SCIP Solves: Multilinear Optimization & Suboptimal Solutions

Hey guys! Ever stumbled upon a situation where your optimization model gives different results based on seemingly equivalent formulations? It's a head-scratcher, right? Today, we're diving deep into one such scenario encountered while using SCIP to solve a continuous multilinear problem. Let's unravel why one formulation can lead to a suboptimal solution while another nails the optimal one, even when SCIP claims optimality in both cases.

The Curious Case of Multilinear Formulations

The core issue revolves around two mathematically equivalent formulations of the objective function in a multilinear problem. The expectation is straightforward: both formulations should yield the same optimal result. However, in practice, one formulation produced a significantly suboptimal solution. This discrepancy raises a critical question: Why do equivalent formulations behave differently in optimization solvers like SCIP?

Let's break down the problem and the PySCIPOpt code snippet that highlights this issue. The code sets up a model with five flows (f0 to f4) and corresponding weights (s0_w to s4_w). The sum of the flows is constrained to 300, and each flow is related to its weight by the equation flows[i] * (weights[i] + 1) == 100. The objective is to minimize a variable obj, which is defined in two ways:

1. RATIO Formulation: obj >= sum([abs(0.9 - weights[i]) for i in range(5)])
2. FLOW Formulation: obj >= sum([abs(0.9 - (100/flows[i] - 1)) for i in range(5)])

Mathematically, these formulations are identical due to the equality constraint linking flows and weights. However, when the solve_problem function is called with obj_func=RATIO, the result is suboptimal (obj=0.84). When called with obj_func=FLOW, the optimal result is achieved (obj=0.78).

This difference is not just a minor variation; it's a significant gap that warrants investigation. The symmetry between variables further complicates the analysis, as comparing variables one-to-one might be misleading.

Diving Deeper: Why the Discrepancy?

So, what's causing this divergence? Several factors could be at play here:

• Numerical Instability: Optimization solvers use numerical methods to find solutions. Slight differences in formulation can lead to variations in numerical stability, especially with multilinear problems. The way SCIP handles floating-point arithmetic might be more sensitive to one formulation over the other.
• Branch-and-Bound Behavior: SCIP, like many Mixed-Integer Programming (MIP) solvers, uses a branch-and-bound algorithm. The branching decisions and bound tightening can be influenced by the formulation. A suboptimal solution might be explored and declared optimal in one formulation while being pruned in the other.
• Constraint Propagation: The solver's ability to propagate constraints and tighten variable bounds can differ based on the formulation. One formulation might allow for tighter bounds, leading to a more efficient search for the optimal solution.
• Solver Heuristics: SCIP employs various heuristics to find good solutions quickly. These heuristics might perform differently depending on the formulation, potentially leading to suboptimal solutions in some cases.

The Impact of Bounds

The original poster also noted that removing the upper bound of 1 on the weights (s0_w to s4_w) resolves the issue. This observation provides a crucial clue. By removing the upper bound, the solution space expands, and the solver can find the optimal solution more easily in both formulations. Similarly, changing the bounds from [0, 1] to [1, 2] (and adapting the constraints and objective function) also seems to fix the problem. This suggests that the original bounds might be interacting with the solver's algorithms in a way that favors the FLOW formulation.

Exploring Potential Solutions and Workarounds

Okay, so we've identified the problem and some potential causes. What can we do about it? Here are some strategies to consider when faced with similar issues:

1. Reformulation Techniques

One of the most effective approaches is to explore alternative formulations of the problem. In this case, the FLOW formulation consistently outperformed the RATIO formulation. If possible, try to stick with formulations that the solver handles more effectively. This might involve rewriting the objective function or constraints in a different form.

For instance, consider using a linear approximation of the absolute value function. Instead of abs(x), you can introduce two new variables, x_pos and x_neg, and add the constraints:

• x = x_pos - x_neg
• x_pos >= 0
• x_neg >= 0

Then, replace abs(x) in the objective function with x_pos + x_neg. This linearization can sometimes improve the solver's performance.

2. Adjusting Solver Parameters

Optimization solvers like SCIP have a plethora of parameters that control their behavior. Experimenting with these parameters can sometimes alleviate issues with suboptimal solutions.

• Emphasis Settings: SCIP has different emphasis settings that prioritize various aspects of the solving process (e.g., optimality, feasibility, speed). Try setting the emphasis to optimality (model.setEmphasis(OPTIMALITY)) to encourage the solver to focus on finding the best possible solution.
• Node Selection and Branching Rules: The way the solver explores the branch-and-bound tree can significantly impact performance. Experiment with different node selection and branching rules to see if they improve the solution quality.
• Presolving and Aggregation: SCIP performs presolving and aggregation steps to simplify the problem. Sometimes, these steps can inadvertently lead to suboptimal solutions. Try disabling or adjusting these features to see if it makes a difference.
• Numerical Tolerances: Adjusting numerical tolerances can sometimes help with numerical instability issues. However, be cautious when modifying these parameters, as it can also affect the solver's ability to find feasible solutions.

3. Tightening Bounds

As we saw in the original example, the bounds on variables can play a crucial role. Tightening the bounds can reduce the search space and help the solver find the optimal solution more quickly. In this case, the original poster found that removing the upper bound on weights improved the results. However, this might not always be feasible or desirable. Try to identify valid bounds that are as tight as possible without eliminating the optimal solution.

4. Symmetry Breaking

Symmetry in a problem can lead to multiple equivalent solutions, which can confuse the solver. Adding symmetry-breaking constraints can help the solver focus on a smaller portion of the solution space. In the original example, the symmetry between the flows might be contributing to the issue. Consider adding constraints that break this symmetry (e.g., by ordering the flows in some way).

5. Using a Different Solver

Sometimes, the best solution is to switch to a different solver. Different solvers use different algorithms and heuristics, and one solver might perform better than another on a particular problem. If you've tried various techniques and are still struggling to get good results, consider trying a different solver (e.g., Gurobi, CPLEX).

Real-World Implications and Best Practices

This deep dive into suboptimal solutions highlights the importance of careful model formulation and solver selection. In real-world applications, the consequences of a suboptimal solution can be significant, leading to increased costs, reduced efficiency, or missed opportunities. Here are some best practices to keep in mind:

• Validate Your Model: Always validate your model thoroughly to ensure that it accurately represents the problem you're trying to solve. This includes checking the constraints, objective function, and data.
• Test with Multiple Formulations: If possible, test your model with multiple formulations to see if they produce consistent results. This can help you identify potential issues with numerical stability or solver behavior.
• Analyze Suboptimal Solutions: When you encounter a suboptimal solution, take the time to analyze it and understand why it's not optimal. This can provide valuable insights into the problem and the solver's behavior.
• Keep Learning: The field of optimization is constantly evolving, with new algorithms and techniques being developed all the time. Stay up-to-date on the latest advances to improve your modeling and solving skills.

Conclusion: Mastering the Art of Optimization

Encountering suboptimal solutions in optimization problems can be frustrating, but it's also an opportunity to learn and grow. By understanding the potential causes of these issues and exploring various solutions and workarounds, you can become a more effective optimization practitioner. Remember, guys, optimization is not just about finding a solution; it's about finding the best solution.

So, the next time you're faced with a tricky optimization problem, don't despair! Take a deep breath, analyze the situation, and apply these techniques to conquer the challenge. Happy optimizing!