SQP Methods for Mathematical Programs With Switching Constraints
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摘要:
存零约束优化(MPSC)问题是近年来提出的一类新的优化问题，因存零约束的存在，使得常用的约束规范不满足，以至于现有算法的收敛性结果大多不能直接应用于该问题。应用序列二次规划(SQP)方法求解该问题，并证明在存零约束的线性独立约束规范下，子问题解序列的聚点为原问题的Karush-Kuhn-Tucker点。同时为了完善各稳定点之间的关系，证明了强平稳点与KKT点的等价性。最后数值结果表明，序列二次规划方法处理这类问题是可行的。
Abstract:The mathematical program with switching constraint (MPSC) problem makes a new-type optimization issue in recent years. Due to the existence of switching constraints, the common constraint specification is not satisfied, so that the convergence results of existing algorithms can not be directly applied to this problem. The sequential quadratic programming (SQP) method was applied to solve the problem, and to prove that the clustering point of the solution sequence of the subproblem is the Karush-Kuhn-Tucker point of the original problem under the linear independent constraint specification with the switching constraint. At the same time, in order to improve the relationship between stationary points, the equivalence between the strong stationary point and the KKT point was proved. Finally, the numerical results show that, the sequential quadratic programming method is feasible to deal with this type of problems.
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表 1 MPSCs数值结果
Table 1. MPSCs numerical results
example ${\boldsymbol{x}}^*$ ${\boldsymbol{x}}'$ S N T 1 (1,1,1) (1.0,1.0,1.0) 1 19 0.0070 2 (0,0) (0.0,0.0) 1 4 0.0043 3 (0,0) (−0.0,−0.0) 1 17 0.0065 4 (0,0) (−0.0,0.0) 1 36 0.0082 
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表 2 投资组合数值结果
Table 2. Portfolio numerical results
$n$ S N T 50 1 24 0.1685 100 1 28 0.6690 200 1 36 3.3620 500 0.96 49 61.2914 800 0.90 59 1387.5846 1000 0.86 63 2336.6510 1200 0.82 69 4425.9890
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