i \in \left( {N \smallsetminus C}\right) \cup \... | Yes |
Consider a set \( S \subseteq \{ 0,1{\} }^{n} \) such that \( S \cap \left\{ {x : {x}_{n} = 1}\right\} \neq \varnothing \) , and let \( \mathop{\sum }\limits_{{i = 1}}^{{n - 1}}{\alpha }_{i}{x}_{i} \leq \beta \) be a valid inequality for \( S \cap \left\{ {x : {x}_{n} = 0}\right\} \) . Then\n\n\[ \n{\alpha }_{n} \mathr... | Proof. The inequality \( \mathop{\sum }\limits_{{i = 1}}^{n}{\alpha }_{i}{x}_{i} \leq \beta \) is valid for \( S \cap \left\{ {x : {x}_{n} = 0}\right\} \) by assumption, and it is valid for \( S \cap \left\{ {x : {x}_{n} = 1}\right\} \) by definition of \( {\alpha }_{n} \) . Thus \( \mathop{\sum }\limits_{{i = 1}}^{n}{... | Yes |
Consider the 0,1 knapsack set\n\n\[ 8{x}_{1} + 7{x}_{2} + 6{x}_{3} + 4{x}_{4} + 6{x}_{5} + 6{x}_{6} + 6{x}_{7} \leq {22} \]\n\n\[ {x}_{j} \in \{ 0,1\} \;\text{ for }j = 1,\ldots ,7. \]\n\nThe index set \( C \mathrel{\text{:=}} \{ 1,2,3,4 \} \) is a minimal cover. The corresponding minimal cover inequality is \( {x}_{1}... | We perform sequential lifting according to the order \( 5,6,7 \) . According to Proposition 7.2, the largest lifting coefficient for \( {x}_{5} \) is\n\n\[ {\alpha }_{5} = 3 - \max \left\{ {{x}_{1} + {x}_{2} + {x}_{3} + {x}_{4} : 8{x}_{1} + 7{x}_{2} + 6{x}_{3} + 4{x}_{4} \leq {22} - 6,{x}_{1},{x}_{2},{x}_{3},{x}_{4} \i... | Yes |
Theorem 7.4. Let \( K \mathrel{\text{:=}} \left\{ {x \in \{ 0,1{\} }^{n} : \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{j}{x}_{j} \leq b}\right\} \), where \( b \geq {a}_{j} > 0 \) for all \( j \in N \) . Let \( C \) be a minimal cover for \( K \), and let\n\n\[ \mathop{\sum }\limits_{{j \in C}}{x}_{j} + \mathop{\sum }\limi... | Proof. Assume that (7.5) is facet-defining for \( \operatorname{conv}\left( K\right) \) and let \( j \in N \smallsetminus C \) . Since \( 0 < {a}_{j} \leq b < {\mu }_{t} \), there exists an index \( h,0 \leq h \leq t - 1 \), such that \( {\mu }_{h} \leq {a}_{j} < {\mu }_{h + 1} \)\n\nBy Proposition 7.2, \( {\alpha }_{j... | Yes |
We illustrate the above theorem on the knapsack set \[ K \mathrel{\text{:=}} \left\{ {x \in \{ 0,1{\} }^{5} : 5{x}_{1} + 4{x}_{2} + 3{x}_{3} + 2{x}_{4} + {x}_{5} \leq 5}\right\} . \] The set \( C \mathrel{\text{:=}} \{ 3,4,5\} \) is a minimal cover. We would like to lift the inequality \( {x}_{3} + {x}_{4} + {x}_{5} \l... | We have \( {\mu }_{0} = 0,{\mu }_{1} = 3 \) , \( {\mu }_{2} = 5,{\mu }_{3} = 6 \) and \( \lambda = 1 \) . Therefore \( {\alpha }_{1} = 2 \) since \( {\mu }_{2} \leq {a}_{1} \leq {\mu }_{3} - \lambda \) . Similarly \( {\alpha }_{2} = 1 \) since \( {\mu }_{1} \leq {a}_{2} \leq {\mu }_{2} - \lambda \) . By Theorem 7.4, th... | Yes |
Consider the 0,1 knapsack set from Example 7.3\n\n\[ 8{x}_{1} + 7{x}_{2} + 6{x}_{3} + 4{x}_{4} + 6{x}_{5} + 6{x}_{6} + 6{x}_{7} \leq {22} \]\n\n\[ {x}_{j} \in \{ 0,1\} \text{ for }j = 1,\ldots ,7. \] | We consider the minimal cover \( C \mathrel{\text{:=}} \{ 1,2,3,4\} \) of Example 7.3 and the corresponding minimal cover inequality is \( {x}_{1} + {x}_{2} + {x}_{3} + {x}_{4} \leq 3 \) . We lift it with the superadditive function \( g \) defined in (7.12). Figure 7.1 plots the function. The lifted minimal cover inequ... | Yes |
Consider the knapsack set \( K \mathrel{\text{:=}} \left\{ {x \in \{ 0,1{\} }^{n} : \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{j}{x}_{j} \leq b}\right\} \) . Note that \( \operatorname{conv}\left( K\right) \) is isomorphic to the face of the single-node flow set \( \operatorname{conv}\left( T\right) \) defined in (7.13), ... | Let \( C \) be a minimal cover for \( K \) . Then \( C \) is a flow cover for \( T \) . Substituting \( {a}_{j}{x}_{j} \) for \( {y}_{j} \), for all \( j = 1,\ldots, n \), in the expression (7.14) of the flow cover inequality relative to \( C \), we obtain the following valid inequality for \( K \n\n\[ \mathop{\sum }\l... | Yes |
Lemma 7.12. Assume that (7.17) is valid for \( \operatorname{conv}\left( {T}^{i - 1}\right) \) . Then (7.18) is valid for \( \operatorname{conv}\left( {T}^{i}\right) \) if and only if \( \left( {{\alpha }_{i},{\beta }_{i}}\right) \) satisfies\n\n\[{\alpha }_{i}{y}_{i} + {\beta }_{i} \leq {f}_{i}\left( {y}_{i}\right) \t... | Proof. By definition of the function \( {f}_{i} \) ,(7.18) is valid for \( \operatorname{conv}\left( {T}^{i}\right) \) if and only if \( \left( {{\alpha }_{i},{\beta }_{i}}\right) \) satisfies \( {\alpha }_{i}{y}_{i} + {\beta }_{i}{x}_{i} \leq {f}_{i}\left( {y}_{i}\right) \) for all \( \left( {{x}_{i},{y}_{i}}\right) \... | Yes |
