SemidefiniteOptimization: False No Solution Claim?

by Luna Greco 51 views

Hey everyone! Ever run into a situation where your optimization tool swears there's no solution, even though you're pretty sure one exists? I recently stumbled upon a fascinating case with the SemidefiniteOptimization function, and I'm excited to share the details and explore potential reasons why this might happen.

The Curious Case of the Missing Solution

So, here’s the deal. I was working with a matrix defined as follows:

mat = {{1, -(3/2), 1 - 1/2 x[2, 2], 0, -x[4, 2],
    1 - 1/2 x[4, 4]}, {-(3/2), x[2, 2], 0,
    x[4, 2], -x[4, 3], -x[5, 4]}, {1 - 1/2 x[2, 2], 0, 1,
    x[4, 3], -(3/2), 1 - 1/2 x[5, 5]}, {0, x[4, 2], x[4, 3], 1,
    1 - 1/2 x[4, 4], -(3/2)}, {-x[4, 2], -x[4, 3], -(3/2),
    1 - 1/2 x[4, 4], x[5, 5], 0}, {1 - 1/2 x[4, 4], -x[5, 4],
    1 - 1/2 x[5, 5], -(3/2), 0, 1}};

This matrix, mat, is a symmetric matrix where some of its elements depend on variables like x[2, 2], x[4, 2], x[4, 4], and so on. The goal here is to find values for these variables such that the matrix is positive semidefinite. In simpler terms, we want to ensure that all the eigenvalues of this matrix are non-negative. This is a common requirement in many optimization problems, especially in areas like control theory and combinatorial optimization. To solve this, I naturally turned to SemidefiniteOptimization, which is designed to handle precisely these kinds of problems.

Here's the command I used:

SemidefiniteOptimization[0, {mat ≽ 0, 0 <= x[2, 2] <= 2, 0 <= x[4, 4] <= 2,
   0 <= x[5, 5] <= 2, -1 <= x[4, 2] <= 1, -1 <= x[4, 3] <= 1,
   -1 <= x[5, 4] <= 1}, {x[2, 2], x[4, 2], x[4, 3], x[4, 4], x[5, 4], 
   x[5, 5]}]

Let's break down this command. The first argument, 0, represents the objective function. In this case, we're not trying to minimize or maximize anything specific; we're just looking for a feasible solution. The second argument is a list of constraints. The crucial constraint here is mat ≽ 0, which tells SemidefiniteOptimization that the matrix mat must be positive semidefinite. The other constraints specify bounds for the variables, ensuring they lie within a certain range (e.g., 0 <= x[2, 2] <= 2). These bounds are important for practical reasons, as they help to narrow down the search space and prevent the variables from taking on unrealistic values. The final argument is a list of the variables we're trying to find.

Now, here's where things get interesting. When I ran this command, SemidefiniteOptimization returned {$Failed, {}}. This result indicates that the function couldn't find any solution that satisfies the given constraints. In other words, it claimed that there are no values for the variables that make the matrix mat positive semidefinite within the specified bounds. But intuitively, and from other methods I tried, I suspected that solutions should exist. This discrepancy got me thinking: why would SemidefiniteOptimization fail to find a solution when one seems plausible?

Digging Deeper: Why No Solution?

There are several potential reasons why SemidefiniteOptimization might fail to find a solution. Let's explore some of the most common ones:

