Linear quasi-shrinkage estimator for high-dimensional optimization with linear constraints

Abstract

In large-scale data-driven optimization problems, parameters are often only known approximately due to noisy and small-sized samples. We consider optimization problems with linear constraints where the true parameter matrix is not precisely known, and the number of constraints and variables are comparable and large. Our goal is to construct a linear estimator of the true parameter matrix by minimizing the Frobenius distance between the estimator and the true parameter matrix. Our method offers three key advantages: 1) the coefficients of the linear estimator are consistently estimated from the observations and require no further calibration; 2) it only requires the sample size to be greater than one and it delivers stable performance across varied sample sizes; and 3) the constraints of the formulated optimization problem using the estimator remain linear, ensuring computational efficiency when the number of constraints and variables are large. Simulation shows that our linear estimator consistently produces stable outcomes in terms of the objective value, the ratio of violated constraints and the magnitude of constraint violation across various scenarios, compared to the nominal and robust methods. Additionally, it demonstrates resilience against high levels of noise, making it a robust choice under uncertainty.