Partial hyperplane activation for generalized intersection cuts

Abstract

The generalized intersection cut paradigm is a recent framework for generating cutting planes in mixed integer programming with attractive theoretical properties. We investigate this computationally unexplored paradigm and observe that a key hyperplane activation procedure embedded in it is not computationally viable. To overcome this issue, we develop a novel replacement to this procedure called partial hyperplane activation (PHA), introduce a variant of PHA based on a notion of hyperplane tilting, and prove the validity of both algorithms. We propose several implementation strategies and parameter choices for our PHA algorithms and provide supporting theoretical results. We computationally evaluate these ideas in the COIN-OR framework on MIPLIB instances. Our findings shed light on the the strengths of the PHA approach as well as suggest properties related to strong cuts that can be targeted in the future.

Appendices

A Additional theory for PHA

1.1 A.1 Existence of strictly dominating GICs

In this section, we provide some theoretical motivation for Algorithm 3 by giving necessary and sufficient conditions for the existence of a GIC that strictly dominates the SIC after activation of a single hyperplane.

Definition 16

([7]) Consider two inequalities that are valid for $P_I$ but not necessarily P. Inequality 2 dominates 1 on P if for every $x \in P$, the fact that x satisfies Inequality 2 implies that x satisfies Inequality 1. Inequality 2 strictly dominates 1 if, in addition, there exists $x \in P$ such that x violates Inequality 2 but satisfies Inequality 1.

The theorem proved in this section strengthens Theorem 5 in Balas and Margot [7] for the case when S is a split disjunction. Theorem 5 of the aforementioned paper gives sufficient conditions for a GIC to strictly dominate the SIC, given that dominance holds. We show that this condition is also necessary for strict dominance when S is a split disjunction. For ease of exposition, Theorem 17 assumes that all rays of C intersect $\mathrm{bd}\,S$ because Proposition 10 shows that intersecting rays cannot lead to deeper points.

Suppose $S = \{x : 0 \le x_k \le 1\}$ is a split disjunction on a variable $x_k$. Let $S^{0}:= \{x : x_k = 0\}$, and $S^{1}:= \{x: x_k = 1\}$. We partition the intersection point set ${{\mathcal {P}}}$ into ${{\mathcal {P}}}^{0}$ and ${{\mathcal {P}}}^{1}$, where ${{\mathcal {P}}}^{0}:= {{\mathcal {P}}}\cap S^{0}$, and ${{\mathcal {P}}}^{1}:= {{\mathcal {P}}}\cap S^{1}$. Recall that ${{\mathcal {P}}}_{0}$ and ${{\mathcal {R}}}_{0}$ are the points and rays obtained from intersecting $\bar{C}$ with $\mathrm{bd}\,S$. We also partition ${{\mathcal {P}}}_{0}$ into ${{\mathcal {P}}}_{0}^{0}:= {{\mathcal {P}}}_{0}\cap S^{0}$ and ${{\mathcal {P}}}_{0}^{1}:= {{\mathcal {P}}}_{0}\cap S^{1}$. Intuitively, the theorem shows that a strictly dominating cut with respect to the SIC must reduce the dimension of ${{\,\mathrm{conv}\,}}({{\mathcal {P}}}_{0}^{0})$ or ${{\,\mathrm{conv}\,}}({{\mathcal {P}}}_{0}^{1})$.

Theorem 17

Suppose that ${{\mathcal {R}}}_{0}= \emptyset$, ${{\mathcal {P}}}_{0}^{0}\ne \emptyset$ and ${{\mathcal {P}}}_{0}^{1}\ne \emptyset$, and $({{\mathcal {P}}}, {{\mathcal {R}}})$ is a proper point-ray collection obtained from activating a single hyperplane H valid for P. There exists a basic feasible solution to (1) corresponding to a cut strictly dominating $\alpha _{0}^{\textsf {T}}x \ge \beta _0$ if and only if ${{\,\mathrm{relint}\,}}(H^+) \cap {{\mathcal {P}}}_{0}^{t} = \emptyset$ and $H^- \cap {{\mathcal {P}}}_{0}^{t} \ne \emptyset$, for at least one side t of the split disjunction, $t \in \{0,1\}$.

