基数约束下连续变量optimization的增广拉格朗日分裂逐次求解法
Splitting and successively solving augmented Lagrangian method for optimization with semicontinuous variables and cardinality constraint
摘要：
提出了一种新的分裂逐次求解增广拉格朗日（SSAL）方法，用于求解具有连续变量和基数约束的最优化问题。这一最优化问题出现在投资组合问题、压缩感知问题和机组组合问题等多个背景下，一般都是NP难问题。在适当的假设下，我们得到了这个最优化问题的最优性条件。通过引入辅助变量和增广拉格朗日函数，将约束分解为两部分。通过固定特定变量，将最优化问题分成两个子问题，交替求解。在适当的假设下，证明了SSAL的收敛性.最后，我们实现了我们的方法的投资组合选择问题和压缩感知问题，分别。实际数据和仿真研究表明，SSAL优于著名的CPLEX 12.6和惩罚分解（PD）方法，而享受类似的决策变量基数。例如，对于投资组合选择问题，数值结果甚至比CPLEX 12.6快近200倍，对于压缩感知问题，数值结果比PD方法快40多倍。特别是当问题规模增大时，SSAL算法具有很强的鲁棒性。
英文摘要：
We propose a new splitting and successively solving augmented Lagrangian (SSAL) method for solving an optimization problem with both semicontinuous variables and a cardinality constraint. This optimization problem arises in several contexts such as the portfolio selection problem, the compressed sensing problem and the unit commitment problem, etc. The problem is in general NP-hard. We derive an optimality condition for this optimization problem, under some suitable assumptions. By introducing an auxiliary variable and using an augmented Lagrangian function, the constraints are decomposed into two parts. By fixing particular variables, the optimization problem is split into two subproblems, which are solved alternatively. Furthermore, we prove the convergence of SSAL, under some suitable assumptions. Finally, we implement our method for the portfolio selection problem and the compressed sensing problem, respectively. Real world data and simulation study show that SSAL outperforms the well-known CPLEX 12.6 and the penalty decomposition (PD) method, while enjoying a similar cardinality of decision variable. For example, the numerical results are even nearly 200 times faster than that of CPLEX 12.6 for the portfolio selection problem, and more than 40 times faster than PD method for the compressed sensing problem, respectively. In particularly, SSAL is powerful when the size of problem increase largely.
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