Differentially Private Linear Programming: Reduced Sub-Optimality and Guaranteed Constraint Satisfaction

TL;DR

This paper introduces a differentially private method for linear programming that guarantees feasible solutions and significantly reduces sub-optimality compared to existing approaches.

Contribution

The paper presents a novel differentially private linear programming approach that ensures solution feasibility and minimizes sub-optimality, addressing open problems in privacy-preserving optimization.

Findings

Solutions always exist for privatized problems.

Solutions satisfy original problem constraints.

65% reduction in sub-optimality compared to state-of-the-art methods.

Abstract

Linear programming is a fundamental tool in a wide range of decision systems. However, without privacy protections, sharing the solution to a linear program may reveal information about the underlying data used to formulate it, which may be sensitive. Therefore, in this paper we introduce an approach for protecting sensitive data while formulating and solving a linear program. First, we prove that this method perturbs objectives and constraints in a way that makes them differentially private. Then, we show that (i) privatized problems always have solutions, and (ii) their solutions satisfy the constraints in their corresponding original, non-private problems. The latter result solves an open problem in the literature. Next, we analytically bound the expected sub-optimality of solutions that is induced by privacy. Numerical simulations show that, under a typical privacy setup, the solution produced by our method yields a $65\%$ reduction in sub-optimality compared to the state of the art.