The problem with multi-objective optimization
Anyone who has run a multi-objective design study knows the concern of not knowing when to stop. In a single-objective optimization study, convergence is easy to reason about: there’s one performance score, and you can watch the best-so-far value flatten out. Multi-objective Pareto studies aren’t so easy. Instead of chasing a single best solution, we are searching for trade-offs between conflicting objectives that are “equally good”, means none of which dominates the others across all objectives. Consequently, we must track the performance of multiple solutions and come up with a procedure to quantify the convergence.
That raises three questions that are hard to answer just by eyeballing a pareto plot:
- Is the optimization actually converging, or is it still exploring?
- Have any feasible solutions been found yet, or is everything still violating constraints?
- Does my study need more compute budget?
Simcenter HEEDS 2604 answers all three with a new run-time analytics feature: the Pareto Convergence plot.
What the Pareto Convergence plot actually shows
The core idea is to track how much the Pareto front (solutions which improve one objective without making another one worse) is still moving, cycle over cycle, and turn that into a single number: the Pareto Drift.
Let’s take a closer look at how the curve is built. Let N be the latest cycle evaluated at the moment the plot is drawn, and let the reference set be the accumulated Pareto solutions (Rank 1) found from the start of the study through cycle N, i.e. the best-so-far front, as of right now. Now we sweep a second index, across every cycle from 1 up to N. For each T:
- Take the accumulated Pareto set found from the start of the study through cycle T (not just the solutions discovered in cycle T).
- Measure its average distance to the reference set (the accumulated front through N).
Plotting that distance against T, for every T from 1 to N, produces the drift curve. Two things fall out of this naturally:
- T = N always sits at (or near) zero drift, since at that point you’re comparing the accumulated front to itself.
- Early T values show the largest drift, since the front was still immature and far from where the study eventually settled.
So, the curve isn’t built once and appended to every time a new cycle finishes, N moves forward, the reference set updates, and the entire curve is recomputed against the new reference. Watching the curve flatten near zero as it approaches the right edge is the signal that the front has stabilized: like watching a residual converge in a CFD run, except here the whole history gets re-evaluated against the newest endpoint each time. The figure below shows two different states of the drift curve, after 600 designs (left) and after 800 designs (right). It noticeably changed magnitudes and the shape is slightly changed as well.
A bonus: feasibility detection for free
Because of how the drift value is normalized, it carries a second piece of information almost as a side effect: feasibility.
- Pareto Drift > 1 → at least one constraint is still violated; the current front includes infeasible solutions.
- Pareto Drift between 0 and 1 → feasible solutions are present.
This means the same plot that tells you whether the optimization has converged also tells you when the study first crossed into the feasible region. For heavily constrained problems this is enormously useful. Rather than scanning through hundreds of designs after the fact to find the first feasible candidate, you can see the exact cycle where the curve drops below the feasibility threshold. The figure above shows the drop of the drift value between the 11th and 12th cycle together with the corresponding pareto plot of the feasible region. In cycle 12 we obtained the first 2 feasible designs in rank 1. Rank 2 remains infeasible
Takeaway
The Pareto Convergence plot in Simcenter HEEDS 2604 turns “is my multi-objective study converging?” from a judgment call into something you can read off a chart, in real time, while the study is still running. For teams running expensive multi-objective studies it’s a small addition with a tremendous effect on how confident you can manage compute budget and communicate progress.
The Author
Florian Vesting, PhD
Contact: support@volupe.com
+46 768 51 23 46
