Normal vector identification and interactive tradeoff analysis using minimax formulation in multiobjective optimization

In multiobjective optimization, tradeoff analysis plays an important role in determining the best search direction to reach a most preferred solution. This paper presents a new explicit interactive tradeoff analysis method based on the identification of normal vectors on a noninferior frontier. The interactive process is implemented using a weighted minimax formulation by regulating the relative weights of objectives in a systematic manner. It is proved under a mild condition that a normal vector can be identified using the weights and Kuhn-Tucker (K-T) multipliers in the minimax formulation. Utility gradients can be estimated using local preference information such as marginal rates of substitution. The projection of a utility gradient onto a tangent plane of the noninferior frontier provides a descent direction of disutility and thereby a desirable tradeoff direction, along which tradeoff step sizes can be decided by the decision maker using an explicit tradeoff table. Necessary optimality conditions are established in terms of normal vectors and utility gradients, which can be used to guide the elicitation of local preferences and also to terminate an interactive process in a rigorous yet flexible way. This method is applicable to both linear and nonlinear (either convex or nonconvex) multiobjective optimization problems. Numerical examples are provided to illustrate the theoretical results of the paper and the implementation of the proposed interactive decision analysis process.