Lemma 7.14. The function \( f \) is superadditive in the interval \( \left\lbrack {0, b}\right\rbrack \) . | The proof of the above lemma can be found in [192]. | No |
Lemma 7.15. Let \( C \) be a flow cover of \( T \) . For \( i = 1,\ldots, n - t \), the function \( {f}_{i} \) defined in (7.19) coincides with the lifting function \( f \) . | Proof. Let \( i \geq 2 \) and assume by induction that \( f = {f}_{1} = \cdots = {f}_{i - 1} \) . Let \( z \in \left\lbrack {0, b}\right\rbrack \) and let \( \left( {{x}^{ * },{y}^{ * }}\right) \) be an optimal solution for (7.19). It follows from the definition of \( {f}_{i}\left( z\right) \) that\n\n\[ 0 \leq {f}_{i}... | Yes |
Theorem 7.16 (Gu et al. [191]). Let \( C \) be a flow cover for \( T \) such that \( \lambda < \mathop{\max }\limits_{{i \in C}}{a}_{j} \) . Let \( r \mathrel{\text{:=}} \max \left\{ {i \in C : {a}_{{j}_{i}} > \lambda }\right\} \) . The inequality (7.16) is facet-defining for \( T \) if and only if, for each \( i \in N... | Proof. By Lemmas 7.12 and 7.15, the inequality (7.16) is facet-defining for \( \operatorname{conv}\left( T\right) \) if and only if, for every \( i \in N \smallsetminus C \), the line of equation \( v = {\alpha }_{i}u + {\beta }_{i} \) lies below the graph of the function \( f \) in the interval \( \left\lbrack {0,{a}_... | Yes |
Consider the single-node flow set\n\n\[ T \mathrel{\text{:=}} \left\{ \begin{array}{ll} \left( {x, y}\right) \in \{ 0,1{\} }^{6} \times {\mathbb{R}}_{ + }^{6} : & {y}_{1} + {y}_{2} + {y}_{3} + {y}_{4} + {y}_{5} + {y}_{6} \leq {20} \\ & {y}_{1} \leq {17}{x}_{1},{y}_{2} \leq 9{x}_{2},{y}_{3} \leq 8{x}_{3} \\ & {y}_{4} \l... | Consider the flow cover \( C \mathrel{\text{:=}} \{ 3,4,5,6\} \) . Note that \( {\mu }_{1} = 8,{\mu }_{2} = {14} \) , \( {\mu }_{3} = {19},{\mu }_{4} = {23},\lambda = 3 \) and \( r = 4 \) . For \( {a}_{1} = {17} \), Case (ii) of the theorem holds for \( h = 2 \) and \( h = 3 \), and Case (iii) holds for \( \ell = 3 \) ... | Yes |
Theorem 7.18. The affine hull of the traveling salesman polytope on \( n \geq 3 \) nodes is \( \left\{ {x \in {\mathbb{R}}^{\left( \begin{matrix} n \\ 2 \end{matrix}\right) } : \mathop{\sum }\limits_{{e \in \delta \left( i\right) }}{x}_{e} = 2}\right\} \) . Furthermore, \( \dim \left( {P}_{\mathrm{{tsp}}}\right) = \lef... | Proof. Note that every point in \( {P}_{\text{tsp }} \) must satisfy the \( n \) degree constraints \( \mathop{\sum }\limits_{{e \in \delta \left( i\right) }}{x}_{e} = 2 \) for \( i \in V \) . We first note that such constraints are linearly independent. Indeed, let \( {Ax} = \mathbf{2} \) be the system formed by the \... | Yes |
Theorem 7.19. For \( S \subset V \) with \( 2 \leq \left| S\right| \leq n - 2 \), the subtour elimination constraint \( \mathop{\sum }\limits_{{e \in \delta \left( S\right) }}{x}_{e} \geq 2 \) defines a facet of the traveling salesman polytope on \( n \geq 4 \) nodes. | Proof. Given \( S \subset V,2 \leq \left| S\right| \leq n - 2 \), let \( F \) be the face defined by \( \mathop{\sum }\limits_{{e \in \delta \left( S\right) }}{x}_{e} \geq 2 \) . Then there exists some valid inequality \( {\alpha x} \leq \beta \) for \( {P}_{\text{tsp }} \) which defines a facet \( \bar{F} \) such that... | Yes |
Proposition 7.20. The comb inequality (7.23) is valid for the traveling salesman polytope. | Proof. We show that (7.23) is a Chvátal inequality for the subtour elimination polytope. Consider the following inequalities, valid for the subtour elimination polytope.\n\n\[ \mathop{\sum }\limits_{{e \in \delta \left( v\right) }}{x}_{e} = 2\;v \in {S}_{0}; \]\n\n\[ \begin{matrix} - {x}_{e} & \leq & 0 & & & & e \in \d... | Yes |
Proposition 7.21. If \( \bar{y} \notin \operatorname{conv}\left( {S}_{i}\right) \), an inequality ay \( \leq b \) separating \( \bar{y} \) from \( \operatorname{conv}\left( {S}_{i}\right) \) can be found by solving a linear program. | Proof. If \( \bar{y} \notin \operatorname{conv}\left( {S}_{i}\right) \), the linear program\n\n\[ \min \;0 \]\n\n\[ \mathop{\sum }\limits_{{h = 1}}^{i}{y}^{h}{u}_{h}\; = \bar{y} \]\n\n\[ \mathop{\sum }\limits_{{h = 1}}^{i}{u}_{h}\; = 1 \]\n\n\[ u \geq 0 \]\n\nhas no solution. Therefore its dual\n\n\[ \max \;a\bar{y} - ... | Yes |
Theorem 7.26. For well-described polyhedra, the separation problem is solvable in polynomial time if and only if the optimization problem is. | Proof. We give the proof for the case when \( P \) is full-dimensional and bounded. The proof is more complicated when \( P \) is not full-dimensional, and we refer the reader to [188] in that case.\n\n\ | No |
