Gradient-based multi-objective optimization with applications to waterflooding optimization

We consider problems where it is desirable to maximize multiple objective functions, but it is impossible to find a single design vector (vector of optimization variables) which maximizes all objective functions. In this case, the solution of the multi-objective optimization problem is defined as the Pareto front. The defining characteristic of the Pareto front is that, given any specific point on the Pareto front, it is impossible to find another point on the Pareto front or another feasible point which yields a greater value of all objective functions. The focus of this work is on the generation of the Pareto front for bi-objective optimization problems with specific applications to waterflooding optimization. The most straightforward way to obtain the Pareto front is by application of the weighted sum method. We provide a procedure for scaling the optimization problem which makes it more straightforward to obtain points which are approximately uniformly distributed on the Pareto front when applying the weighted sum method. We also compare the performance of implementations of the weighted sum and normal boundary intersection (NBI) procedures where, with both methodologies, a gradient-based algorithm is used for optimization. The vector of objective functions maps the set of feasible design vectors onto a set Z, and it is well known that all points on the Pareto front are on the boundary of Z. The weighted sum method cannot find points which are on the concave part of the boundary of Z, whereas the NBI method can be used to find all points on the boundary of Z, even though all points on this boundary may not correspond to Pareto optimal points. We develop and implement an NBI algorithm based on the augmented Lagrange method where the maximization of the augumented Lagrangian in the inner loop of the augmented Lagrange method is accomplished by a gradient-based optimization algorithm with the necessary gradients computed by the adjoint method. Two waterflooding optimization problems are considered where we wish to optimize (maximize) two conflicting objectives. In the first, the two objectives are to maximize the life-cycle net present value (NPV) of production and to maximize the short-term NPV of production. In the second application, given an uncertain reservoir description, we wish to maximize the expected value of the NPV of life-cycle production and minimize the standard deviation of NPV over the ensemble of geological realizations.