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

Ilya Bolotov ()
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Ilya Bolotov: Faculty of International Relations, Prague University of Economics and Business, W. Churchill Sq. 1938/4, Žižkov, 130 67 Prague, Czech Republic

Mathematics, 2025, vol. 13, issue 21, 1-37

Abstract: 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 ) = ( A Z ( 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 R 2 , 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.

Keywords: convex optimization; modular estimators; least squares; generalized inverse; regularization; normalized RMSE; Monte Carlo simulation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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