Theorem 7.27 (Grötschel et al. [186]). There exists an oracle-polynomial time algorithm that solves the weak optimization problem for every circumscribed convex set \( \\left( {K;n, R}\\right) \) given by a weak separation oracle and every choice of \( c \\in {\\mathbb{Q}}^{n} \) and \( \\varepsilon > 0 \) . | There exists an oracle-polynomial time algorithm that solves the weak separation problem for every circumscribed convex set \( \\left( {K;n, R}\\right) \) given by a weak optimization oracle and every choice of \( y \\in {\\mathbb{Q}}^{n} \) and \( \\delta > 0 \) . The equivalence hinges on an approximate version of th... | No |
Proposition 8.1. \( {z}_{LR}\left( \lambda \right) \geq {z}_{I} \) for every \( \lambda \in {\mathbb{R}}_{ + }^{{m}_{1}} \) . | Proof. The result holds when \( {z}_{I} = - \infty \), so we may assume that a feasible solution of (8.1) exists. Let \( \bar{x} \) be any feasible solution of (8.1). Since \( \bar{x} \in Q \) , \( \bar{x} \) is feasible to \( \operatorname{LR}\left( \lambda \right) \) and since \( {A}_{1}\bar{x} \leq {b}^{1} \) and \(... | Yes |
Theorem 8.2. Assume \( \left\{ {x : {A}_{1}x \leq {b}^{1}, x \in \operatorname{conv}\left( Q\right) }\right\} \neq \varnothing \) . Then \( {z}_{LD} = \) \( \max \left\{ {{cx} : {A}_{1}x \leq {b}^{1}, x \in \operatorname{conv}\left( Q\right) }\right\} . \) | Proof. Since \( {A}_{2}x \leq {b}^{2} \) is a rational system, by Meyer’s theorem (Theorem 4.30), \( \operatorname{conv}\left( Q\right) \) is a rational polyhedron. Let \( {Cx} \leq d \) be a system of linear inequalities such that \( \operatorname{conv}\left( Q\right) = \left\{ {x \in {\mathbb{R}}^{n} : {Cx} \leq d}\r... | Yes |
Corollary 8.3. The function \( {z}_{LR} \) defined in (8.2) is a piecewise linear convex function of \( \lambda \) over its domain. | Proof. By (8.4) the function \( {z}_{LR} \) is the maximum of a finite number of affine functions, therefore it is convex and piecewise linear. | Yes |
Corollary 8.4. \( {z}_{I} \leq {z}_{LD} \leq {z}_{LP} \) | Proof. \( \operatorname{conv}\left( S\right) \subseteq \operatorname{conv}\left( Q\right) \cap \left\{ {x \in {\mathbb{R}}_{ + }^{n} : {A}_{1}x \leq {b}^{1}}\right\} \subseteq \left\{ {x \in {\mathbb{R}}_{ + }^{n} : {Ax} \leq b}\right\} \). Maximizing the linear function \( {cx} \) over these three sets gives the desir... | Yes |
Corollary 8.5. \( {z}_{LD} = {z}_{LP} \) for all \( c \in {\mathbb{R}}^{n} \) if \( \operatorname{conv}\left( Q\right) = \left\{ {x \in {\mathbb{R}}_{ + }^{n} : {A}_{2}x \leq {b}^{2}}\right\} \) . | In particular, for pure integer programs, when \( {A}_{2} \) is totally unimodular and \( {b}^{2} \) is an integral vector, we have \( {z}_{LD} = {z}_{LP} \), i.e., the Lagrangian dual bound is no better that the usual linear programming relaxation bound. | No |
Proposition 8.8. For any given \( \lambda \in {\mathbb{R}}^{m} \), the following hold.\n\n(i) \( {z}_{LR}\left( \lambda \right) = \mathop{\sum }\limits_{j}{\left( \mathop{\sum }\limits_{i}{\left( {c}_{ij} - {\lambda }_{i}\right) }^{ + } - {f}_{j}\right) }^{ + } + \mathop{\sum }\limits_{i}{\lambda }_{i} \).\n\n(ii) \( {... | Proof. (i) follows from Proposition 8.7. (ii) follows from Corollary 8.5 and the fact that the constraints of (8.6) are totally unimodular. | No |
Proposition 8.9. The 1-tree polytope of \( G \) is described by the linear relaxation of the constraints in (8.7). | from \( \mathop{\sum }\limits_{{e \in E}}{x}_{e} = \left| V\right| \), one obtains the degree constraint \( \mathop{\sum }\limits_{{e \in \delta \left( 1\right) }}{x}_{e} = 2 \)\n\n\( \left| V\right| - 2 \), which is expressed only in terms of the variables relative to edges of \( G \smallsetminus \{ 1\} \) . It follow... | Yes |
Proposition 8.10. If \( {x}^{ * } \) is an optimal solution of \( {LR}\left( {\lambda }^{ * }\right) \) as defined in (8.2), then \( {s}^{ * } \mathrel{\text{:=}} {b}^{1} - {A}_{1}{x}^{ * } \) is a subgradient of the function \( {z}_{LR} \) at point \( {\lambda }^{ * } \) . | Proof. For all \( \lambda \geq 0 \) ,\n\n\[ \n{z}_{LR}\left( \lambda \right) \geq c{x}^{ * } + \lambda \left( {{b}^{1} - {A}_{1}{x}^{ * }}\right) = c{x}^{ * } + {\lambda }^{ * }\left( {{b}^{1} - {A}_{1}{x}^{ * }}\right) + \left( {\lambda - {\lambda }^{ * }}\right) \left( {{b}^{1} - {A}_{1}{x}^{ * }}\right) \n\]\n\n\[ \... | Yes |
