Effective Linear Calculational Method for Nonlinear Optimization with a Convex Polyhedral Objective Function and Linear Constraints

Abstract

This research proposes an effective linear calculational method based on convex cone concept for solving non-linear optimization problems with a convex polyhedral objective function and linear constraints. One familiar type of convex polyhedral objective functions is a triangular type, which can be normally solved by weighted goal programming (WGP).
The necessary preference information of WGP is weights of positive and negative deviational variables. Alternatively, the linear calculational method prefers the other operational way in designing of an objective function by deviational constants, which is practical for the decision maker. For convex polyhedral type objective function problems, conventionally separable convex programming and goal programming (GP) are applied. By separable convex programming, it needs to separate the objective function into line segments before solving the problem, which means increasing of variables and constraints. In case of GP each breakpoint is determined as a goal so the number of constraints and the deviational variables are drastically increased.
By the effective linear calculational method proposed in this paper, the problem could be simply formulated to the linear programming problem, which is easy for the decision maker to apply. Moreover this method has lower number of constraints and variables than existing methods so the calculational time can also be reduced.