Enhancements

Benders decomposition is a powerful technique for solving large-scale optimization problems, but its convergence can be slow. To address this, various acceleration strategies have been developed, which can be broadly categorized into two groups: those that speed up the solution of each iteration and those that reduce the total number of iterations required to find a solution.

        graph LR

    A(Benders Decomposition Acceleration Strategies);

    B(Speed Up Each Iteration);
    C(Reduce Number of Iterations);

    A --> B;
    A --> C;

    B1(Accelerate Master Problem Solving);
    B2(Accelerate Subproblem Solving);

    B --> B1;
    B --> B2;

    B1 --> B1b(Warm Starting);
    B1 --> B1a(Cut Management);

    B2 --> B2d(Warm Starting);
    B2 --> B2a(Parallelization);
    B2 --> B2b(CP and Algorithmic Solver);

    C1(Generate Stronger Cuts and Bounds);
    C2(Improve Convergence Quality and Stability);
    C3(Optimize Algorithm Control Flow);

    C --> C1;
    C --> C2;
    C --> C3;

    C1 --> C1b(Global Valid Inequalities);
    C1 --> C1a(Pareto-optimal Cuts);
    C1 --> C1c(Multi-cut Generation);
    C2 --> C2b(Cut Normalization);
    C2 --> C2a(Trust Region);
    C3 --> C3a(Local Branching);
    C3 --> C3c(Branch and Check);
    C3 --> C3b(Early Stopping);

    classDef root fill:#0066CC,stroke:#333,color:#fff;
    classDef mainBranch fill:#5c9ce6,stroke:#333,color:#fff;
    classDef midLevel fill:#a9c9f7,stroke:#333,color:#000;
    classDef leaf fill:#fff,stroke:#333,color:#000;

    class A root;
    class B,C mainBranch;
    class B1,B2,C1,C2,C3 midLevel;
    class B1a,B1b,C3c,B2a,B2b,B2d,C1a,C1b,C1c,C2a,C2b,C3a,C3b leaf;

Benders Decomposition Acceleration Strategies

BendersLib is designed with a flexible and extensible architecture that facilitates the implementation of these acceleration strategies. The library’s callback system allows for the integration of custom logic at various stages of the decomposition process. This enables users to implement techniques such as warm starting, cut management, and custom termination criteria. Additionally, the modular design of BendersLib allows for the extension and customization of core components, such as custom cut generators and custom solvers, making it possible to incorporate advanced methods like Pareto-optimal cuts and local branching.

Introductions to some of these strategies are provided below. Though they are not exhaustive, they serve as examples of how BendersLib can be used to implement various acceleration techniques.

Warm Starting

Warm starting [1] is a technique used to improve the performance of iterative optimization algorithms by providing a good initial solution or starting point. In the context of Benders decomposition, warm starting can be applied to both the master problem and the subproblems. By providing a high-quality initial solution (e.g., from the previous iteration), the effort required to solve the master problem and subproblems can be reduced.

Example

This example (Warm Start) demonstrates how to implement warm starting in BendersLib using Callbacks.

Cut Management

As the number of iterations increases, the master problem can become encumbered by a large number of cuts, many of which may no longer be relevant or effective. This can significantly slow down the solution process. Cut management, or cut pooling, techniques aim to control the size of the master problem by selectively removing insignificant cuts. By keeping the master problem lean, the overall performance of the algorithm can be improved.

Example

This example (Cut Management) demonstrates how to implement cut management in BendersLib using Callbacks.

Parallelization

In Stochastic Programming, the L-shaped methods (L-shaped Method, Integer L-shaped Method) often yield subproblems that can be solved in parallel. This can significantly reduce the solution time per iteration. BendersLib supports this feature through the parallel_sub parameter. However, the overhead of parallelization may outweigh the benefits if subproblems are solved very quickly. It is advisable to benchmark to see if parallelization is advantageous for your specific problem.

Example

These examples (L-shaped Method, Integer L-shaped Method, L-shaped Method with Convex Recourse, Integer L-shaped Method (IIS)) demonstrate how to enable parallel subproblem solving in BendersLib.

CP and Algorithmic Solvers

When subproblems exhibit special combinatorial structures, using specialized algorithms instead of general-purpose Mathematical Programming solvers can be significantly more efficient. This is the core idea behind Logic-based Benders Decomposition and Combinatorial Benders Decomposition. Constraint Programming (CP) is a powerful paradigm for solving combinatorial problems. It can serve as a general purpose subproblem solver. BendersLib facilitates this by providing backends for CP solvers, allowing users to seamlessly integrate them for solving subproblems within the Benders decomposition framework.

