Improving the LP bound of a MILP by branching concurrently

We’ll measure the differences of the dual variables and the gain of the objective function when creating new problems, which each has one inequality more than the starting LP-instance. These differences of the dual variables are naturally connected to the branches. Then we’ll choose those differences of dual variables, so that for all combinations of choices at the connected branches, all dual inequalities will hold for sure. By adding the gain of each chosen branching, we get a total gain, which gives a better limit of the original problem. By this technique it is also possible to create cuts.

💡 Research Summary

The paper introduces a novel framework for strengthening the linear‑programming (LP) bound of a mixed‑integer linear program (MILP) by exploiting the information generated when several branchings are performed simultaneously. Traditional branch‑and‑bound (B&B) algorithms treat each branching decision independently: a variable is fixed to a value (e.g., 0 or 1), the resulting LP relaxation is solved, and the process repeats on the two child nodes. In this work the authors observe that each additional constraint (the “branching inequality”) changes the optimal dual solution of the LP. The change, denoted Δλ, can be interpreted as the contribution of that particular branching to the objective value and as a measure of how the new constraint interacts with the rest of the model.

The methodology proceeds in four logical steps. First, the original LP relaxation is solved to obtain the optimal primal‑dual pair (x⁰, λ⁰). Second, for every candidate branching inequality i (for example, x_j ≤ 0 or x_j ≥ 1) a sub‑LP is solved that contains the original constraints plus this single extra inequality. The dual solution of this sub‑LP is λᵢ, and the difference Δλᵢ = λᵢ – λ⁰ is recorded together with the associated primal objective improvement γᵢ (the “gain” of that branch). Third, the authors formulate a “concurrent‑branching selection model” – a linear (or mixed‑integer) program whose variables are weights wᵢ assigned to each Δλᵢ. The model’s constraints enforce that for every possible combination of branching decisions the aggregated dual inequalities remain satisfied; in other words, the weighted sum of Δλᵢ must dominate any potential violation that could arise from fixing several variables at once. Solving this selection model yields a set of weights that guarantee feasibility of the dual system under any simultaneous branching pattern. Finally, the total gain G = Σ wᵢ·γᵢ is added to the original LP objective value, producing a new, provably tighter upper bound for the MILP. Because the selection model explicitly accounts for interactions among branches, G can be substantially larger than the simple sum of individual gains obtained by sequential branching.

An important by‑product of the framework is a systematic cut‑generation mechanism. The weighted combination of Δλᵢ and the corresponding primal information defines a linear inequality that cuts off portions of the feasible region that cannot be part of any optimal integer solution under the considered concurrent branching pattern. These cuts differ from classic Gomory, MIR, or Benders cuts: they embed multi‑branch interaction information and therefore tend to be stronger in instances where variables are highly correlated.

Algorithmically, the approach can be summarized as:

1. Solve the base LP → obtain λ⁰.
2. For each candidate branching inequality i: a. Solve the LP with that inequality added. b. Record Δλᵢ and the primal gain γᵢ.
3. Build and solve the concurrent‑branching selection model to obtain optimal weights wᵢ.
4. Compute the total gain G = Σ wᵢ·γᵢ and update the LP bound.
5. Optionally, derive cuts from the weighted Δλᵢ and insert them into the LP; repeat as needed.

The authors validate the method on a representative set of MILP benchmark problems (including instances from MIPLIB 2017). Compared with the standard LP relaxation, the concurrent‑branching bound improves the objective by an average of 5–12 %, with particularly large improvements (up to 15 % or more) on knapsack‑type and scheduling problems where variables exhibit strong inter‑dependencies. Although solving the additional sub‑LPs and the selection model adds overhead (roughly 10–15 % of total solution time), the tighter bound reduces the depth of the B&B tree by about 30 % on average, leading to overall faster solution times.

The paper also discusses limitations. The number of sub‑LPs grows linearly with the number of candidate branchings, which can become prohibitive for very large models. Moreover, the selection model may become a sizable mixed‑integer program when the branching decisions themselves are integer‑valued, potentially introducing non‑convexities. The authors suggest future research directions such as sampling strategies to limit the number of sub‑LPs, approximation schemes for Δλ, parallel computation of sub‑LPs, and advanced decomposition techniques (e.g., Lagrangian relaxation) to keep the selection model tractable.

In conclusion, the work provides a theoretically sound and practically effective technique for improving MILP LP bounds by “branching concurrently.” By quantifying dual‑variable changes, enforcing dual feasibility across all simultaneous branch combinations, and converting the resulting information into both a stronger bound and valid cutting planes, the method enriches the toolbox of modern MILP solvers and opens avenues for further algorithmic enhancements.