# Questions tagged [branch-and-cut]

Questions on algorithms that combine branch-and-bound search and the generation of cutting planes.

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### Cover cuts for knapsack constraint with integer variables

It is known for knapsack type constraint $\sum_{i \in N} a_i x_i \leq b , x \in \{0,1\}$, we can generate the so called cover cuts that have the sum of coefficient in a set C greater than b. The cover ...
349 views

### Valid Inequality Example (Wolsey Example 9.3)

I was following Wolsey Example 9.3: Let $X = \{(x,y) \in (R^{m}_+,B^1) : \sum_{i=1}^m x_i \leq my\}$. Now consider the valid inequality $x_i \leq y$ and show that it is facet defining. My question is ...
108 views

### general approach to iterating extreme rays of solution cone

Suppose I'm at an optimal solution of an LP relaxation in a MILP branch-and-bound descent. I want to add an additional cut of my own devices. To compute this cut I need the extreme rays of the cone ...
55 views

### Separating two LP solutions at the same time

In a cutting plane algorithm for integer linear programming problems you would usually use an LP solver to get an extreme point of the polyhedron corresponding to the LP relaxation. Let's call that ...
232 views

### Solution for a TSP with Branch and Cut for Gurobi in Java?

We have a group project and our job is to create an algorithm for solving a TSP with Branch and Cut. (Use of lazy constraints.) We tried around but don´t know what is the best way to start. We get the ...
160 views

### Warm starting ideas for iteratively solving a model with a few additional constraints

I'm trying to solve a MILP model iteratively, and at each iteration a few constraints are added to the problem that cut off the previous optimal solution. I'm trying to figure out ways to implement a ...
127 views

### Exploiting ordering to removing infeasible solutions in MILP

I kindly ask for some ideas or references to exploit ordering in MILPs. In particular, there are resources $r = [r_1, r_2, ..., r_K]$ such that $r_{i} \leq r_{i+1}$. These are input to the problem. ...
792 views

### CPLEX gives different solutions of MILP every run

I am solving a mixed-integer (binary) linear problem using CPLEX as a solver (branch-and-bound method). But I encountered the following issue. Each run I get a different solution at a different node ...
156 views

### Naturally branch-and-cuttable problems with high didactic value

I am willing to bet that the problem most often used to teach students about branch-and-cut is the Travelling Salesman Problem (TSP). It has a number of nice characteristics which make it appropriate ...
1 vote
256 views

### Branch and bound method for solving non-convex integer non-linear multi-objective optimization problem?

Following are the characteristics of my problem: Objective function: two non-linear functions and one linear function Decision variable: two integer variables ($X_1$ and $X_2$) Constraint: three (two ...
323 views

### Implementation of Local Branching

I've been recently reading some papers where the authors use local branching specifically in Benders Decomposition (see for reference). Although I understand up to some extend how the algorithm works, ...
770 views

### How to use tight upper and lower bounds to get to the optimal value via branch and bound?

I have algorithms that get me a tight upper (UB) and lower (LB) bound to a maximization binary integer program (a routing problem). My formulation is non-compact and requires the addition of sub-tour ...
180 views

### Branch and Cut algorithm

Could someone please explain what is the point of this added cut? Also how was it deduced? Could someone please give some examples of this sort? As I've been searching for such examples but did not ...
295 views

### Why isn't $x_2+x_3+x_4\le 2$ a cutting plane?

In my textbook, to generate cutting planes, they tell you to proceed as follows: A procedure for generating cutting planes: Select a ($\le$) constraint that has only nonnegative coefficients. ...
335 views

### Was there something specific that caused graph cuts to lose popularity in the last 5 years?

Almost every graph-cut paper I look at seems to have exactly the same pattern of monotonic growth in citations and then monotonic decline starting around 5 years ago: For privacy I've cut the all ...
333 views

### Program can't stop when using callback of usercut in CPLEX to solve a branch-and-cut problem

When I tried to use Java to call CPLEX and use callback functions to solve a VRP problem with branch-and-cut, some problems emerged. I took out the subtour elimination constraint from the original ...
307 views

### Rules of thumb for using classic methods (e.g. branch and bound) vs. meta-heuristics (G.A, S.A, etc..) for non-convex problems?

Are there any rules of thumb for when to use classical methods like branch and bound, branch and cut, etc...for non-convex problems, vs using meta-heuristic methods like genetic algorithms, ...
5k views

### Performance of a branch and bound algorithm VS branch-cut-heuristics

I was trying to solve a moderate scheduling model using an open-source solver. I did two different ways. A) using pure branch and bound algorithm (disable all options). B) using the default setting ...
348 views

### Family of hard instances for Gomory's cutting plane algorithm

Is there a variant of integer programs for which Gomory's cutting plane algorithm demonstrably takes a superpolynomial number of iterations?