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学术讲座通知报告题目：A new theoretical framework for numerical methods in optimization problems with equality constraints 主讲人：B.S. Goh（Mathematics Dept, Nanjing University）时间：2010年11月15日（星期一）下午 3: 30--4: 30 地点：教二楼424会议室主要内容：Current numerical methods, like the trust region method in solving optimization problems with equality constraints, use a quadratic model to construct an iteration. This is motivated by the Newton method which works well near a solution. We show that the use of quadratic model at a point far away from a solution needs careful justification. We describe a new way to define properly an iteration in a spherical local search region at a point which does nor contain a solution. At such point the Lagrange multipliers should not used. This conclusion is surprising given the dominance of Lagrange multipliers in the theory of optimization problems with equality constraints. We study long term optimal numerical methods for nonlinear optimization problems with equality constraints. For a nonconvex objective function it may be possible to construct an accessory function which is a linear combination of the objective function and the constraint violation functions such that the accessory function is a global merit function with a unique minimum point. As the parameters of this accessory function are varied the minimum point of the accessory function generates a sequence of points which move towards a neighborhood of the minimum point of the constrained optimization problem. We develop a two phases method. In Phase I some constraints are not approximately satisfied or the current point is not close to the solution. Then the Lagrange multipliers are all set equal to zero. Once all the constraint equations are approximately satisfied, the initial values of the Lagrange multipliers for the damped Newton iterative equations are chosen to minimize a second merit function. A test with the second merit function is used to determine whether or not the current point and the Lagrange multipliers are both close to the optimal solution. If not, Phase I is continued. If otherwise, Phase II is activated and a Newton method is used to compute the optimal solution. In this way, fast convergence is achieved near the solution. Keywords: Optimization; equality constraints; approximate greatest descent; SUMT: Newton. 此讲座为前沿课题讲座，欢迎全校师生踊跃参加。校学术委员会信通院 2010年11月10日