CLSP: Linear Algebra Foundations of a Modular Two-Step Convex Optimization-Based Estimator for Ill-Posed Problems

This paper develops the linear-algebraic foundations of the Convex Least Squares Programming (CLSP) estimator and constructs its modular two-step convex optimization framework, capable of addressing ill-posed and underdetermined problems. After reformulating a problem in its canonical form, A(r)z(r)=b, Step 1 yields an iterated (if r>1) minimum-norm least-squares estimate z^(r)=(AZ(r))†b on a constrained subspace defined by a symmetric idempotent Z (reducing to the Moore–Penrose pseudoinverse when Z=I). The optional Step 2 corrects z^(r) by solving a convex program, which penalizes deviations using a Lasso/Ridge/Elastic net-inspired scheme parameterized by α∈[0,1] and yields z^*. The second step guarantees a unique solution for α∈(0,1] and coincides with the Minimum-Norm BLUE (MNBLUE) when α=1. This paper also proposes an analysis of numerical stability and CLSP-specific goodness-of-fit statistics, such as partial R2, normalized RMSE (NRMSE), Monte Carlo t-tests for the mean of NRMSE, and condition-number-based confidence bands. The three special CLSP problem cases are then tested in a 50,000-iteration Monte Carlo experiment and on simulated numerical examples. The estimator has a wide range of applications, including interpolating input–output tables and structural matrices.