CPLEX Error 5002求助：高维度MIQP问题非凸判定异常

Hey there, let's break down how to tackle that CPLEX Error 5002 you're facing when your MIQP problem scales beyond 600 variables. Even though you've confirmed your matrix is semi-positive definite (semi-PSD), high-dimensional problems often introduce numerical quirks that trip up solvers like CPLEX. Here are actionable steps to diagnose and fix the issue:

• Double-check numerical semi-PSDness of your matrix
  Theoretical semi-PSD doesn't always translate to numerical semi-PSD, especially with high-dimensional matrices. Run eig(Hprim'*Hprim) to inspect the eigenvalues. If you see tiny negative values (like -1e-10 or smaller), that's likely the culprit—CPLEX uses strict checks for convexity, and even minuscule numerical errors can make it flag the matrix as non-convex.

• Add a tiny regularization term to your Hessian
  To fix numerical semi-PSD issues, tweak your matrix to be strictly positive definite with a negligible diagonal term:

  H = Hprim'*Hprim + 1e-8*eye(size(Hprim,2));
  

  Test values between 1e-9 and 1e-7 to find a balance that passes CPLEX's convexity check without altering your problem's optimal solution.

• Verify your ctype variable type definitions
  Even if small problems work, it's easy to mess up ctype when scaling variables. Make sure the length of ctype matches your total number of variables, and each character correctly marks variables as 'C' (continuous) or 'B' (binary). A single misplaced character could throw off CPLEX's problem classification.

• Adjust CPLEX parameters for convexity checks
  CPLEX has settings that can relax or refine convexity validation:

  • Set the optimalitytarget to 3, which is designed for convex MIQPs:

    cplex.Param.optimalitytarget = 3;
    

  • Widen the convexity tolerance to allow minor numerical deviations:

    cplex.Param.qp.tolerances.convexity = 1e-6;
    

  (Note: Don't make this tolerance too large, as it might let truly non-convex problems slip through.)

• Improve numerical stability for high dimensions
  High-dimensional matrices often have poor condition numbers, leading to numerical instability:

  • Convert dense matrices to sparse storage—CPLEX handles sparse data more reliably for large problems;
  • Scale your variables or matrix entries to reduce the range of values (e.g., normalize columns to have similar magnitudes), which improves numerical precision.

• Check your inequality constraints for numerical issues
  Even with empty Aeq, your Aineq and bineq might have problems in high dimensions: linear dependent rows, or entries with vastly different magnitudes. Use rank(Aineq) to check for linear dependence, and normalize constraint rows to have consistent scales—this can indirectly help CPLEX correctly assess the problem's convexity.

内容的提问来源于stack exchange，提问作者lmbd_a