mwang@watmath.UUCP (mwang) (02/23/84)
_D_E_P_A_R_T_M_E_N_T _O_F _C_O_M_P_U_T_E_R _S_C_I_E_N_C_E _U_N_I_V_E_R_S_I_T_Y _O_F _W_A_T_E_R_L_O_O _S_E_M_I_N_A_R _A_C_T_I_V_I_T_I_E_S _N_U_M_E_R_I_C_A_L _A_N_A_L_Y_S_I_S _S_E_M_I_N_A_R - Friday, March 2, 1984. Dr. M.L. Overton of Courant Institute, New York Univer- sity, will speak on ``Projected Hessian Updating Algo- rithms for Nonlinearly Constrained Optimization.'' TIME: 2:30 PM (Please Note) ROOM: MC 6091A ABSTRACT Consider the nonlinearly constrained optimization prob- lem: min f(x) subject to c(x) = 0 , i = 1,...,m. where the minimization is over x in R and where f(x) and the {c(x)} are smooth functions. We first discuss several Newton-type methods which have been proposed for solving this problem. We then con- sider quasi-Newton methods, which avoid forming Hessian matrices but update an appropriate approximating ma- trix. The standard method updates an approximation to the full Hessian of the Lagrangian function. We sug- gest a new Broyden-type method which was suggested by the work of Goodman. This method updates an n x (n-m) matrix which approximates a ``one-sided projected Hes- sian'' of a Lagrangian function. The method is easily shown to have a local one-step Q-superlinear conver- gence property. We also give a new proof of the Boggs-Tolle-Wang necessary and sufficient condition for Q-superlinear convergence of a class of quasi-Newton methods for solving this problem. Finally, we describe an algorithm which updates an approximation to a ``two-sided projected Hessian,'' a symmetric matrix of order n-m which can be expected to be positive definite near a solution. This idea was suggested in 1978 by Murray and Wright and a particular variant has recently been analysed by Coleman and Conn. We present several new variants of this algorithm and show that under cer- tain conditions they all have a local two-step Q- superlinear convergence property, even though only one set of gradients is evaluated per iteration. February 23, 1984