Theorem 8.11 (Poljak [310]). Assume that problem (8.9) has finite value \( {g}^{ * } \), and that the length of all subgradients of \( g \) is bounded by a constant \( S \in {\mathbb{R}}_{ + } \) . If the sequence \( {\left( {\alpha }_{t}\right) }_{t = 1}^{\infty } \) converges to 0 and \( \mathop{\sum }\limits_{{t = 1... | Proof. The key to the proof is to study the Euclidean distance of the current point \( \lambda \) to an optimal solution \( {\lambda }^{ * } \) of (8.9).\n\n\[ \parallel {\lambda }^{t + 1} - {\lambda }^{ * }{\parallel }^{2}\; = \;\parallel {\mathrm{{proj}}}_{P}\left( {{\lambda }^{t} - {\alpha }_{t}{s}^{t}}\right) - {\l... | Yes |
Consider the Lagrangian relaxation of the traveling salesman problem proposed in (8.7). As discussed in Sect. 8.1.1, the bound provided by (8.7) is equal to\n\n\\[ \min \\;\\mathop{\\sum }\\limits_{{e \\in E}}{c}_{e}{x}_{e} \\]\n\n\\[ \\mathop{\\sum }\\limits_{{e \\in \\delta \\left( i\\right) }}{x}_{e} = 2\\;i \\in V ... | ∎ | No |
Consider the cutting stock problem discussed in Sect. 2.3, where we need to cut at least \( {b}_{i} \) rolls of width \( {w}_{i} \) out of rolls of width \( W \) , \( i = 1,\ldots, m \), while minimizing the number of rolls of width \( W \) used. Consider the formulation (2.1), where \( p \) is an upper bound on the nu... | \[ \min \;\mathop{\sum }\limits_{{j = 1}}^{p}{y}_{j} \] \[ \begin{matrix} \mathop{\sum }\limits_{{i = 1}}^{m}{w}_{i}{z}_{ij} & \leq & W{y}_{j} & j = 1,\ldots, p \end{matrix} \] \[ \mathop{\sum }\limits_{{j = 1}}^{p}{z}_{ij}\; \geq \;{b}_{i}\;i = 1,\ldots, m \] (8.19) \[ {y}_{j} \in \{ 0,1\} \;j = 1,\ldots, p \] \[ {z}_... | Yes |
In Example 8.15, at each iteration the master problem is defined by a subset \( {\mathcal{S}}^{\prime } \subseteq \mathcal{S} \) of the patterns.\n\n\[ \min \;\mathop{\sum }\limits_{{a \in {\mathcal{S}}^{\prime }}}{x}_{a} \]\n\n\[ \mathop{\sum }\limits_{{a \in {\mathcal{S}}^{\prime }}}{a}_{i}{x}_{a}\; \geq {b}_{i}\;\ma... | Given an optimal solution \( \bar{\pi } \in {\mathbb{R}}^{m} \) to the dual of the master problem, namely,\n\n\[ \max \;\mathop{\sum }\limits_{{i = 1}}^{m}{b}_{i}{\pi }_{i} \]\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{a}_{i}{\pi }_{i}\; \leq 1\;\text{for}\;a \in {\mathcal{S}}^{\prime } \]\n\n\[ \pi \geq 0 \]\n\nthe pri... | Yes |
We consider the Dantzig-Wolfe relaxation of the generalized assignment problem discussed in Example 8.14. When solving by column generation, at each iteration the master problem is defined by subsets \( {S}_{j} \subseteq {Q}_{j}, j = 1,\ldots, n \) . | \[ \max \;\mathop{\sum }\limits_{j}\mathop{\sum }\limits_{{v \in {S}_{j}}}\left( {\mathop{\sum }\limits_{i}{c}_{ij}{v}_{i}}\right) {\lambda }_{v}^{j} \] \[ \begin{matrix} \mathop{\sum }\limits_{j}\mathop{\sum }\limits_{{v \in {S}_{j}}}{v}_{i}{\lambda }_{v}^{j} & \leq & 1 & i = 1,\ldots, m \end{matrix} \] \[ \mathop{\su... | Yes |
Theorem 8.18 (Benders). Problem (8.26) can be reformulated as\n\n\\[ \n{z}_{I} = \\max \\;\\eta + {cx} \n\\]\n\n\\[ \n\\eta \\leq {u}^{k}\\left( {b - {Ax}}\\right) \\;\\text{ for all }k \\in K \n\\]\n\n(8.27)\n\n\\[ \n{r}^{j}\\left( {b - {Ax}}\\right) \\geq 0\\;\\text{ for all }j \\in J \n\\]\n\n\\[ \nx \\in X,\\;\\eta... | Proof. Let \\( P \\mathrel{\\text{:=}} \\left\\{ {\\left( {x, y}\\right) \\in {\\mathbb{R}}^{n} \\times {\\mathbb{R}}^{p} : {Ax} + {Gy} \\leq b, y \\geq 0}\\right\\} \\) . Then (8.26) can be rewritten as\n\n\\[ \n{z}_{I} = \\max \\left\\{ {{cx} + {z}_{LP}\\left( x\\right) : x \\in {\\operatorname{proj}}_{x}\\left( P\\r... | Yes |
Theorem 9.1. Two nonsingular square matrices \( A \) and \( B \) generate the same lattice if and only if there exists a unimodular matrix \( U \) such that \( B = {AU} \) . | Proof. Assume \( B = {AU} \) for some unimodular matrix \( U \) . Since \( U \) is integral, by definition every column of \( B \) is in the lattice generated by \( A \) . On the other hand, \( A = B{U}^{-1} \) and, by Lemma \( {1.16},{U}^{-1} \) is unimodular. So every column of \( A \) is in the lattice generated by ... | Yes |
Theorem 9.2. Let \( B = \left( {{b}^{1},\ldots ,{b}^{n}}\right) \) be a square nonsingular matrix. If \( B \) is reduced, then \( \begin{Vmatrix}{b}^{1}\end{Vmatrix}\cdots \begin{Vmatrix}{b}^{n}\end{Vmatrix} \leq {2}^{n\left( {n - 1}\right) /4}\det \left( B\right) \) . | Proof. Let \( {g}^{1},\ldots ,{g}^{n} \) be the Gram–Schmidt orthogonalization of \( {b}^{1},\ldots ,{b}^{n} \) . Since \( {g}^{1},\ldots ,{g}^{n} \) are pairwise orthogonal, it follows that \( {\begin{Vmatrix}{g}^{j} + {\mu }_{j, j - 1}{g}^{j - 1}\end{Vmatrix}}^{2} = \) \( {\begin{Vmatrix}{g}^{j}\end{Vmatrix}}^{2} + {... | Yes |
