An Efficient Bi-Objective Optimization Workflow Using the Distributed Quasi-Newton Method and Its Application to Well-Location Optimization

Summary Although it is possible to apply traditional optimization algorithms to determine the Pareto front of a multiobjective optimization problem, the computational cost is extremely high when the objective function evaluation requires solving a complex reservoir simulation problem and optimization cannot benefit from adjoint-based gradients. This paper proposes a novel workflow to solve bi-objective optimization problems using the distributed quasi-Newton (DQN) method, which is a well-parallelized and derivative-free optimization (DFO) method. Numerical tests confirm that the DQN method performs efficiently and robustly. The efficiency of the DQN optimizer stems from a distributed computing mechanism that effectively shares the available information discovered in prior iterations. Rather than performing multiple quasi-Newton optimization tasks in isolation, simulation results are shared among distinct DQN optimization tasks or threads. In this paper, the DQN method is applied to the optimization of a weighted average of two objectives, using different weighting factors for different optimization threads. In each iteration, the DQN optimizer generates an ensemble of search points (or simulation cases) in parallel, and a set of nondominated points is updated accordingly. Different DQN optimization threads, which use the same set of simulation results but different weighting factors in their objective functions, converge to different optima of the weighted average objective function. The nondominated points found in the last iteration form a set of Pareto-optimal solutions. Robustness as well as efficiency of the DQN optimizer originates from reliance on a large, shared set of intermediate search points. On the one hand, this set of searching points is (much) smaller than the combined sets needed if all optimizations with different weighting factors would be executed separately; on the other hand, the size of this set produces a high fault tolerance, which means even if some simulations fail at a given iteration, the DQN method’s distributed-parallelinformation-sharing protocol is designed and implemented such that the optimization process can still proceed to the next iteration. The proposed DQN optimization method is first validated on synthetic examples with analytical objective functions. Then, it is tested on well-location optimization (WLO) problems by maximizing the oil production and minimizing the water production. Furthermore, the proposed method is benchmarked against a bi-objective implementation of the mesh adaptive direct search (MADS) method, and the numerical results reinforce the auspicious computational attributes of DQN observed for the test problems. To the best of our knowledge, this is the first time that a well-parallelized and derivative-free DQN optimization method has been developed and tested on bi-objective optimization problems. The methodology proposed can help improve efficiency and robustness in solving complicated bi-objective optimization problems by taking advantage of model-based search algorithms with an effective information-sharing mechanism. NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

An approximation algorithm for multi-objective optimization problems using a box-coverage

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin.

A note on completely positive relaxations of quadratic problems in a multiobjective framework

In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.

A Comparison of Benson’s Outer Approximation Algorithm with an Extended Version of Multiobjective Simplex Algorithm

The multiple objective simplex algorithm and its variants work in the decision variable space to find the set of all efficient extreme points of multiple objective linear programming (MOLP). Other approaches to the problem find either the entire set of all efficient solutions or a subset of them and also return the corresponding objective values (nondominated points). This paper presents an extension of the multiobjective simplex algorithm (MSA) to generate the set of all nondominated points and no redundant ones. This extended version is compared to Benson’s outer approximation (BOA) algorithm that also computes the set of all nondominated points of the problem. Numerical results on nontrivial MOLP problems show that the total number of nondominated points returned by the extended MSA is the same as that returned by BOA for most of the problems considered.

A general branch-and-bound framework for continuous global multiobjective optimization

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.

Enumeration of the Nondominated Set of Multiobjective Discrete Optimization Problems

In this paper, we propose a generic algorithm to compute exactly the set of nondominated points for multiobjective discrete optimization problems. Our algorithm extends the ε-constraint method, originally designed for the biobjective case only, to solve problems with two or more objectives. For this purpose, our algorithm splits the search space into zones that can be investigated separately by solving an integer program. We also propose refinements, which provide extra information on several zones, allowing us to detect, and discard, empty parts of the search space without checking them by solving the associated integer programs. This results in a limited number of calls to the integer solver. Moreover, we can provide a feasible starting solution before solving every program, which significantly reduces the time spent for each resolution. The resulting algorithm is fast and simple to implement. It is compared with previous state-of-the-art algorithms and is seen to outperform them significantly on the experimented problem instances.

Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms

The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in [Formula: see text] time. This improves upon the result of [Formula: see text] in Bader ( 2009 ), and slightly improves upon the result of [Formula: see text] in Bringmann et al. ( 2014b ), which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.

A Pseudo-Polynomial Time Algorithm for a Special Multiobjective Order Picking Problem

In this study, we work on the order picking problem (OPP) in a specially designed warehouse with a single picker. Ratliff and Rosenthal [Operations Research31(3) (1983) 507–521] show that the special design of the warehouse and use of one picker lead to a polynomially solvable case. We address the multiobjective version of this special case and investigate the properties of the nondominated points. We develop an exact algorithm that finds any nondominated point and present an illustrative example. Finally we conduct a computational test and report the results.