Proof

For the “if” direction of the proof, suppose without loss of generality that ${{\,\mathrm{relint}\,}}(H^+) \cap {{\mathcal {P}}}_{0}^{0}= \emptyset$ and $H^- \cap {{\mathcal {P}}}_{0}^{0}\ne \emptyset$. Any point in ${{\mathcal {P}}}^{0}$ lying on the SIC is in ${{\,\mathrm{conv}\,}}({{\mathcal {P}}}_{0}^{0})$. Because ${{\,\mathrm{relint}\,}}(H^+) \cap {{\mathcal {P}}}_{0}^{0}= \emptyset$, it holds that $({{\mathcal {P}}}^{0}{\setminus } {{\mathcal {P}}}_{0}^{0}) \cap {{\,\mathrm{conv}\,}}({{\mathcal {P}}}_{0}^{0}) = \emptyset$. This implies that any point p in $\mathcal {P}^0 {\setminus } \mathcal {P}^0_0$ satisfies $\alpha _{0}^{{\textsf {T}}} p > \beta _0$. Recall that $|{{\mathcal {P}}}_{0}^{0} | + |{{\mathcal {P}}}_{0}^{1} | = n$. Since some point of ${{\mathcal {P}}}_{0}^{0}$ lies in $H^-$, $|{{\mathcal {P}}}_{0}^{0}\cap H^+ | \le |{{\mathcal {P}}}_{0}^{0} | - 1$. It follows that at most $n-1$ intersection points from ${{\mathcal {P}}}_{0}$ remain in ${{\mathcal {P}}}$. This added degree of freedom and the aforementioned depth of points in ${{\mathcal {P}}}^{0}{\setminus } {{\mathcal {P}}}_{0}^{0}$ allows the SIC to be tilted to obtain a GIC that strictly dominates the SIC. The “only if” direction follows from Theorem 5 in Balas and Margot [7]. $\square$

The above result shows that any single hyperplane is unlikely to directly lead to a strictly dominating cut. Instead of looking for one such hyperplane, in our implementation we focus on activating a set of hyperplanes that together cut away large parts of ${{\,\mathrm{conv}\,}}({{\mathcal {P}}}_{0}^{0})$ and ${{\,\mathrm{conv}\,}}({{\mathcal {P}}}_{0}^{1})$. We do this by targeting each of the intersection points in ${{\mathcal {P}}}_{0}$ one at a time in step 6 of Algorithm 3.

Although strict dominance is difficult to attain, our next result shows that activating hyperplanes is monotonic in the sense that the lower bound implied by the point-ray collection can only be improved by activating hyperplanes. This complements Theorem 3 of Balas and Margot [7], in which it is shown that activating hyperplanes increases the depth of points with respect to the SIC. This leaves open the question of whether the lower bound on the objective value (implied by the points) improves after activating hyperplanes, which Proposition 18 resolves. Using the notation from Sect. 5.2, we show that when the ray creating the least cost intersection point $\underline{p}^k$ is cut by a hyperplane, the objective value implied by the new intersection points is greater than or equal to $\underline{z}$.

Proposition 18

Let r be the edge of C that leads to $\underline{p}^k$, i.e., $\underline{p}^k = r \cap \mathrm{bd}\,S_{k}$, H be a hyperplane intersected by r before $\mathrm{bd}\,S_{k}$, and ${{\mathcal {P}}}'$ denote the set of intersection points originating at $r \cap H$, obtained by activating H. Then $\min _p \{c^{{\textsf {T}}} p : p \in {{\mathcal {P}}}'\} \ge \underline{z}$.

Proof

Suppose $S_{k}^{0}$ is the facet of $S_{k}$ containing $\underline{p}^k$, and let $S_{k}^{1}$ be the opposite facet. We have that $\underline{z} = \min _x \{ c^{{\textsf {T}}} x : x \in C \cap S_{k}^{0}\} \le \min _x \{ c^{{\textsf {T}}} x : x \in C \cap S_{k}^{1}\}$. Since each of the points $p \in {{\mathcal {P}}}'$ is either (possibly strictly) in $C \cap S_{k}^{0}$ or $C \cap S_{k}^{1}$, the result follows. $\square$

1.2 A.2 Characterizing bounded objective functions for (PRLP)

We turn to an analysis of (PRLP). It is possible for the optimal solution to (PRLP) to be unbounded, a behavior we have in fact observed in our numerical implementation. To better understand this, in this section we present some structural properties of $\text {CutRegion}(\bar{\beta }, {{\mathcal {P}}}, {{\mathcal {R}}})$, the feasible region to (PRLP) for a fixed right-hand size $\bar{\beta }$, that characterize the objective function choices leading to unboundedness.