Lemma 9.3. The Gram-Schmidt basis remains unchanged in Step 1 of the basis reduction algorithm. | Proof. Let \( \widetilde{B} \) be the basis obtained from \( B \) after Step 1. Since \( {\widetilde{b}}^{j} \) equals \( {b}^{j} \) plus a linear combination of \( {b}^{1},\ldots ,{b}^{j - 1} \), it follows that \( {b}^{1},\ldots ,{b}^{j} \) and \( {\widetilde{b}}^{1},\ldots ,{\widetilde{b}}^{j} \) generate the same v... | Yes |
Theorem 9.4. When the basis reduction algorithm terminates, the basis \( {b}^{1},\ldots ,{b}^{n} \) is reduced. | Proof. The algorithm stops in Step 2 when Condition (9.4)(ii) is satisfied. To show that the basis is reduced, it suffices to show that, at each iteration, at the end of Step 1 the current basis always satisfies Condition (9.4)(i).\n\nBy Lemma 9.3, the Gram-Schmidt basis \( {g}^{1},\ldots ,{g}^{n} \) remains unchanged ... | Yes |
Lemma 9.11. Let \( A \in {\mathbb{Q}}^{m \times n}, b \in {\mathbb{Q}}^{m},\alpha \in {\mathbb{Z}}^{n},\beta \in \mathbb{Z} \) be such that the entries of \( \alpha \) are relatively prime. There exists a matrix \( D \in {\mathbb{Z}}^{n \times \left( {n - 1}\right) } \) and a vector \( {b}^{\prime } \in {\mathbb{Q}}^{m... | Proof. Since all entries of \( \alpha \) are relatively prime, by Corollary 1.9, \( {\alpha x} = \beta \) has an integral solution \( \bar{x} \), and there exists a unimodular matrix \( U \) such that \( {\alpha U} = {e}^{1} \), where \( {e}^{1} \) denotes the first unit vector in \( {\mathbb{R}}^{n} \) . If we define ... | Yes |
Proposition 9.13. If node \( {N}_{a} \) has an isomorphic node to its left in the enumeration tree, then node \( {N}_{a} \) is pruned by isomorphism. | Proof. Suppose \( {N}_{a} \) has an isomorphic node \( {N}_{b} \) to its left. Then there exists a permutation \( \pi \in \Gamma \) such that \( \pi \left( {F}_{b}^{1}\right) = {F}_{a}^{1} \) and \( \pi \left( {F}_{b}^{0}\right) = {F}_{a}^{0} \) . Let \( {N}_{d} \) be the common ancestor of \( {N}_{a} \) and \( {N}_{b}... | Yes |
We illustrate orbital fixing on the following 0,1 linear program, where \( n \geq 3 \) .\n\n\[ \max \;\mathop{\sum }\limits_{{j = 1}}^{n}{x}_{j} \]\n\n\[ {x}_{i} + {x}_{j} \leq 1\;\text{ for all }i, j \]\n\n(9.15)\n\n\[ x \in \{ 0,1{\} }^{n}\text{.} \] | A branch-and-bound algorithm first solves the linear programming relaxation. The optimum of this linear program puts all variables at \( \frac{1}{2} \) for an objective value of \( \frac{n}{2} \) . Branching is done on variable \( {x}_{1} \), say. The branch where \( {x}_{1} \) is fixed to 1 can be pruned by integralit... | Yes |
Theorem 10.1. If the primal problem (10.1) (resp. the dual problem (10.2)) is strictly feasible and bounded, then the dual (resp. the primal) problem admits an optimal solution, and the optimal values of (10.1) and (10.2) coincide. | See Theorem 1.4.2 (3a) in [49] for a proof. | No |
Proposition 10.2. Let \( A \in {\mathbb{R}}^{n \times n} \) be a symmetric matrix. The following are equivalent.\n\n(i) \( A \) is positive semidefinite.\n\n(ii) There exists \( U \in {\mathbb{R}}^{d \times n} \), for some \( d \leq n \), such that \( A = {U}^{T}U \) .\n\n(iii) All principal submatrices of \( A \) have... | The proof can be found in [207] for example. | No |
Theorem 10.3 (Goemans and Williamson [173]). \( \frac{{z}_{I}}{{z}_{\text{sdp }}} > {0.87856} \). | Proof. Consider an optimal solution \( Y \) to the semidefinite relaxation. Since \( Y \succcurlyeq 0 \), by Proposition 10.2 we can write \( Y = {U}^{T}U \) where \( U \) is a \( d \times n \) matrix for some \( d \leq n \). Let \( {u}_{j} \in {\mathbb{R}}^{d} \) denote the \( j \) th column of \( U \). Note that \( {... | Yes |
Theorem 10.4. For any graph \( G,\operatorname{STAB}\left( G\right) \subseteq \operatorname{TH}\left( G\right) \subseteq \operatorname{QSTAB}\left( G\right) \) . | Proof. Let \( x \) be the characteristic (column) vector of a stable set. Define \( Y \mathrel{\text{:=}} \left( \begin{array}{l} 1 \\ x \end{array}\right) \left( {1{x}^{T}}\right) \) . Then \( Y \) satisfies all the properties needed in (10.4). In particular \( {y}_{jj} = {x}_{j}^{2} = {x}_{j} \) since \( {x}_{j} = 0 ... | Yes |
