Multi-objective Pareto-frontier examples (workforce + portfolio)

[cafzal]

Multi-objective (Pareto frontier) examples — companion to the cuopt-multi-objective-exploration skill

Two examples that extend existing folders to demonstrate multi-objective Pareto-frontier exploration with cuOpt — the workflow added as the cuopt-multi-objective-exploration skill in NVIDIA/cuopt#1355 (discussion NVIDIA/cuopt#1351).

• workforce_optimization/workforce_optimization_multiobjective.ipynb (MILP) — turns the cost-minimizing workforce model into a tradeoff surface: cost vs. coverage (relax coverage == required and sweep a coverage floor) and cost vs. fairness (promote the base model's fixed max_shifts cap to a swept ε-constraint — a fixed constraint treated as a candidate objective). Reads the frontier as an exchange rate (marginal $ per shift), caps every MILP solve with a time_limit, and shows that an integer program has no constraint duals. ε-constraint is the default; weighted-sum is a one-line method note, not a demo.

• portfolio_optimization/QP_portfolio_frontier_duals.ipynb (QP) — recognizes the base QP notebook's hand-coded target-return loop as the ε-constraint method, rebuilds it as the named workflow, and adds the piece the manual sweep omits: the return-constraint dual (shadow price d(variance)/d(return)) along the efficient frontier, with the PDLP-tolerance caveat.

Both reuse the base notebooks' data, run on cuOpt alone (Colab GPU), and follow the repo's notebook idiom (GPU check → cuopt-cu12 install → solve). Notebooks ship output-stripped (repo convention — every existing cuopt-examples notebook has 0 cell outputs); the run evidence is below.

User testing

Both notebooks were built by following the skill, then run end-to-end on Colab T4, cuOpt 26.4.0 — clean, no API errors:

• Workforce — cost vs. coverage (MILP). Cheapest full-coverage plan = $468 (all 51 shifts staffed). The ε-constraint sweep traces a 52-point frontier (coverage 0–51); the marginal cost read off the curve rises ~$8 → $12 per added shift; all 52 solved to Optimal (0 FeasibleFound). A single solve was only ever the right-most point.
• Workforce — cost vs. fairness (MILP). Sweeping the max_shifts cap (the constraint-as-objective move): full coverage holds at $468 down to a cap of 11, then $470 / $473 / $484 at caps 10 / 9 / 8, and goes infeasible at ≤ 7 — a clean price-of-fairness curve plus a feasibility cliff.
• Portfolio (QP). The efficient frontier plus the return-constraint dual, which rises ~0 → 13.2 as the required return climbs (the marginal risk cost of return). All points solved to Optimal (0 PrimalFeasible), so the PDLP-tolerance caveat is mild here.

Notes

• Synthetic/toy data — same as the base notebooks; this demonstrates the method, not a staffing or investment study.
• MILP points all solved to Optimal (0 FeasibleFound) — optimal to cuOpt's gap tolerance (exact here, since labor cost is integer-valued); the per-solve time_limit is a guard that didn't bind.
• QP duals are optimal to PDLP tolerance (first-order solver); any PrimalFeasible point is flagged (none here).

Companion to the now-merged cuopt-multi-objective-exploration skill (NVIDIA/cuopt#1355).

[rgsl888prabhu]

Few minor suggestions, but test looks good. Awesome work @cafzal.

[mlubin]

I may be missing something but the notebook doesn't really show how the skill was used, it just shows the output. What is the take home for readers of the notebook?

[cafzal]

Reworked both intros around the method as explicit steps (recognize → constrain one → sweep → read the dual) and added a "Takeaway" section; cut the "follows the skill" framing. The key point is now explicit: a hand-coded target-return loop already is an ε-constraint sweep.

[mlubin]

What does it mean for the notebook to follow the skill? The notebook doesn't show how to use the skill; it just shows the output.

[cafzal]

Same resolution. Please let me know if the reframe better illustrates the workflow.