Sparse multi-term disjunctive cuts for the epigraph of a function of binary variables

Abstract

We propose a new method for separating valid inequalities for the epigraph of a function of binary variables. The proposed inequalities are disjunctive cuts defined by disjunctive terms obtained by enumerating a subset I of the binary variables. We show that by restricting the support of the cut to the same set of variables I, a cut can be obtained by solving a linear program with $2^{|I|}$ constraints. While this limits the size of the set I used to define the multi-term disjunction, the procedure enables generation of multi-term disjunctive cuts using far more terms than existing approaches. We present two approaches for choosing the subset of variables. Experience on three MILP problems with block diagonal structure using |I| up to size 10 indicates the sparse cuts can often close nearly as much gap as the multi-term disjunctive cuts without this restriction and in a fraction of the time. We also find that including these cuts within a cut-and-branch solution method for these MILP problems leads to significant reductions in solution time or ending optimality gap for instances that were not solved within the time limit. Finally, we describe how the proposed approach can be adapted to optimally “tilt” a given valid inequality by modifying the coefficients of a sparse subset of the variables.

Appendix

1.1 Test problems

1.1.1 SNIP problem

The SNIP problem [48] is a two-stage stochastic integer program with pure binary first-stage and continuous second-stage variables. In this problem, by installing sensors on some arcs of a directed network to in the first stage, the defender tries to find the attacker and minimize the probability that the attacker travels from the origin to the destination undetected. In the second stage, the origin and destination of the attacker are observed and the attacker chooses to travel on the maximum reliability path from its origin to its destination. Let N and A denote the node set and the arc set of the network and let $D\subseteq A$ denote the set of interdictable arcs. The first-stage variables are denoted by $\textbf{x}$, where $x_a=1$ if and only if the defender installs a sensor on arc $a\in D$. Each scenario $s \in S$ is associated with a possible origin/destination combination of the attacker, with $u^s$ representing the origin and $v^s$ representing the destination of the attacker for each scenario $s \in S$. The second-stage variables are denoted by $\pi$, where $\pi ^s_i$ denotes the maximum probability of reaching destination $v^s$ undetected from node i in scenario s. The budget for installing sensors is b, and the cost of installing a sensor on arc a is $c_a$ for each arc $a\in D$. For each arc $a\in A$, the probability of traveling on arc a undetected is $r_{a}$ if the arc is not interdicted, or $q_a$ if the arc is interdicted. Parameter $\bar{\pi }_j^s$ denotes the maximum probability of reaching the destination undetected from node j when no sensors are installed. The extensive formulation of the problem is as follows:

$$\begin{aligned} \begin{aligned} \min _{\textbf{x},\mu ^s}\ {}&\sum _{s\in S}p_s\pi _{u^s}^s\\ \text {s.t. }\ {}&\sum _{a\in A}c_ax_a\le b,\\&\pi _i^s-r_a\pi _j^s\ge 0,{} & {} a=(i,j)\in A\setminus D,\ s\in S,\\&\pi _i^s-r_a\pi _j^s\ge -(r_a-q_a)\bar{\pi }_j^sx_a,{} & {} a=(i,j)\in D,\ s\in S,\\&\pi _i^s-q_a\pi _j^s\ge 0,{} & {} a=(i,j)\in D,\ s\in S,\\&\pi _{v^s}^s=1,{} & {} s\in S,\\&x_a\in \{0,1\},{} & {} a\in D. \end{aligned} \end{aligned}$$

We use the SNIP instances from [48]. We consider 40 instances with snipno$\in \{3,4\}$, and budget $b\in \{30,50,70,90\}$. All instances have 320 first-stage binary variables, 2586 s-stage continuous variables per scenario and 456 scenarios.

1.1.2 LLA problem

The LLA problem is introduced in [45]. In this problem, a retailer chooses a set of items to display for customers to purchase. The model assumes all customers are from a set of customer segments. For each customer segment k, the customers arrive according to a Poisson process with rate $\lambda _k$ and only purchases products in the consideration set $C_k$. For item i, the cost for displaying it is $c_i$, the relative attractiveness of it to customers in segment k is $v_i^k$, and the retailer earns a positive profit $w_i$ for each purchase of it by the customers. The preference of not purchasing anything is denoted by $v_0^k$ for customers in segment k. The retailer can choose items with a total cost up to $\rho \sum _{i=1}^n c_i$ to display. The problem can be formulated as an MINLP as follows:

$$\begin{aligned} \max _{\textbf{x}}\left\{ \sum _{k=1}^{N}\sum _{i\in C_k}\frac{\lambda _kv_i^kw_ix_i}{\sum _{i\in C_k}v_i^kx_i+v_0^k}:\sum _{i=1}^nc_ix_i\le \rho \sum _{i=1}^nc_i, \textbf{x}\in \{0,1\}^n \right\} . \end{aligned}$$

