一类稀疏约束非线性规划的约束规格

王鑫 彭定涛 周倩

摘 要 研究一类带有闭凸集约束的稀疏约束非线性规划问题，这类问题在变量选择、模式识别、投资组合等领域具有广泛的应用.首先引进了限制性Slater约束规格的概念，证明了该约束规格强于限制性M-F约束规格，然后在此约束规格成立的条件下，分析了其局部最优解成立的充分和必要條件.最后，对约束集合的两种具体形式，指出限制性Slater约束规格必满足，并给出了一阶必要性条件的具体表达形式.

关键词 稀疏约束非线性规划;限制性约束规格;最优性条件

中图分类号 O224  文献标识码 A

Abstract A class of sparse nonlinear programming was studied， whose feasible set is the intersection of a closed convex set and a sparse set. This model is a typical sparse optimization problem which has wide applications in variable selection， pattern recognition， portfolio management and other fields. We defined  the restricted Slater constraint qualification for this sparse nonlinear programming and proved  that this restricted Slater constraint qualification is stronger than the restricted M-F constraint qualification. Under this restricted Slater constraint qualification， we analyzed  the necessary or sufficient optimality conditions for the local solutions. Finally， we provided  the specific expressions of the first-order necessary optimality condition for the model with two specific constraint sets.

Key words sparse constraint nonlinear programming; restricted constraint qualification; optimality condition

5 总 结

本文对一类带有闭凸集约束的稀疏约束非线性规划问题引进了限制性Slater约束规格的概念，分析表明该约束规格强于限制性M-F约束规格且更容易验证，此约束规格可保证局部最优解是M-KKT点、C-KKT点和B-稳定点.最后，对约束集合的两种具体形式，指出限制性Slater约束规格必满足，并给出了一阶必要性条件的具体表达形式.本文的结果对于设计和分析有效算法提供了理论基础.

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