Double Secant Method for Nonlinear Mathematical Programming

Abstract

This papaer presented a method to solve the general nonlinear programming. The nonlinear constraints and the objective function are linearized and solved by the simplex method with consideration of upper bounds for the move limits in each iteration. For each design variable, we use two variables in linearized programming problem. One variable is for the increased value, the other variable is for the decreased value. For the increased variable, x i + , its coefficient in the constraint equation is the divided difference of the values of constraints at (x 1, ..., x i + d i + ..., x n ) and (x 1, ..., x i , ..., x n ). Similarly, for the decreased variable, x i − , its coefficient is the divided difference of the values of constraints at (x 1, ..., x i − d i − , ..., x n ) and (x 1, ..., x i , ..., x n ). Where the d i + and d i − can be different for two directions, and are chosen to be the move limits for the next step. This situation is much different from the so called cutting plane method. In the proposed method, each constraint surface is represented by a surface formed by many plane surfaces parallel to the planes cut through the initial point. Therefore, the linearized constraint surface is not a plane in the original n-dimensional design space as in the cutting plane method but is a multiple plane surface in the original n-dimensional design space. Although the linearized constraint surface is not a plane in the original n-d space, it is indeed a plane in the 2n-dimensional enlarged super-space. The advantage is the error of the linearization from the proposed method is quite smaller than that from the cutting plane method.