A polynomial-based QCQP solver for encrypted optimization

TL;DR

This paper introduces a polynomial-based method for solving quadratically constrained quadratic problems using only addition and multiplication, enabling privacy-preserving optimization with homomorphic encryption.

Contribution

It proposes a novel polynomial penalty approach combined with gradient descent for encrypted constrained optimization, compatible with homomorphic encryption schemes.

Findings

Method converges to the true constrained minimum.

Feasible set remains invariant during iterations.

Demonstrated on a cryptographic problem involving encrypted data.

Abstract

In this paper, we present a novel method for solving a class of quadratically constrained quadratic optimization problems using only additions and multiplications. This approach enables solving constrained optimization problems on private data since the operations involved are compatible with the capabilities of homomorphic encryption schemes. To solve the constrained optimization problem, a sequence of polynomial penalty functions of increasing degree is introduced, which are sufficiently steep at the boundary of the feasible set. Adding the penalty function to the original cost function creates a sequence of unconstrained optimization problems whose minimizer always lies in the admissible set and converges to the minimizer of the constrained problem. A gradient descent method is used to generate a sequence of iterates associated with these problems. For the algorithm, it is shown that the iterate converges to a minimizer of the original problem, and the feasible set is positively invariant under the iteration. Finally, the method is demonstrated on an illustrative cryptographic problem, finding the smaller value of two numbers, and the encrypted implementability is discussed.