完全逐段线性多目标优化

Since the existing construction theory of the Pareto solution set and optimal value set of a piecewise multiobjective optimization problem requires all constrained functions are linear, this proposal plans to establish the construction theory of Pareto solution set and optimal value set for a fully piecewise multiobjective optimization problem with both the objective function and all constrained functions are piecewise linear. Noting that a multiobjective optimization problem has possibly many abnormal optimal values and that an iteration sequence corresponding to a normal Pareto optimal value is often needed to converge to a solution corresponding this Pareto optimal value, we plan to consider a new weak sharp minima, which is of uniformity with respect to all Pareto optimal values. We also plan to study the stability, such as continuity and metric regularity, of feasible set mapping and solution mapping when the piecewise linear objective and constrained functions undergo small perturbations. Moreover, in terms of some vector-valued subdifferentials of the objective function and the normal cone of the feasible set and using techniques of variational analysis, we consider KKT conditions for fully piecewise linear multiobjective optimization problems.

动机于现有逐段线性多目标优化问题的解集结构理论均建立在约束函数的线性性假设之下，本项目拟就目标和约束函数均为逐段线性函数的完全逐段线性多目标优化问题建立帕累托解集和优化值集的结构。考虑到多目标优化问题可能有许多反常优化值且实际问题常常需要对应于某一“好优化值”的迭代序列收敛于对应于该优化值的解，我们拟就完全逐段线性多目标优化问题建立关于优化值有一致性特征的weak sharp minima并考虑基于此的收敛性分析。鉴于实际问题的数学建模中使用的逐段线性函数不可避免地有误差，本项目拟研究逐段线性约束函数和逐段线性目标函数经扰动后对应的可行集映射和解映射的连续性及度量正则性等稳定性。此外，本项目拟通过目标函数的向量次微分及可行集的法锥并使用变分分析的方法研究完全逐段线性多目标优化问题的KKT条件。

本项目主要研究了目标和约束都是逐段线性的多目标优化问题。在一般赋范空间框架下，研究了逐段线性向量值函数的分类和分解， 建立了无穷维空间上完全逐段多目标优化问题Pareto解集、弱Pareto解集和弱Pareto优化值集的结构，给出了求完全逐段线性多目标优化问题弱Pareto解集和Pareto解集具算法功能的构造性program。 发现欧氏空间X上的逐段线性多值映射F在其整个值域F(X)上不必有一致全局和一致有界度量次正则性，并证明赋范空间上的逐段线性多值映射在其值域F(X)上总是“几乎”一致有界度量次正则的。进一步地，研究了逐段线性多值映射有一致全局度量次正则性的可验证特征。作为应用，建立了完全逐段线性向量优化问题的一致有界和一致全局weak sharp minima性质刻画。使用变分分析方法和技术，建立了目标函数经tilt扰动及参变量联合扰动对应带参数优化问题的一般fully Holder稳定解理论。建立了多值映射度量次正则性关于Lipschitz扰动和calm扰动有稳定性的特征。对于赋范空间上一般的非凸向量值映射f，引进并研究了其Clarke切导数(或Bouligand切导数)关于序锥的Slater条件。特别地，扩展凸优化中关于Slater条件理论到非凸情形。此外，建立了闭凸集上凸优化问题的各种“好的”可解性。作为经典分离性定理和James定理的补充，还建立了闭凸集的可达分离性及可达严格分离性。