We begin by studying when the system $\text {CutRegion}(\bar{\beta },{{\mathcal {P}}},{{\mathcal {R}}})$ has valid cuts for a given proper point-ray collection. Recall that $\mathcal {K}'$ denotes the connected component of the skeleton of P that includes $\bar{x}\cap \mathrm{int}\,S$, and any inequality feasible to (1) that cuts a point $v \in \mathcal {K}'$ is valid. We will consider the system

$$\begin{aligned} {{\mathcal {G}}}^\# := \{\alpha : \alpha \in \text {CutRegion}(\bar{\beta },{{\mathcal {P}}},{{\mathcal {R}}});\ v^{{\textsf {T}}} \alpha < \bar{\beta }\}. \end{aligned}$$

Theorem 19

Let $({{\mathcal {P}}},{{\mathcal {R}}})$ be a proper point-ray collection and let $v \in \mathcal {K}'$. The system $\text {CutRegion}(\bar{\beta }, {{\mathcal {P}}}, {{\mathcal {R}}})$ has valid cuts as feasible solutions in the following cases: (1) for $\bar{\beta } = 1$ if and only if $0 \not \in {{\mathcal {G}}}$ and $v \notin {{\,\mathrm{conv}\,}}({{\mathcal {P}}}) + {{\,\mathrm{cone}\,}}({{\mathcal {P}}}\cup {{\mathcal {R}}}) = {{\mathcal {G}}}+ {{\,\mathrm{cone}\,}}({{\mathcal {P}}})$, (2) for $\bar{\beta } = -\,1$ if and only if $v \not \in {{\,\mathrm{conv}\,}}({{\mathcal {G}}}\cup \{0\})$, and (3) for $\bar{\beta } = 0$ if and only if $v \not \in {{\,\mathrm{cone}\,}}({{\mathcal {P}}}\cup {{\mathcal {R}}})$.

Proof

Let Q be the $|{{\mathcal {P}}} | \times n$ matrix containing the intersection points in ${{\mathcal {P}}}$ as its rows, and R be the $|{{\mathcal {R}}} | \times n$ matrix with rows comprised of the rays in ${{\mathcal {R}}}$. Let e denote the n-vector of all ones. Using the nonhomogeneous Farkas’ lemma [32], ${{\mathcal {G}}}^\#$ has a feasible solution if and only if the following two systems are infeasible:

$$\begin{aligned} \begin{matrix} \left\{ \lambda , \mu \ge 0 : \begin{array}{l} Q^{{\textsf {T}}} \lambda + R^{{\textsf {T}}} \mu = v \\ \bar{\beta }e^{{\textsf {T}}} \lambda \ge \bar{\beta }\end{array}\right\} &{} \qquad \qquad &{} \left\{ \lambda , \mu \ge 0 : \begin{array}{l} Q^{{\textsf {T}}} \lambda + R^{{\textsf {T}}} \mu = 0 \\ \bar{\beta }e^{{\textsf {T}}} \lambda > 0 \end{array}\right\} \end{matrix} \end{aligned}$$

When $\bar{\beta }= 1$, the first system is infeasible if and only if $v \notin {{\mathcal {G}}}+ {{\,\mathrm{cone}\,}}({{\mathcal {P}}})$, and the second system is infeasible if and only if $0 \notin {{\mathcal {G}}}$, since the existence of a solution $(\lambda ,\mu )$ implies $(\lambda /e^{{\textsf {T}}} \lambda , \mu )$ is also feasible. When $\bar{\beta }= -\,1$, the first system is infeasible if and only if $v \notin {{\,\mathrm{conv}\,}}({{\mathcal {G}}}\cup \{0\})$, and the second system is always infeasible, since $\lambda \ge 0$. When $\bar{\beta }= 0$, the first system is infeasible if and only if $v \not \in {{\,\mathrm{cone}\,}}({{\mathcal {P}}}\cup {{\mathcal {R}}})$, and the second system is again always infeasible. $\square$