Let \( P \mathrel{\text{:=}} \left\{ {x \in {\left\lbrack 0,1\right\rbrack }^{4} : {x}_{1} - 2{x}_{2} + 4{x}_{3} + 5{x}_{4} \geq 3}\right\} \) , and \( S \mathrel{\text{:=}} P \cap {\mathbb{Z}}^{4} \) . Constructing a nonlinear system as in Step 1 of the Lovász-Schrijver procedure and then linearizing, we obtain | \[ {x}_{1}\left( {{x}_{1} - 2{x}_{2} + 4{x}_{3} + 5{x}_{4} - 3}\right) \geq 0 \] \[ - 2{x}_{1} - 2{y}_{12} + 4{y}_{13} + 5{y}_{14} \geq 0 \] \[ 3{x}_{1} - 2{x}_{2} + 4{x}_{3} + 5{x}_{4} + 2{y}_{12} - 4{y}_{13} - 5{y}_{14} \geq 3 \] \[ {x}_{2}\left( {{x}_{1} - 2{x}_{2} + 4{x}_{3} + 5{x}_{4} - 3}\right) \geq 0 \] \[ - 5{... | Yes |
The relaxation \( \mathrm{{TH}}\left( G\right) \) of the stable set polytope defined in Sect. 10.2.2 is related to the Lovász-Schrijver relaxation \( {N}_{ + }\left( {\operatorname{FRAC}\left( G\right) }\right) \) : Let us apply the Lovász-Schrijver procedure to \( P \mathrel{\text{:=}} \operatorname{FRAC}\left( G\righ... | For every \( i, j \in V \), linearizing \( {x}_{i}{x}_{j} \geq 0 \) we obtain \( {y}_{ij} \geq 0 \) . For every \( {ij} \in E \), linearizing \( {x}_{i}\left( {1 - {x}_{i} - {x}_{j}}\right) \geq 0 \) we obtain \( {y}_{ij} \leq 0 \), thus implying \( {y}_{ij} = 0 \) for all \( {ij} \in E \) . Therefore \( {N}_{ + }\left... | No |
Lemma 10.8. \( N \subseteq { \cap }_{j = 1}^{n}{P}_{j} \subseteq P \) | Proof. The linear inequalities defining \( {Q}_{j} \) are a subset of those defining \( M \) . Therefore \( N \subseteq {P}_{j} \) for \( j = 1,\ldots, n \) .\n\nThe inclusion \( {P}_{j} \subseteq P \) follows by observing that the inequalities \( {Ax} \geq b \) can be obtained by summing up the constraints defining \(... | No |
Theorem 10.9. \( {P}_{j} = \operatorname{conv}\left\{ {\left( {P \cap \left\{ {x : {x}_{j} = 0}\right\} }\right) \cup \left( {P \cap \left\{ {x : {x}_{j} = 1}\right\} }\right) }\right\} \) | Proof. The linear system produced at Step 2 of the lift-and-project procedure is\n\n\[ A{x}^{1} \geq {\lambda b} \]\n\n\[ {x}_{j}^{1}\; = \;\lambda \]\n\n\[ A{x}^{2} \geq \left( {1 - \lambda }\right) b \]\n\n\[ {x}_{j}^{2} = 1 - \lambda \]\n\n\[ {x}^{1} + {x}^{2} = x \]\n\n\[ 0 \leq \lambda \leq 1 \]\n\nwhere we introd... | Yes |
Theorem 10.10. \( P \supseteq {N}^{1} \supseteq {N}^{2} \supseteq \ldots \supseteq {N}^{n} = \operatorname{conv}\left( S\right) \) | Proof. The inclusions follow from Lemma 10.7. As a consequence of Lemma 10.8, we have \( {N}^{1} \subseteq {P}_{1},{N}^{2} \subseteq {P}_{2}\left( {P}_{1}\right) \) , \( \ldots ,{N}^{n} \subseteq {P}_{n}\left( {\ldots {P}_{2}\left( {P}_{1}\right) }\right) \). By Theorem 10.9, \( {P}_{j} = \operatorname{conv}\left\{ \le... | Yes |
Theorem 10.11. \( {S}_{t} \subseteq {N}^{t} \) . | Proof. We already observed that \( {S}_{1} = {N}^{1} \) . We prove the theorem by induction. Assume \( {S}_{t - 1} \subseteq {N}^{t - 1} \) for some \( t \geq 2 \) . We have \( N\left( {S}_{t - 1}\right) \subseteq {N}^{t} \) . Therefore, to prove the theorem, it suffices to show that \( {S}_{t} \subseteq N\left( {S}_{t... | Yes |
Proposition 10.13. Let \( y \in {K}_{t} \) . Then\n\n(i) Given \( I, J \subseteq \{ 1,\ldots, n\} \) such that \( \left| I\right| ,\left| J\right| \leq t + 1 \), if \( {y}_{I} = 0 \), then \( {y}_{I \cup J} = 0.\n\n(ii) Given \( I, J \subseteq \{ 1,\ldots, n\} \) such that \( \left| I\right| ,\left| J\right| \leq t + 1... | Proof. (i) If \( {y}_{I} = 0 \), then the \( 2 \times 2 \) principal submatrix of \( {M}_{t + 1}\left( y\right) \) indexed by \( I, J \) has determinant \( - {y}_{I \cup J}^{2} \), which must be nonnegative since \( {M}_{t + 1}\left( y\right) \succcurlyeq 0 \) . Thus \( {y}_{I \cup J} = 0 \).\n\n(ii) Let \( I, J \subse... | Yes |
Theorem 10.14. \( {L}_{t} \subseteq {N}_{ + }^{t} \) . | Proof. We already observed that \( {L}_{1} \subseteq {N}_{ + } \) . To prove \( {L}_{t} \subseteq {N}_{ + }^{t} \) for \( t \geq 2 \), we will show by induction the stronger claim that \( {L}_{t} \subseteq {N}_{ + }\left( {L}_{t - 1}\right) \) . Let \( \widehat{x} \in {L}_{t} \) . Then there exists \( \widehat{y} \in {... | Yes |
Lemma 10.16. Let \( S \) be a finite set. Then \( \mathop{\sum }\limits_{\substack{{H, K \subseteq S} \\ {H \cup K = S} }}{\left( -1\right) }^{\left| H\right| + \left| K\right| } = {\left( -1\right) }^{\left| S\right| } \) . | Proof. By induction on \( \left| S\right| \), the statement being trivial for \( \left| S\right| = 0 \) . Let \( S \neq \varnothing \), and let \( a \in S \) . Let \( \mathcal{F} \mathrel{\text{:=}} \{ \left( {H, K}\right) : H \cup K = S \smallsetminus \{ a\} \} \) . Then\n\n\[ \mathop{\sum }\limits_{\substack{{H, K \s... | Yes |
