An evolutionary algorithm for multi-criteria inverse optimal value problems using a bilevel optimization model

Given a linear program, a desired optimal objective value, and a set of feasible cost vectors, one needs to determine a cost vector of the linear program such that the corresponding optimal objective value is closest to the desired value. The problem is always known as a standard inverse optimal value problem. When multiple criteria are adopted to determine cost vectors, a multi-criteria inverse optimal value problem arises, which is more general than the standard case. This paper focuses on the algorithmic approach for this class of problems, and develops an evolutionary algorithm based on a dynamic weighted aggregation method. First, the original problem is converted into a bilevel program with multiple upper level objectives, in which the lower level problem is a linear program for each fixed cost vector. In addition, the potential bases of the lower level program are encoded as chromosomes, and the weighted sum of the upper level objectives is taken as a new optimization function, by which some potential nondominated solutions can be generated. In the design of the evolutionary algorithm some specified characteristics of the problem are well utilized, such as the optimality conditions. Some preliminary computational experiments are reported, which demonstrates that the proposed algorithm is efficient and robust.