AI 中文总结
研究双层优化正则化间隙函数重构中乘子无界问题，证明近似平稳序列聚点的C平稳性，给出反例说明标准重构可能不保证M平稳性，引入新惩罚公式并开发方法，使其聚点满足M平稳性。
AI 中文摘要
值函数类型的重构为双层优化产生了广泛的方法。然而，相应的值函数类型约束本质上是退化的，通常不满足标准约束资格，因此相关的乘子序列可能是无界的，有界乘子收敛分析变得不适用。我们研究了具有约束凸下层规划的双层问题的正则化间隙函数重构的这个问题。我们证明，即使与正则化间隙函数约束相关的乘子序列是无界的，近似平稳序列的聚点对于相应的基于Karush-Kuhn-Tucker的带互补约束的数学规划（MPCC）是C平稳的。该结果在上下层约束系统的Mangasarian-Fromovitz约束资格（MFCQ）和极限MPCC点的MPCC-MFCQ下成立，而对正则化间隙函数约束本身没有任何约束资格要求。我们进一步提供了一个例子，表明标准正则化间隙函数重构的近似平稳点可能收敛到一个是C平稳但不是M平稳的点。为了保证M平稳性，我们引入了一种基于松弛的双参数惩罚公式，保持精确的乘子-松弛互补性，并在惩罚参数的支配条件下建立M平稳性。我们开发了一种具有自适应惩罚更新和可行性校正的不精确松弛惩罚方法，在所述假设下其聚点是M平稳的。
英文摘要
Value-function-type reformulations have generated a broad class of methods for bilevel optimization. However, the corresponding value-function-type constraints are inherently degenerate and generally fail to satisfy standard constraint qualifications, so the associated multiplier sequences may be unbounded and bounded-multiplier convergence analyses become inapplicable. We study this issue for the regularized gap-function reformulation of bilevel problems with constrained convex lower-level programs. We prove that accumulation points of approximate stationary sequences are C-stationary for the corresponding Karush-Kuhn-Tucker-based mathematical program with complementarity constraints (MPCC), even when the multiplier sequence associated with the regularized gap-function constraint is unbounded. The result holds under Mangasarian-Fromovitz constraint qualification (MFCQ) for the upper- and lower-level constraint systems and MPCC-MFCQ at the limiting MPCC point, without any constraint qualification on the regularized gap-function constraint itself. We further provide an example showing that approximate stationary points of the standard regularized gap-function reformulation may converge to a point that is C-stationary but not M-stationary. To guarantee M-stationarity, we introduce a slack-based two-parameter penalty formulation preserving exact multiplier-slack complementarity and establish M-stationarity under a domination condition on the penalty parameters. We develop an inexact slack-penalty method with adaptive penalty updates and feasibility correction, whose accumulation points are M-stationary under the stated assumptions.