Example

This example (Facility Location (Gurobi)) demonstrates how to implement customized subproblem solvers within the Logic-based Benders Decomposition framework using BendersLib.

Global Valid Inequalities

Global valid inequalities are constraints added to the master problem to strengthen the formulation and accelerate convergence. Unlike traditional Benders cuts, which are derived from specific subproblem solutions, these inequalities are valid for the overall problem. They often encapsulate a relaxation of the subproblem, expressed in terms of master problem variables [2]. In stochastic programming, a global bound for the estimator variables of the recourse function can tighten the master problem by providing stronger lower bound [3]. Symmetry breaking constraints also fall into this category.

Note

Global valid inequalities can be added at the beginning of the algorithm when building the master problem, using solver-specific APIs. They can also be added dynamically during the algorithm, using Callbacks.

Pareto-optimal Cut

When a subproblem is degenerate, its dual problem may have multiple optimal solutions, leading to different Benders cuts. A Pareto-optimal cut, also known as a Magnanti-Wong cut [4], is a non-dominated Benders cut, meaning no other cut generated from the same subproblem solution is stronger than it. Generating these cuts usually requires solving an additional problem relying on the subproblem solution. This extra effort can be avoided by using approximate core points [5]. Comprehensive reviews on cut selection can be found in the literature review paper [6].

Example

This example (Pareto-optimal Cut) demonstrates how to implement Pareto-optimal cut in BendersLib using Callbacks.

Multi-cut Generation

For stochastic programming problems, multiple cuts can be generated from the subproblems at each iteration. This includes multiple optimality cuts [7] from different subproblem solutions and multiple feasibility cuts from different sources of infeasibility. This approach, can accelerate convergence by providing more information to the master problem. However, it can also increase the size of the master problem and the computational burden per iteration. BendersLib supports this through the multi_opti_cut and multi_feas_cut parameters.

Example

See the examples with tags benders: l-shaped and benders: integer l-shaped for enabling multi-cut generation in BendersLib and for a performance comparison.

Cut Normalization

Cut normalization enhances numerical stability in Benders decomposition by scaling cuts that have large coefficients and right-hand side (RHS) values. Multiplying the cut by a factor produces an equivalent but more manageable cut, which can improve convergence speed and reliability. A common normalization method is to divide the cut by the norm of its coefficients.

Example

Trust Region Method

The trust region method is a stabilization technique used to address the oscillatory behavior often observed in Benders decomposition, where master variable solutions can jump between distant points in the feasible region, slowing convergence. The core idea is to restrict the master problem’s solution to a trust region around the current incumbent solution. This is achieved by adding a constraint that limits the distance between a new solution and the current best solution. By preventing drastic changes in the master variables, the trust region method promotes a more stable convergence.

Example

Using Callbacks provided by BendersLib, the trust region methods are implement in Trust Region Method (L1 Norm), Trust Region Method (Box Constraints), and Trust Region Method (Hamming Distance) (with Hamming distance [8]).

Local Branching

Local branching [9] is a heuristic that explores the neighborhood of a good incumbent solution to find improved solutions. It works by adding a local branching constraint that limits the number of binary variables that can change their value from the incumbent, creating an easier-to-solve problem. In the context of Benders decomposition, this helps to find better upper bounds and generate different cuts to obtain better lower bounds, which can accelerate convergence [10].

Example

This example (Local Branching) demonstrates how to implement local branching in BendersLib using Callbacks.

Branch-and-check Method

The Branch-and-check method [11] [12], also known as Branch-and-Benders-cut method [13], integrates Benders decomposition into a branch-and-bound framework. Instead of repeatedly solving the master problem to optimality, it generates cuts at branch-and-bound nodes, avoiding the need to build a new tree at each iteration. Cuts can be generated from feasible or incumbent solutions. This strategy requires the underlying solver’s callback functionality [14] and allows for a more efficient exploration of the solution space.

Example

  • BendersLib’s implementation can be used via bnc_solve().

  • These examples (tag: branch-and-check) demonstrate how to enable Branch-and-check method in BendersLib.

Early Stopping

Early stopping is a practical strategy to terminate the Benders decomposition process before the theoretical optimality gap is closed. This is often done when the improvement in the objective function becomes negligible over several iterations. This can save significant computation time, especially when finding the true optimal solution is not critical, and a good-enough solution is acceptable.

Example

This example (Early Stop) demonstrates how to implement early stopping in BendersLib using Callbacks.

References

We recommend these papers [15] [16] [17] [18] [19] [20], these slides [21] [22], and this blog post [23] for discussions on enhancement strategies for Benders decomposition.