The feasible region to (PRLP) is $\text {CutRegion}(\bar{\beta },{{\mathcal {P}}},{{\mathcal {R}}})$, not ${{\mathcal {G}}}^\#$. However, if we assume that v is used as the objective to (PRLP), then Theorem 19 can be used to show when (PRLP) has a finite solution. Observe that (PRLP) implicitly ranks valid inequalities and picks the most violated cut with respect to v. If there exists a homogeneous inequality valid for ${{\mathcal {G}}}$ that cuts off v, this ranking breaks down, since all homogeneous inequalities can be scaled to have arbitrarily large violation and hence are unbounded directions in (PRLP).

From the $\bar{\beta } = 0$ case in Theorem 19, it follows that the linear program (PRLP) is bounded if and only if v belongs to ${{\,\mathrm{cone}\,}}({{\mathcal {P}}}\cup {{\mathcal {R}}})$. Corollary 20 characterizes the two open objective function sets within ${{\,\mathrm{cone}\,}}({{\mathcal {P}}}\cup {{\mathcal {R}}})$ that admit valid cuts of only one type, either with right-hand side 1 or $-1$.

Corollary 20

The system (PRLP) has valid inequalities that cut off a point v only for

• 1. $\bar{\beta } = 1$ if and only if $0 \notin {{\mathcal {G}}}$ and $v \in {{\,\mathrm{conv}\,}}({{\mathcal {G}}}\cup \{0\}){\setminus } {{\mathcal {G}}}$.

• 2. $\bar{\beta } = -\,1$ if and only if $v \in ({{\mathcal {G}}}+ {{\,\mathrm{cone}\,}}({{\mathcal {P}}})) {\setminus } {{\mathcal {G}}}$.

Proof

Notice that

$$\begin{aligned} {{\,\mathrm{conv}\,}}({{\mathcal {G}}}\cup \{0\}) {\setminus } {{\mathcal {G}}}\subseteq {{\,\mathrm{conv}\,}}({{\mathcal {G}}}\cup \{0\}) \subseteq {{\,\mathrm{cone}\,}}({{\mathcal {P}}}\cup {{\mathcal {R}}}) \end{aligned}$$

and $\{{{\,\mathrm{conv}\,}}({{\mathcal {G}}}\cup \{0\}) {\setminus } {{\mathcal {G}}}\} \cap \{{{\mathcal {G}}}+ {{\,\mathrm{cone}\,}}({{\mathcal {P}}})\} = \emptyset$. Therefore, the result in part 1 follows from Theorem 19. The proof of the second part is similar. $\square$

B Tilting for degenerate hyperplanes

We previously showed how to tilt a hyperplane H defining P that is not degenerate, i.e., $\bar{x}$ does not lie on H. We defined H using n affinely independent points obtained by intersecting the n affinely independent rays of $\bar{C}$ with H. These points all lie on one-dimensional faces of $\bar{C}$. In the case that H is degenerate, we will instead use two-dimensional faces of $\bar{C}$ to define the hyperplane, which we will then modify to define a targeted tilting.

When a degenerate hyperplane H is activated on $\bar{C}$, each of the extreme rays of the new cone $\bar{C}\cap H^+$ that lie on H can be defined by H and $n-2$ hyperplanes of $\bar{C}$ that are not redundant for $\bar{C}\cap H^+$. Thus, each ray of the new cone lying on H is on a two-dimensional face of $\bar{C}$. Let ${{\mathcal {R}}}^{H*}$ denote an affinely independent set of $n-1$ of these rays lying on H. Let be the rays of $\bar{C}$ that are cut by $H^+$. For each $r \in {{\mathcal {R}}}^{H*}$, since it lies on a two-dimensional face of $\bar{C}$, we can define $(\bar{r}^1,\bar{r}^2) \in \bar{\mathcal {R}}^c \times (\bar{\mathcal {R}} {\setminus } \bar{\mathcal {R}}^c)$ such that r can be expressed as a convex combination of $\bar{r}^1$ (which is cut by H) and $\bar{r}^2$ (which is not cut by H); in particular, let $\lambda ^H_r$ be the multiplier such that $r = \lambda ^H_r \bar{r}^1 + (1-\lambda ^H_r) \bar{r}^2$. Figure 4 depicts this construction.