Theorem 10.17 (Hilbert’s Nullstellensatz). Let \( \mathbb{K} \) be a finite field and \( \overline{\mathbb{K}} \) its algebraic closure. Given \( {f}_{1},\ldots ,{f}_{m} \in \mathbb{K}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \), the system of polynomial equations \( {f}_{1}\left( x\right) = 0,\ldots ,{f}_{m}... | Note that the \ | No |
Theorem 1.3 (Szemerédi). If a set \( A \subseteq \mathbb{N} \) has upper density\n\n\[ \overline{\mathrm{d}}\left( A\right) \mathrel{\text{:=}} \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{\operatorname{card}\left( {A\cap \{ 1,\ldots, n\} }\right) }{n} > 0, \]\n\nthen it contains arbitrarily long arithmetic ... | A complete proof of this theorem remains beyond the reach of this book. However, we shall provide some of the necessary tools. | No |
Example 2.4 (Finite-Dimensional Contractions). Let \( \parallel \cdot \parallel \) be a norm on \( {\mathbb{R}}^{d} \) and let \( T : {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{d} \) be linear and contractive with respect to the chosen norm, i.e., \( \parallel {Tx}\parallel \leq \parallel x\parallel, x \in {\mathbb{R}}... | As a more concrete example we choose the norm \( \parallel x{\parallel }_{\infty } = \max \left\{ {\left| {x}_{1}\right| ,\ldots ,\left| {x}_{d}\right| }\right\} \) on \( {\mathbb{R}}^{d} \) and the linear operator \( T \) given by a row-substochastic matrix \( {\left( {t}_{ij}\right) }_{i, j = 1,\ldots, d} \) , i.e., ... | Yes |
Example 2.6 (Cantor System). The Cantor set is\n\n\[ \nC = \\left\\{ {x \\in \\left\\lbrack {0,1}\\right\\rbrack : x = \\mathop{\\sum }\\limits_{{j = 1}}^{\\infty }\\frac{{a}_{j}}{{3}^{j}},{a}_{j} \\in \\{ 0,2\\} }\\right\\} ,\n\]\n\n(2.1)\n\ncf. Appendix A.8. As a closed subset of the unit interval, the Cantor set \( ... | The continuity of \( \\varphi \) is clear, and a close inspection using (2.1) reveals that \( \\varphi \) maps \( C \) to itself. Hence, \( \\left( {C;\\varphi }\\right) \) is a topological system, called the Cantor system. | No |
Consider the interval \( K \mathrel{\text{:=}} \lbrack 0,1) \) and define\n\n\[ d\left( {x, y}\right) \mathrel{\text{:=}} \left| {{\mathrm{e}}^{{2\pi }\mathrm{i}x} - {\mathrm{e}}^{{2\pi }\mathrm{i}y}}\right| \;\left( {x, y \in \lbrack 0,1}\right) ). \]\n\nBy Exercise \( 3, d \) is a metric on \( K \), continuous with r... | By Exercise \( 3, d \) is a metric on \( K \), continuous with respect to the standard one, and turning \( K \) into a compact metric space. By Exercise \( 3,\varphi \) is continuous with respect to the metric \( d \), hence it gives rise to a topological system on \( \lbrack 0,1) \) . | No |
Example 2.12 (Group Rotation on \( \mathbb{R}/\mathbb{Z} \) ). Consider the additive group \( \mathbb{R} \) with the standard topology and its closed (normal) subgroup \( \mathbb{Z} \) . The homogeneous space \( \left( { = \text{factor group}}\right) \mathbb{R}/\mathbb{Z} \) is compact since \( \mathbb{R}/\mathbb{Z} = ... | \[ q\left( x\right) \mathrel{\text{:=}} x + \mathbb{Z},\;\mathbb{R} \rightarrow \mathbb{R}/\mathbb{Z} \] is the canonical homomorphism. Each \( \alpha \in \mathbb{R} \) acts by translation \( x + \mathbb{Z} \mapsto \alpha + x + \mathbb{Z} \) and gives rise to the homogeneous (viz. group rotation) system \( \left( {\mat... | Yes |
Example 2.13 (Heisenberg System). In the following we describe an important example of a homogeneous system which is in general not a group rotation. The non-Abelian group \( G \) of upper triangular real matrices with all diagonal entries equal to 1 , i.e., | \[ G = \left\{ {\left( \begin{array}{lll} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array}\right) : x, y, z \in \mathbb{R}}\right\} \] is called the Heisenberg group. We introduce the notation \[ \left\lbrack {x, y, z}\right\rbrack \mathrel{\text{:=}} \left( \begin{array}{lll} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{ar... | Yes |
Consider a left group rotation system \( \left( {G;a}\right) \) . Then any right rotation \( {\rho }_{h}, h \in G \), is an automorphism of \( \left( {G;a}\right) \). | \[ {\rho }_{h}\left( {a \cdot g}\right) = \left( {ag}\right) h = a\left( {gh}\right) = a \cdot \left( {{\rho }_{h}\left( g\right) }\right) \;\left( {g \in G}\right) . \] | Yes |