In [45], the authors reformulate the MINLP as an MILP by introducing a variable $y_k$ to represent the value of $1/(\sum _{i\in C_k}v_i^kx_i+v_0^k)$ and a variable $z_{ik}$ to linearize the product $x_iy_k$. The MILP reformulation of the problem is as follows:

$$\begin{aligned} \max _{\textbf{x},\textbf{y},\textbf{z}}~&\sum _{k=1}^N\sum _{i\in C_k}\lambda _kv_i^kw_i z_{ik}\\ \text {s.t. }&v_0^ky_k+\sum _{i\in C_k}v_i^kz_{ik}=1,{} & {} \hspace{-2cm}k\in [N],\\&v_0^ky_k-v_0^kz_{ik}\le 1-x_i,{} & {} \hspace{-2cm}k\in [N],i\in C_k,\\&z_{ik}\le y_k,{} & {} \hspace{-2cm}k\in [N],i\in C_k,\\&(v_0^k+v_i^k)z_{ik}\le x_i,{} & {} \hspace{-2cm}k\in [N],i\in C_k,\\&\sum _{i=1}^nc_ix_i\le \rho \sum _{i=1}^nc_i,\\&\textbf{x}\in \{0,1\}^n,~\textbf{y}\ge \textbf{0},~\textbf{z}\ge \textbf{0}. \end{aligned}$$

For the generation of test instances, we follow the basic scheme of generating type-1 problems in [45], but increase the value of some of the parameters to make the instances harder. We set $n=200$ and $N=200$. For each segment k, its arrival rate $\lambda _k$ and the preference of no purchase $v_0^k$ are randomly generated according to the uniform distributions Uniform([0, 1]) and Uniform([0, 4]), respectively. The preference $v_i^k$ for segment k of purchasing product i is randomly generated according to the discrete uniform distribution Uniform$(\{0,1,\ldots ,10\})$. For each odd $k\in K$, $C_k$ is a independently randomly chosen subset of [n] with size $p\in \{12,16,20\}$. For each even $k\in [N]$, $C_k$ is a random subset of $C_{k-1}$ of size p/2. The profit $w_i$ of product i is independently randomly generated according to the uniform distribution Uniform([100, U]) with $U\in \{150,350\}$. The capacity parameter $\rho$ is chosen from $\{20\%,50\%,100\%\}$. We consider both cases with uniform and nonuniform costs for generating the cost parameters $c_i$:

• (Uniform cost) $c_i=1$ for $i\in [n]$;

• (Nonuniform cost) $c_i=f_i\cdot |\{k\in [N]:i\in C_k\}|$ for $i\in [n]$, where each $f_i$ is independently randomly generated according to the normal distribution with mean 1 and standard deviation 0.1.

We generate one instance with uniform costs and one instance with nonuniform costs for each combination of $(p,U,\rho )$. Therefore, we have 36 LLA instances in total.

1.1.3 CAP problem

The stochastic CAP problem [14] is a generalization of the deterministic CAP problem [43], which can be formulated as a stochastic two-stage integer program. In this problem, the decision maker chooses to open a set of facilities to meet uncertain customer demands. The first-stage variables are denoted by $\textbf{x}$ with $x_i=1$ if and only if facility i is chosen to be opened. The second-stage variables are denoted by $\textbf{y}$, where $y_{ij}^k$ is the amount of the jth customer’s demand met by facility i in scenario k. For each facility i, the associated opening cost and its capacity are denoted by $f_i$ and $s_i$, respectively. The cost associated with satisfying a unit of the jth customer demand using facility i (sending a unit of flow from facility i to customer j) is denoted by $q_{ij}$. The jth customer’s demand under scenario k is denoted by $\lambda _j^k$. The extensive formulation of the problem is as follows:

$$\begin{aligned} \min _{\textbf{x},\textbf{y}}~&\sum _{i=1}^nf_ix_i+N^{-1}\sum _{k=1}^N{} & {} \hspace{-3.25cm}\sum _{i=1}^n\sum _{j=1}^m q_{ij}y_{ij}^k\\ \text {s.t. }&\sum _{i=1}^ny_{ij}^k\ge \lambda _j^k,{} & {} \hspace{-3.3cm}j\in [m],~k\in [N],\\&\sum _{j=1}^my_{ij}^k\le s_ix_i,{} & {} \hspace{-3.3cm}i\in [n],~k\in [N],\\&\sum _{i=1}^ns_ix_i\ge \max _{k\in [N]}\sum _{j=1}^m\lambda _j^k,\\&\textbf{x}\in \{0,1\}^n,~\textbf{y}\ge \textbf{0}. \end{aligned}$$

There are in total 32 CAP test instances all taken from [14] with $n\in \{25,50\}$, $m=50$ and $N=250$.

1.2 Additional root node results

Table 10 Number of cutting planes generated for SNIP instances. Results in each row are averages over five instances

Table 11 Number of cutting planes generated for LLA instances. Results in each row are averages over six instances

Table 12 Number of cutting planes generated for CAP instances. Results in each row are averages of eight instances

To provide further insights into the Benders model at the root node, in Tables 10, 11, and 12, we present the number of I-sparse cuts generated in IBC and the numbers of three major types of solver cutting planes (MIR, flow cover and relax-and-lift) generated in BBC and IBC, as reported by Gurobi. It is observed that a smaller number of solver cuts are generated in IBC than in BBC. This is due to the fact that the LP relaxation of the Benders model in IBC is stronger than that of the Benders model in BBC. Consequently, even though the LP relaxation of IBC includes a great number I-sparse cuts, processing one node in the branch-and-bound tree is not significantly more challenging in IBC than in BBC, as fewer solver cutting planes are introduced there.