Fig. 4

Illustration of targeted tilting construction for a degenerate hyperplane

If we know $\lambda ^H_r$ and the rays $\bar{r}^1$ and $\bar{r}^2$ for every $r \in {{\mathcal {R}}}^{H^*}$, then we can use this to give an alternate definition of H. To define a targeted tilting of H, we can modify the values $\lambda ^H_r$ for each $r \in {{\mathcal {R}}}^{H^*}$ using some $\delta _r$. In order to coordinate with the definition of a targeted tilting that we gave in Sect. 4, the intersection points and rays created by activating the tilted hyperplane should either be identical to those obtained from activating H or coincide with some initial intersection point or ray. It is not difficult to see that this means the allowed values for $\delta _r$ are 0 and $-\lambda ^H_r$. With such a tilting, Theorem 11 will still hold, i.e., activations can be performed by using H and ${{\mathcal {R}}}_A$ without a need for explicitly computing the tilted hyperplane.

Fig. 5

Feasible region of P and $\bar{C}$

C Tilting example

In this section, we demonstrate an example in which a valid tilting combined with the implicit computation used in Theorem 11 leads to invalid cuts. The example additionally provides intuition for the targeted tilting rule that if a hyperplane cuts a ray of $\bar{C}$ that has already been cut, then we should not tilt the hyperplane along that ray. This is sufficient, as we have shown, to allow us to apply the implicit computation scheme.

The left panel of Fig. 5 shows the feasible region of $P := \{x \in \mathbb {R}^3 : -2 x_2 + x_3 \le 0; -2 x_1 + x_3 \le 0; 12 x_1 + 10 x_2 - 5 x_3 \le 9; 10 x_1 + 12 x_2 - 5 x_3 \le 9; x_1 + x_2 + x_3 \le 1\}$, and the right panel of the same figure shows the cone $\bar{C}$. The cut-generating set S is the unit box, $\{x \in \mathbb {R}^3: 0 \le x_1 \le 1;\ 0 \le x_2 \le 1\}$. The hyperplane activations are of $H_4$ and then $H_5$. We tilt $H_4$ along $r^2$ so that the intersection of this ray with $\widetilde{H}_4$ coincides $r^2 \cap \mathrm{bd}\,{S}$, and $H_5$ will similarly be tilted along $r^1$ so that the intersection of $r^1$ with $\widetilde{H}_5$ is at the point $r^1 \cap \mathrm{bd}\,{S}$. The tilted hyperplanes are shown in the top panel of Fig. 6.

The tilting defined above is clearly valid. It also satisfies all but one of the conditions of being a targeted tilting; it does not meet the requirement that $H_5$ and $\widetilde{H}_5$ must intersect $r^1$ at the same point, as a result of the activation of $H_4$ on $r^1$ prior to the activation of $H_5$.

However, as shown in the bottom panel of Fig. 6, when the implicit computation algorithm is applied to these tilted hyperplanes, a point of ${{\,\mathrm{conv}\,}}(P {\setminus } \mathrm{int}\,{S})$ is cut. This is because $\widetilde{H}_5$ intersects $r^1$ outside of the interior of S. Hence, the intersection point $\widetilde{H}_5 \cap r^1$ is not added to the point collection, as a result of step 5 of Algorithm 1. This intersection point is the same as $H_4 \cap r^1$, which has already been removed from the point collection during the activation of $\widetilde{H}_4$.

Fig. 6

Hyperplane activations leading to cutting a point in ${{\,\mathrm{conv}\,}}(P {\setminus } \mathrm{int}\,S)$

There may be many approaches in which a tilted hyperplane activation can be computed implicitly using only information from the non-tilted hyperplane. Our method prevents the situation in this example from occurring by requiring that $r^1$ intersects $\widetilde{H}_5$ at the same point as it intersects $H_5$. An example of an alternative would be to add the intersection point $\widetilde{H}_5 \cap r^1 = H_4 \cap r^1$ back into the point collection when activating $\widetilde{H}_5$, but then the intersection points obtained from activating $H_4$ on $r^1$ would be redundant.