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.

翻译：本文提出了一种仅使用加法和乘法求解一类二次约束二次规划问题的新方法。由于所涉及的操作与同态加密方案的能力兼容，该方法能够在私有数据上求解约束优化问题。为求解约束优化问题，我们引入了一系列递增次数的多项式罚函数，这些函数在可行集边界处具有足够陡峭的特性。将罚函数加入原始成本函数后，产生了一系列无约束优化问题，其极小值点始终位于容许集内，并收敛于约束问题的极小值点。采用梯度下降法生成与这些问题相关的迭代序列。对于该算法，我们证明了迭代点收敛于原问题的极小值点，且可行集在该迭代下具有正不变性。最后，该方法在一个示例性密码学问题（求两数较小值）上得到验证，并讨论了其加密可实施性。