Smooth Exact Penalty Functions: A General Approach
M. V. Dolgopolik
备注：
This is a slightly edited post-peer-review, pre-copyedit version of an article published by Springer in Optimization Letters. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11590-015-0886-3
In this article we present a new perspective on the smooth exact penalty function proposed by Huyer and Neumaier that is becoming more and more popular tool for solving constrained optimization problems. Our approach to Huyer and Neumaier's exact penalty function allows one to apply previously unused tools (namely, parametric optimization) to the study of the exactness of this function. We give a new simple proof of the local exactness of Huyer and Neumair's penalty function that significantly generalizes all similar results existing in the literature. We also obtain new necessary and sufficient conditions for the global exactness of this penalty function.
A unifying theory of exactness of linear penalty functions II: parametric penalty functions
In this article we develop a general theory of exact parametric penalty functions for constrained optimization problems . The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions , and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parame…
Exact-Penalty Prox-Linear Methods for Bilevel Optimization with $\ell_1$ Lower-Level Gradient Penalty
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology（北京理工大学计算机辅助设计与图形学北京市重点实验室/信息安全数学理论计算北京市重点实验室）
An Alternating Direction Method with Increasing Penalty for Stable Principal Component Pursuit
稳定主元追踪（Stable Principal Component Pursuit，SPCP）是一个非光滑的凸优化问题，它的解使得即使只有一小部分数据是可观测的，也能可靠地恢复被密集噪声矩阵破坏的数据矩阵的低秩和稀疏分量。本文提出了一种求解SPCP的新算法。该算法是一个修改的交替方向法的乘法器（ADMM），我们使用一个递增序列的罚参数，而不是一个固定的罚款。该算法是基于部分变量分裂和工作直接与非光滑目标函数。我们证明了原始序列和对偶序列在罚参数序列上在较弱的条件下收敛。据我们所知，这是第一个收敛结果的可变罚款ADMM时，罚款是没有界的，目标函数是非光滑的，它的次微分不是一致有界的。使用部分变量分裂和采用递增序列的惩罚乘数，一起，显着减少所需的迭代次数，以实现实际的可行性。我们的初步计算测试表明，该算法在实践中非常好，并优于ASALM，一个国家的最先进的ADMM算法的SPCP问题与一个常数罚参数。
Exact Penalty Algorithm of Strong Convertible Nonconvex Optimization
This paper defines a strong convertible nonconvex(SCN) function for solving the unconstrained optimization problems with the nonconvex or nonsmooth(nondifferentiable) function. First, many examples of SCN function are given, where the SCN functions are nonconvex or nonsmooth. Second, the operational properties of the SCN functions are proved, including addition, multiplication, compound operations and so on. Third, the SCN forms of some special functions common in machine learning and engineering applications are presented respectively where these SCN function optimization problems can be tran…
The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues
School of System Science and Statistics, Beijing Wuzi University（北京物资大学系统科学与统计学院）；LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院教育部重点实验室）
Exact Quadratic Penalty Function for Symplectic Eigenvalue Problem
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
Exact penalty decomposition method for zero-norm minimization based on MPEC formulation
我们将零范数最小化问题转化为一个等价的具有平衡约束的数学规划，并证明了在目标函数中加入互补约束所得到的惩罚问题是精确的.然后，利用精确罚问题的特殊结构，提出了一种分解方法，在[M]中的零空间条件下，可以求出零范数极小化问题的全局最优解。A. Khajehnejad等人，IEEE Trans. Signal.过程.，59（2011），pp. 1985-2001]通过求解有限个加权$l_1$ -范数最小化问题。为了处理加权$l_1$ -范数子问题，我们提出了一种部分邻近点法，其中子问题可以用有限记忆BFGS（L-BFGS）或半光滑Newton-CG近似求解.最后，将L-BFGS与半光滑Newton-CG相结合求解加权$l_1$ -范数子问题的精确罚分解方法应用于几类稀疏最优化问题，并与罚分解方法[Z. Lu和Y. Zhang，SIAM J. Optim.，23（2013），pp. 2448- 2478]、迭代支撑检测方法[Y. L. Wang和W. T. Yin，SIAM J. Sci.计算：3（2010），pp. 462-491]和现有技术代码FPC_AS [Z. W. Wen等人SIAM J. Sci.计算：32（2010），pp. 1832-1857年]。数值比较表明，所提出的方法是非常有效的可恢复性和所需的计算时间。
