Application of Lexicographic Goal Programming with Convex Optimization in Control Systems

Abstract

Goal Programming (GP) is a multi-objective optimization technique that is used when there are multiple conflicting goals in the cost function of an optimization problem. Lexicographic Goal Programming (LGP) is most commonly used form of GP when there are clear priorities in the goals that leads to a lexicographic order, i.e., priroritization. We can solve LGP problems by solving a sequence of optimization problems, where the optimal cost of an optimization problem in the sequence becomes a constraint for all optimization problems that are solved afterwards. In this paper, we present the basic LGP framework where each problem in the sequence is a convex optimization problem. We provide examples from control systems where LGP is a natural choice due to the clear priorities of objectives. LGP formulation leads to a sequence of convex optimization problems, which enables us to use polynomial time Interior Point Method (IPM) algorithms to solve these problems efficiently and potentially in real-time. In some cases, we also show that this prioritization provides convexification of the problem at hand, which would have otherwise required the solution of a non-convex optimization problem. Our primary objective in writing this paper is to illustrate the usefulness LGP formulation of control problems that can be encountered in aerospace engineering.