Decision SupportA mesh adaptive direct search algorithm for multiobjective optimization☆
Introduction
A common goal in multiobjective optimization is to identify the best trade-off solutions between different criteria. In this paper, we consider the multiobjective problem under general constraints, which may be stated aswithwhere is a nonempty subset of is the number of variables, and p is the number of objective functions. In the context of nonsmooth optimization, the set is typically defined through blackbox constraints given by an oracle, such as a computer code may even return a yes or no indicating whether or not a specified trial point is feasible. Examples of such blackboxes in the single-objective case are described in [6], [19]. There is usually no single solution that simultaneously minimizes each of the p objective functions, and a notion of trade-off between solutions is required. Pareto dominance [25] are used to define the set of optimal trade-off solutions of a MOP problem: Definition 1.1 Let be two decision vectors. Then, (u weakly dominates v) if and only if for all . (u dominates v) if and only if and for at least one . (u is indifferent to v) if and only if u does not dominate v, and v does not dominate u. is called Pareto optimal if there is no that dominates u.
The aim of this paper is to propose an algorithm for multiobjective blackbox optimization. The algorithm relies on the recent mesh adaptive direct search (Mads) algorithm for single-objective optimization [5], and on the natural boundary intersection method of [13] to handle the multiple objectives. The convergence of the proposed method is analyzed, and numerical results are presented on five test problems.
The paper is organized as follows. An overview of pointwise approximation methods is presented in Section 2. In particular, two methods are described: the natural boundary intersection (NBI) method [13] for multiobjective optimization and the biobjective mesh adaptive direct search (BiMads) algorithm [8] for biobjective optimization. Then, new single-objective formulations are introduced and the algorithm MultiMads is detailed in Section 3. MultiMads combines techniques from both NBI and BiMads. MultiMads extends BiMads in terms of scalability to any number of objectives. It is applicable to a wider class of problems than NBI since it allows blackbox functions whereas NBI requires problems whose objectives and constraints are twice continuously differentiable. In Section 4, MultiMads is tested on problems from [15], [14] with various complexities and landscapes designed to highlight difficulties which may be encountered in real-world problems. Some test problems are scalable to an arbitrary number of variables. Finally, the algorithm is tested on a tri-objective problem from chemical engineering with eight variables and nine general constraints.
Section snippets
Pointwise approximation methods
This section describes two different pointwise approximation methods. The NBI method [13] for multiobjective optimization is presented in Section 2.1, and the BiMads algorithm [8] for biobjective optimization is presented in Section 2.2. Ideas from both methods are combined in the new algorithm proposed in the present paper.
The MultiMads algorithm for multiobjective optimization
The property that Pareto points may be ordered in BOP is exploited by BiMads to define the reference points. This strategy cannot be easily implemented in MOP due to the lack of ordering property. Hence, we propose in this section another strategy to generate the reference point from an alternate simplex to the Convex Hull of Individual Minima (CHIM). Section 3.1 proposes a new single-objective formulation, and the general scheme of Mmads is presented in Section 3.2.
Numerical results
The performance of MultiMads is evaluated using test problems from Deb et al. [15], on a problem with a discontinuous objective and a discontinuous Pareto front and finally, on a problem from the chemical engineering. The simplicity of construction, the scalibility to any number of variables and objectives, the a priori knowledge of the Pareto front and the introduction of various complexities are the main features of the test suite [15]. Indeed, the problems test the ability of an algorithm to
Discussion
We proposed a new solution approach for constrained multiobjective optimization MOP ensuring some first-order necessary optimality conditions for nonsmooth functions. Our algorithm MultiMads combines strategies from both the NBI and BiMads algorithms. Our approach is not limited to blackbox optimization, but in the present work we chose to measure its effect on such problems using our Mads single-objective optimization method. We believe that using the approach on smooth problems would benefit
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Work of the first author was supported by Fcar grant Nc72792, Nserc grant 239436-0, Afosr FA9550-07-1-0302 and ExxonMobil Upstream Research Company.