There are many important practical optimization problems whose feasible regions are not known to be nonempty or not. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation, which is called the optimization problem with the least constraint violation (OLCV). Firstly, we introduce the constrained convex optimization. The solvability of the dual of the optimization problem with the least constraint violation is investigated. If the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, we establish the optimality conditions and propose the augmented Lagrangian method for the convex optimization with the least constraint violation, which has the linear convergence rate under an error bound condition. Secondly, we focus on the constrained nonconvex optimization. Various types of stationary points are presented for the MPCC reformulation problem. The penalty method and the smoothing function method for solving nonlinear programming case are constructed. For solving the inequality constrained case, the smoothing Barrier augmented Lagrangian method, combining the augmented Lagrangian and the interior-point technique, is proposed with the locally linearly convergent to the KKT point. Finally, we discuss some specific optimization with the least constraint violation including the quadratic programming, the linear semidefinite programming and the multi-objective programming.
戴彧虹研究员，博士生导师,中国科学院数学与系统研究院副院长, 中国运筹学会理事长，亚太运筹学会联合会主席。戴彧虹教授长期从事优化方法的理论及应用研究，在连续优化、整数规划和应用优化等方面作出了系统的创造性工作。曾或正主持国家杰出青年科学基金、国家基金委创新研究群体项目、“十四五”国家重点研发计划项目等多项基金项目。应邀在2022年国际数学家大会做45分钟邀请报告，在第24届国际数学规划大会作一小时邀请报告。 曾获国家自然科学二等奖、中国青年科技奖、钟家庆数学奖、冯康科学计算奖、陈省身数学奖和首届萧树铁应用数学奖等奖项。