【作者】 张绍武；

【导师】 张立卫；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2012， 博士

【摘要】 本论文主要研究约束Lipschitz优化问题的约束规范,约束DC优化问题的序列凸近似方法和邻近点方法,以及作为应用的联合机会约束优化问题的序列凸近似方法,取得的主要结果可概述如下：第二章对于约束Lipschitz优化问题提出弱于[1]中的广义Robinson约束规范(GRCQ)的三个新的约束规范：弱广义Robinson约束规范,有界约束规范,广义Abadie约束规范,研究了这些约束规范与解映射平稳性条件之间的联系.研究结果表明,弱广义Robinson约束规范和有界约束规范是容易验证的保证解映射平稳性条件的充分性条件,而用变分分析中的图导数刻画的广义Abadie约束规范,弱于解映射的平稳性条件.我们把这些约束规范应用到带有互补约束的数学规划(MPCC)问题中,得到了保证MPCC问题C-稳定点的新的约束规范.第三章考虑目标函数和不等式约束函数均为DC函数的DC优化问题.分非光滑DC优化和光滑DC优化两种情况,研究序列凸近似方法的收敛性.首先,将非光滑DC优化的稳定点条件表示为一单调集值映射的广义方程.构造非光滑DC优化问题的一个序列凸近似方法,它生成可行的使目标函数值下降的点列.基于序列凸问题可行域的连续收敛性和Klatte and Li(1999)[2]的渐近约束规范,证得序列凸近似方法生成序列的任何聚点均是非光滑DC优化问题的稳定点.类似地,我们构造光滑DC优化问题的一个序列凸近似方法,在广义Slater条件下,得到凸问题约束集合序列的连续收敛性,同样证得序列凸近似方法生成序列的任何聚点均是光滑DC优化问题的稳定点.第四章考虑的是目标函数为一光滑函数与—DC函数之和,不等式约束函数为DC函数的非光滑优化问题.用—严格凸的二次函数(称为迫近项)来近似目标函数中的光滑函数,用线性函数近似所有DC函数的第二个凸函数,得到确定搜索方向的凸优化问题.首先研究搜索方向为凸问题的精确解,步长采用Armijo线搜索原则得到的迫近次梯度方法的收敛性；其次研究搜索方向为凸问题的非精确解,步长采用Armijo线搜索原则得到的近似迫近次梯度方法的收敛性；收敛性定理表明,两种方法生成的序列的任何聚点均是稳定点.第五章考虑目标函数是—DC函数的联合机会约束优化问题.我们采用Hong, Yang and Zhang(2011)[3]对约束的处理方法,用DC函数近似约束函数,得到一依赖于参数ε>O的问题(Pε)来近似原来的概率约束问题.在合适的假设条件下,证明ε—O时,(Pε)的全局最优解到原问题全局最优解的收敛性以及(Pε)的稳定点的收敛性.采用序列凸优化方法求解每一个(Pε),并证明了收敛性定理.由于序列凸优化方法涉及的凸规划问题是数学期望函数定义的问题,我们采用Monte Carlo方法求解这些凸问题,并用得到的序列凸近似Monte Carlo方法求解随机L1范数极小化问题和非凸随机二次规划问题,报告了得到的数值结果.同Hong, Yang and Zhang (2011)[3]不同的是,我们讨论的问题的目标函数是一光滑函数与一DC函数之和,形式更一般,而且不要求极大值随机函数c(x,ξ)=max{c1(x,ξ),…,cP(x,ξ)}的可微性,也不要求c(x,ξ)的积累函数F(,,x)的连续可微性.更多还原

【Abstract】 This dissertation focuses on the study of constraint qualifications for Lipschitz optimization problems, sequential convex approximation methods and proximal point methods for optimization problems of DC functions, and as an application the sequential convex approximation approach for joint chance constrained optimization problems. The main results, obtained in this dissertation, may be summarized as follows:Chapter2introduces three constraint qualifications weaker than the generalized Robinson constraint qualification (GRCQ) proposed by [1] for constrained Lipschitz optimization. The three constraint qualifications are the weak generalized constraint qualification, the bounded constraint qualification and the generalized Abadie constraint qualification. The relationships among these three constraint qualifications are studied, showing that the first two constraint qualifications are verifiable sufficient conditions for the calmness of the solution mapping, whereas the generalized Abadie constraint qualification is weaker than the calmness. The three constraint qualifications are applied to mathematical programs with complementarity constraints (MPCC) yielding new constraint qualifications for C-stationary points of MPCC.Chapter3considers the DC optimization problems with DC objective and DC constraint functions. The sequential approximation methods are constructed and their convergence properties are studied for both nonsmooth DC and smooth DC optimization problems. The set of stationary point conditions for the nonsmooth DC optimization problem is reformulated as a generalized equation of a monotone set-valued mapping. The sequential convex approximation method generates a sequence of points whose objective values are decreasing. Based on the continuous continuity of feasible regions of convex optimization problems and the asymptotical constraint qualification in Klatte and Li (1999)[2], we prove that any accumulation point of the sequence generated by the sequential convex approximation approach is a stationary point of the nonsmooth DC optimization problem. Similarly, for smooth DC optimization, under the generalized Slater condition, we demonstrate the continuous continuity of the feasible region for convex optimization problems, and that any accumulation point of the sequence generated by the sequential convex approximation approach is a stationary point of the smooth DC optimization problem.Chapter4considers a nonsmooth optimization problem whose objective is the sum of a smooth function and a DC function, and inequality constraints are DC function constraints. Search directions are determined by solving convex optimization problems whose objectives are obtained by using a strictly convex quadratic function (or a proximal term) and linear functions to approximate the smooth function and the second convex functions, respectively. If the search direction is the exact solution to the convex problem and the stepsize is generated by Armijo rule, we obtain the proximal subgradient method. Otherwise, if the search direction is an inexactly solution, then we obtain the inexact proximal point method. For both methods, we demonstrate that any accumulation point of the sequence generated by the method is a stationary point of the DC optimization problem.Chapter5deals with a nonsmooth optimization problem whose objective function is a nonsmooth DC function and the constraint set is defined by a joint chance constraint. We adopt the similar method as in Hong, Yang and Zhang (2011)[3] to deal with the joint chance constraint, in which a DC function is constructed to approximate the probabilistic function so that a parameter ε>0depended problem (Pε) is obtained to approximate the original chance constrained problem. It is demonstrated that, under mild conditions, the convergence of the global solutions and the stationary points of (Pε) to the global solution set and the set of stationary points of the original problem, respectively when ε→0. For a specific problem (Pε), the sequential convex approximation approach in Chapter3is employed and the convergence theorem is proved. As the convex problems in the sequential convex approach are defined by expectation functions, we use the Monte Carlo method to solve them. The sequential convex approximation Monte Carlo approach is used to solve stochastic L1-norm minimizing problem and the nonconvex stochastic quadratic programming problem, and the numerical results are reported. Different from the work by Hong, Yang and Zhang (2011)[3], the objective function in our problem is a DC function and the assumptions about the differentiability of the maximum random function c(x,ξ)=max{c1(x,ξ),…cp(x,ξ)} and the continuous differentiability of F{t, x) the accumulation function of c(x,ξ), are not required in our analysis.更多还原