An information-passing strategy for achieving Pareto optimality in the design of complex systems

Abstract

As engineering systems grow in complexity, it becomes more challenging to achieve system-level designs that effectively balance the trade-offs among subsystems. Lewis and others have developed a well-known, traditional game-theoretic approach for formally modeling complex systems that can locate a Nash equilibrium design with a minimum of information sharing in the form of a point design. This paper builds on Lewis’ work by proposing algorithms that are capable of converging to Pareto-optimal system-level designs by increasing cooperation among subsystems through additional passed information. This paper investigates several forms for this additional passed information, including both quadratic and eigen-based formulations. Such forms offer guidance to designers on how they should change parameter values to better suit the overall system by providing information on directionality and curvature. Strategies for representing passed information are examined in three case studies of 2- and 3-player scenarios that cover a range of system complexity. Depending on the scenario, findings suggest that passing more information generally leads to convergence to a Pareto-optimal set. However, more iterations may be required to reach the Pareto set than if using a traditional game-theoretic approach.

Appendix: Remarks on the algorithm and its convergence

Definition 1

(Problem statement) Determine the Pareto set for the problem.

$$ \mathop{\hbox{min}}\limits_{{\bf x}\in {\mathcal{U}}} F_k \quad \hbox{for}\,1\leq k \leq N $$

Definition 2

(Pareto-optimal points) ${\bf x}^{\ast} \in \mathcal{U}$ is Pareto-optimal, if it does not exist an ${\bf y} \in \mathcal{U}$ such that F _j (y) ≤ F _j (x ^*) for all j such that 1 ≤ j ≤ N and F _k (y) ≠ F _k (x ^*) for some k such that 1 ≤ k ≤ N.

Definition 3

(Pareto set) The Pareto set is defined as the set of points ${\bf x}\in \mathcal{U}$ that are Pareto-optimal.

We observe that in practical terms, a point is Pareto-optimal if there exists no other point that decreases some objective function without causing a simultaneous increase in at least one objective function.

Definition 4

(EXACT algorithm)

• 1. All designers compute their respective minima ${\bf x}_{k} =\hbox{arg min}_{{\bf x} \in \mathcal{U}} F_k$;

• 2. The designers select a starting point x ₀ s.t. it is in the convex hull of the Pareto minima;

• 3. The distributed design process starts from designer p, all other designers k ≠ p pass to p their objective function at that point, i.e., $\bar{F}_k = F_k({\bf x}_0)$;

• 4. The designer p computes ${\bf x}_{new} = \hbox{arg min}_{{\bf x} \in \mathcal{U}} F_p({\bf x})$ such that $F_k({\bf x}_{new}) = \bar{F}_k$ for all k ≠ p;

• 5. The design process restarts from $p^{\prime}\neq p$, goes back to point 3 and continues until convergence is reached.

Remark 1 (Dominated inequalities)

The next point found by the algorithm is never worse than the previous. It is sufficient to note that by construction $F_k^{(n)}\leq F_k^{(n-1)}\leq \cdots \leq F_k^{(0)}$ for all k, where the upper index indicates the iteration number.

Proposition 1

If the Pareto set is convex and of dimension N − 1, then the algorithm is feasible and it converges to the Pareto Set in at most 1 iteration.

Proof

Since the Pareto set exists then minima and maxima for the F _j exist in $\mathcal{U}$ and are in the Pareto set. Since the Pareto set is convex then x such that ${\hbox{min}_{\mathcal{U}} F_j\leq F_j({\bf x})\leq \hbox{max}_{\mathcal{U}} F_j}$ for all j. If there exists one point in the Pareto set such that $\bar{F}_j=F_j({\bf x})$ for j ≠ k, then that point in the Pareto set is found by taking x = arg min F _k with $\bar{F}_j=F_j({\bf x})$. $\square$

Proposition 2

Suppose the Pareto exists and is of dimension N − 1 and there exists a point in the Pareto such that F _k  = F _k (x ₀) for all 1 ≤ k ≤ N 's but one, then the algorithm is feasible and it converges to the Pareto Set in at most N trials.

Proof

In order to show convergence, it is sufficient to apply the argument used in the earlier proof for all 1 ≤ p ≤ N. $\square$

Remark 2

On the Propositions

• The nature of the hypothesis for the two propositions is shown in Fig. 7. In the convex case, Fig. 7a, the projection of the starting point onto the Pareto set is guaranteed to occur. Convexity is not necessary as shown in Fig. 7b. In the 2 objective functions case, in order to achieve convergence in one iteration, it is sufficient that there exist two points in the Pareto set such that $F_1(x_{Pareto}^{\prime})= F_1({\bf x}_0)$ and $F_2({\bf x}_{Pareto}^{\prime\prime})= F_2({\bf x}_0)$. A similar argument can be used in the case that the Pareto set is not connected (Fig. 7c). In this case, the algorithm will take at most 2 trials to converge.

• Realization of equality constraints We observe that enforcing equality constraints is costly, especially in terms of shared information. Throughout the article, we compared the exact algorithm with the approximation of the shared objective function. In particular, we took that $F_k({\bf x}) -\bar{F}_k \approx \hbox{approx}\left(F_k \right) -\bar{F}_k$, where approx indicates an approximation operator. We note that $F_k({\bf x}) = \hbox{approx}\left(F_k\right) + \hbox{error approx.}$, where the error can be quantified.

• Estimates of the approximation errors In the article, we utilized a polynomial approximation operator. For this type of approximation strategy, we can provide the error estimate on the basis of the generalized mean value theorem. In particular, if we assume that $F_k\in C^{m+1}$ then the error at a point x due to truncation at order m will be at most $\frac{1}{(m+1)!}\left[\sup\limits_{{\bf x} \in \mathcal{U}} F_k^{(m+1)}(x)\right]({\bf x}- {\bf x}_0)^{(m+1)}$.

• Relaxation of shared constraints As shown in the previous remark, the shared constraint can be approximated numerically by means of polynomials. However, it is clear that feasibility of the design cannot always be guaranteed unless the approximation error is small.

Fig. 7

Visual explanation of Propositions 1 and 2