Computing the Lipschitz constant of the solution map of a multi-parametric quadratic program is important for the analysis of optimization-based control. This problem is governed by three factors: the parameter dimension, the number of decision variables, and the number of constraints. While empirical evidence has long suggested exponential complexity, a rigorous complexity-theoretic proof has been lacking. In this paper, we fill this gap by proving that this problem is not only NP-hard but also APX-hard. Furthermore, we reveal that: (a) the problem becomes polynomial-time solvable when the number of constraints or decision variables is fixed; and (b) both NP-hardness and APX-hardness persist even in the scalar parameter case. These results confirm that the complexity stems from the number of constraints and variables, rather than the parameter dimension. Numerical experiments further validate these theoretical findings.

翻译：计算多参数二次规划解映射的Lipschitz常数对于基于优化的控制分析具有重要意义。该问题受三个因素制约：参数维度、决策变量数量和约束条件数量。尽管长期以来的经验证据表明其具有指数复杂度，但一直缺乏严格的复杂性理论证明。本文通过证明该问题不仅是NP难的，更是APX难的，填补了这一空白。此外，我们揭示出：（a）当约束条件或决策变量数量固定时，该问题可在多项式时间内求解；（b）即使在标量参数情形下，NP难和APX难性依然存在。这些结果证实，问题的复杂性源于约束条件和变量的数量，而非参数维度。数值实验进一步验证了这些理论发现。