In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relationships are imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The $\ell_1$ penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the $\ell_1$ penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problem in network transportation. We first provide an instance from this class, the exactness of whose $\ell_1$ penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.
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…
A sequential linear complementarity problem method for generalized Nash equilibrium problems
State Key Laboratory of Mathematical Sciences, 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）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University（东北大学工业智能与系统优化前沿科学中心）；Key Laboratory of Data Analytics and Optimization for Smart Industry (Northeastern University), Ministry of Education（教育部数据智能与优化重点实验室（东北大学））
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…
A Lasry-Lions envelope approach for mathematical programs with complementarity constraints
KU Leuven（荷语鲁汶大学）；Kyushu University（九州大学）；International Centre for Numerical Methods in Engineering (CIMNE)（国际数值工程中心）；Catalan Institution for Research and Advanced Studies (ICREA)（加泰罗尼亚研究与高级研究学院）
Randomized Branch Methods with Inexact Subproblems for Bouligand Stationarity in Linear and Quadratic Programs with Complementarity Constraints
Southern University of Science and Technology（南方科技大学）；Institute of Computational Mathematics and Scientific/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研究所）；University of Chinese Academy of Sciences（中国科学院大学）；National Center for Applied Mathematics Shenzhen（深圳应用数学国家研究中心）
Fast and Effective Computation of Generalized Symmetric Matrix Factorization
School of Computer Science and Engineering（计算机科学与工程学院）；Guangdong Province Key Laboratory of Computational Science（广东省计算科学重点实验室）；Sun Yat-sen University（中山大学）；School of Mathematics and Information Science（数学与信息科学学院）；Guangzhou University（广州大学）；Department of Mathematics（数学系）；The University of Tennessee, Knoxville（田纳西大学，诺克斯维尔）
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。