  • Infeasibility: The most straightforward reason is that the problem is genuinely infeasible. In other words, there might not be any combination of variable values that satisfies all the constraints simultaneously. This could happen if the constraints are too restrictive or if they contradict each other. However, in my case, I had a strong suspicion that the problem wasn't inherently infeasible, given the structure of the matrix and the nature of the constraints.
  • Numerical Issues: Semidefinite optimization, like many numerical methods, relies on iterative algorithms that approximate the solution. These algorithms can be sensitive to numerical issues, such as rounding errors or ill-conditioning. If the problem is poorly scaled or if the matrix has a high condition number, the algorithm might struggle to converge to a solution or might even produce incorrect results. This is a common pitfall in numerical optimization, and it often requires careful attention to the problem formulation and solver settings.
  • Solver Limitations: Different semidefinite programming solvers use different algorithms and have different strengths and weaknesses. Some solvers might be better suited for certain types of problems than others. It's possible that the default solver used by SemidefiniteOptimization in Mathematica isn't the most appropriate for this particular problem. This could be due to the size of the problem, the structure of the matrix, or the specific constraints involved. In such cases, trying a different solver or adjusting the solver settings might help.
  • Problem Formulation: Sometimes, the way we formulate the problem can affect the solver's ability to find a solution. For example, introducing redundant constraints or using a particular parametrization might make the problem harder to solve. In my case, I had carefully considered the problem formulation, but it's always worth double-checking to see if there are alternative ways to express the constraints or variables.

Exploring Potential Fixes

Given these potential reasons, I started to explore ways to address the issue. Here are some of the approaches I considered:

  1. Verifying the Problem Formulation: The first step was to double-check my problem formulation. I wanted to ensure that I hadn't made any mistakes in defining the matrix or the constraints. This involved carefully reviewing the expressions and making sure they accurately represented the problem I was trying to solve. It's always a good practice to start with the basics and eliminate any potential errors in the problem setup.
  2. Tightening the Bounds: Since numerical issues can sometimes arise from unbounded variables, I considered tightening the bounds on the variables. By reducing the range of possible values, I hoped to make the problem more well-conditioned and easier for the solver to handle. This is a common technique in optimization, as it can help to reduce the search space and improve the convergence of the algorithm. However, it's important to strike a balance, as overly tight bounds might exclude the true solution.
  3. Trying a Different Solver: As mentioned earlier, different solvers have different strengths and weaknesses. Mathematica's SemidefiniteOptimization function might have a default solver that isn't the best fit for my problem. So, I decided to explore other solvers that are available in Mathematica or through external libraries. This involved researching different semidefinite programming solvers and experimenting with their settings to see if they could find a solution.
  4. Reformulating the Problem: Sometimes, a slight reformulation of the problem can make a big difference in the solver's performance. This might involve introducing auxiliary variables, changing the way the constraints are expressed, or using a different parametrization. The goal is to find an equivalent problem that is easier for the solver to handle. This often requires a good understanding of the underlying mathematical structure of the problem.

The Quest for a Solution Continues

As of now, I'm still in the process of investigating this issue. I've tried a few different approaches, but I haven't yet found a definitive solution. However, I'm confident that by systematically exploring the potential reasons for the failure and trying different techniques, I'll eventually be able to find the missing solution. This is the nature of problem-solving in mathematical optimization – it often involves a bit of detective work and a willingness to experiment.

I'm planning to try the suggestions, such as using FindInstance to find a feasible starting point and then using SemidefiniteOptimization with adjusted settings. I'll also explore alternative solvers and consider reformulating the problem if necessary. It's a journey of discovery, and I'm excited to see where it leads.

Seeking Your Insights and Experiences

I'm sharing this experience with you guys because I believe that collaboration and shared knowledge are essential in tackling complex problems. Have you ever encountered a similar situation where an optimization solver claimed no solution when you suspected one existed? What strategies did you use to overcome the challenge? I'd love to hear your insights and experiences.

Maybe you've worked with SemidefiniteOptimization before and have some tips or tricks to share. Or perhaps you've encountered similar issues with other optimization tools and have learned valuable lessons along the way. Whatever your experience, I encourage you to join the conversation and share your thoughts.

In the meantime, I'll keep you updated on my progress as I continue to investigate this puzzle. Together, we can unravel the mysteries of semidefinite optimization and learn how to find those elusive solutions that seem to vanish into thin air!

Repair Input Keyword

Why does the SemidefiniteOptimization function claim there is no solution when a solution actually exists for the given code?

Title

SemidefiniteOptimization: No Solution When It Exists?