For \( \alpha \in \lbrack 0,1) \) let \( a \mathrel{\text{:=}} {\mathrm{e}}^{{2\pi }\mathrm{i}\alpha } \) . Then the three topological systems\n\n1) \( \left( {\lbrack 0,1}\right) ;\alpha ) \) from Example 2.7,\n\n2) \( \left( {\mathbb{T};a}\right) \) from Example 2.8,\n\n3) and \( \left( {\mathbb{R}/\mathbb{Z};\alpha ... | Proof. The (well-defined!) map\n\n\[ \Phi : \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{T},\;x + \mathbb{Z} \mapsto {\mathrm{e}}^{{2\pi }\mathrm{i}x} \]\n\nis a group isomorphism. It is continuous since \( \Phi \circ q \) is continuous and \( q : \mathbb{R} \rightarrow \mathbb{R}/\mathbb{Z} \) is a quotient map (Appendix... | No |
Consider a group rotation \( \left( {G;a}\right) \) and let \( \Gamma \) be a closed subgroup of \( G \) . The equivalence relation\n\n\[ x \sim y\;\overset{\text{ Def. }}{ \Leftrightarrow }\;{y}^{-1}x \in \Gamma \]\n\nis a congruence since \( {\left( ay\right) }^{-1}\left( {ax}\right) = {y}^{-1}{a}^{-1}{ax} = {y}^{-1}... | The set of corresponding equivalence classes is simply the homogeneous space\n\n\[ G/\Gamma = \{ {g\Gamma } : g \in G\} \]\n\nof left cosets, and the induced dynamics on it is given by \( {g\Gamma } \mapsto {ag\Gamma } \) . In this way we recover the homogeneous system \( \left( {G/\Gamma ;a}\right) \), cf. Example 2.1... | Yes |
Lemma 2.25. Let \( \left( {K;\varphi }\right) \) be a topological system and let \( A \subseteq K \) . Then the following assertions hold:\n\na) If \( A \) is bi-invariant or stable, then \( A \) is invariant.\n\nb) If \( A \) is stable and \( \varphi \) is injective, then \( A \) is bi-invariant.\n\nc) If \( A \) is b... | Proof. This is Exercise 9. | No |
Lemma 2.26. Suppose that \( \left( {K;\varphi }\right) \) is a topological system and \( \varnothing \neq A \subseteq K \) is closed and invariant. Then there is a nonempty, closed set \( B \subseteq A \) such that \( \varphi \left( B\right) = B \) . | Proof. Since \( A \subseteq K \) is invariant,\n\n\[ A \supseteq \varphi \left( A\right) \supseteq {\varphi }^{2}\left( A\right) \supseteq \cdots \supseteq {\varphi }^{n}\left( A\right) \;\text{ holds for all }n \in \mathbb{N}. \]\n\nAll these sets are compact and nonempty since \( A \) is closed and \( \varphi \) is c... | Yes |
Example 2.28 (Minimal Invertible Extension). Let \( \left( {K;\varphi }\right) \) be a surjective system and consider the infinite product \( {K}^{\infty } \mathrel{\text{:=}} \mathop{\prod }\limits_{{j \in \mathbb{N}}}K \) together with the map\n\n\[ \psi : {K}^{\infty } \rightarrow {K}^{\infty },\;\psi \left( {{x}_{1... | The extension \( \pi : \left( {L;\psi }\right) \rightarrow \left( {K;\varphi }\right) \) is called the (minimal) invertible extension of \( \left( {K;\varphi }\right) \) . See Exercise 19 for more information. | No |
We claim that each subsystem of \( \left( {{\mathcal{W}}_{k}^{ + };\tau }\right) \) arises in this way. | To prove this claim, let \( \left( {F;\tau }\right) \) be a subsystem and consider the set \( B \) of finite sequences that are not present in any of the words in \( F \), i.e., \[ B \mathrel{\text{:=}} \{ y : y\text{is a finite sequence and not contained in any}x \in F\} \text{.} \] We have \( F \subseteq {\mathcal{W}... | Yes |
Proposition 2.33. Let \( \left( {K;\varphi }\right) \) be a topological system and consider the following assertions:\n\n(i) \( \left( {K;\varphi }\right) \) is forward transitive, i.e., there is a point \( x \in K \) with \( \overline{{\mathrm{{orb}}}_{ + }}\left( x\right) = K \) .\n\n(ii) For all open sets \( U, V \n... | Proof. The proof of the equivalence of (ii) and (iii) is left to the reader.\n\n(i) \( \Rightarrow \) (ii): Suppose that \( K \) has no isolated points and that \( x \in K \) has dense forward orbit. Let \( U, V \) be nonempty open subsets of \( K \) . Then certainly \( {\varphi }^{k}\left( x\right) \in U \) for some \... | No |
Theorem 2.36 (Rotation Systems). Let \( \left( {G;a}\right) \) be a left rotation system. Then the following statements are equivalent:\n\n(i) \( \left( {G;a}\right) \) is topologically forward transitive.\n\n(ii) Every point of \( G \) has dense forward orbit.\n\n(iii) \( \left( {G;a}\right) \) is topologically transi... | Proof. Since every right rotation \( {\rho }_{h} : g \mapsto {gh} \) is an automorphism of \( \left( {G;a}\right) \), we have\n\n\[{\rho }_{h}\left( {{\operatorname{orb}}_{ + }\left( g\right) }\right) = {\operatorname{orb}}_{ + }\left( {gh}\right) \;\text{ and }\;{\rho }_{h}\left( {\operatorname{orb}\left( g\right) }\r... | Yes |
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