Read epsilon-constraint duals as frontier slope / exchange rate in multi-objective skill

skills/cuopt-multi-objective-exploration/SKILL.md

@@ -92,10 +92,12 @@ subject to  f2(x) ≤ ε2
 
 Sweep each `ε_k` across the range from the payoff table. Each `(ε2, ε3, …)` combination is a single standard cuOpt solve. This recovers the **full** frontier, including the concave regions weighted-sum cannot reach, which is why it's the default when completeness matters. The cost is more solves (a grid over the constrained objectives) and bookkeeping of the ε values.
 
-ε-constrain *linear* objectives directly. A quadratic objective (e.g. risk `xᵀΣx`) is simplest kept as the objective `f1` while you ε-constrain the linear ones. A **convex** quadratic objective *can* instead be ε-constrained directly: add it as a quadratic constraint `xᵀQx ≤ ε` (Q positive semidefinite, inequality only), which cuOpt routes through the barrier solver as a second-order cone. Non-convex or equality quadratic constraints are unsupported, and the MILP path stays linear-constraint only.
+ε-constrain *linear* objectives directly. A quadratic objective (e.g. risk `xᵀΣx`) is simplest kept as the objective `f1` while you ε-constrain the linear ones. A **convex** quadratic objective *can* instead be ε-constrained directly: add it as a quadratic constraint `xᵀQx ≤ ε`, which cuOpt handles as a second-order cone. Non-convex or equality quadratic constraints are unsupported, and the MILP path stays linear-constraint only.
 
 Spot it in existing code: a hand-coded loop over a target or budget value (a return target, a cost cap) is already the ε-constraint method — name it as such, filter dominated points, and read the swept constraint's dual (LP/QP only).
 
+**That dual is the frontier's local slope.** The dual on a swept ε-constraint is how much the kept objective `f1` moves per unit of that bound — the exchange rate between the two objectives, and the frontier's tangent at that point, read off the solve for free (differencing two solves gives only a secant across any bend between them). A **zero** dual means the bound is slack, not binding — no tradeoff there, so the sweep has run past the frontier's edge. It applies to **LP/QP** off a **linear** ε-constraint only: MILP optima have no duals, and any quadratic constraint voids them for the whole solve, so keep a quadratic objective as `f1` and ε-constrain the linear ones; where no dual is available, difference adjacent points instead. (`cuopt-numerical-optimization-formulation` covers what duals mean and which problem types expose them.)
+
 **Picking a method:** weighted-sum for a quick convex sketch or when you know the frontier is convex (e.g. a pure-LP/QP tradeoff); ε-constraint when the problem is MILP, when the frontier may be non-convex, or when the user needs a faithful and complete curve.
 
 ## Step 4 — sweep, collect, and filter
@@ -117,14 +119,15 @@ Practical notes:
 - **Cap each MILP solve.** Set a per-solve time limit on MILP sweeps (see `cuopt-numerical-optimization-api-python`) — a sweep is many solves, and branch-and-bound can over-spend certifying optimality past a tiny gap, while cuOpt sets no limit by default and won't warn. Report the points as optimal *to the gap you set*, not certified optimal.
 - **Filter dominated points.** A correct sweep can still emit dominated points (especially weighted-sum near the hull, or MILP). Drop them; they are not part of the frontier.
 - **Resolution is a budget.** Curve fidelity trades against solve count. Start coarse to see the shape, then refine the grid only where the curve bends.
+- **Spend the budget where the slope changes (LP/QP).** Because the ε-constraint dual is the frontier's local slope, compare it across solved points: where it barely changes, the curve is nearly straight — interpolate rather than add solves; where it jumps by more than the solve tolerance, the frontier bends between those points — refine there (smaller differences are solver noise, not curvature). This concentrates solves where the curve actually bends instead of spreading them over a uniform grid. MILP has no duals, so judge where to refine from the gaps between primal objective values instead.
 - **Verify, don't assume.** When you claim one method beats another, measure it — e.g. count the efficient points ε-constraint recovered that weighted-sum missed — rather than asserting it; and flag any solve returning feasible-but-not-`Optimal` so a non-certified point is never read as exact.
 
 ## Step 5 — interpret the frontier
 
-- **Report tradeoffs, not single numbers.** A frontier point means nothing in isolation. Quote the exchange rate — "≈ $4k of extra cost per 1% of added coverage in this region" — so the user can judge whether a move is worth it.
-- **Flag knee points; don't auto-pick them.** The "knee" is where the curve bends most sharply — beyond it you pay a lot for a little. It's often the best-balanced compromise and worth highlighting, but the final choice is the user's preference, not a rule.
+- **Report tradeoffs, not single numbers.** A frontier point means nothing in isolation. Quote the exchange rate — "≈ $4k of extra cost per 1% of added coverage in this region" — so the user can judge whether a move is worth it. On an LP/QP frontier this exchange rate is the swept constraint's dual at that point — the local slope of the frontier, accurate to the solve's optimality tolerance (tighten it before relying on a dual); on MILP, estimate it from the gap to the adjacent frontier point.
+- **Flag knee points; don't auto-pick them.** The "knee" is where the curve bends most sharply — beyond it you pay a lot for a little. It's often the best-balanced compromise and worth highlighting, but the final choice is the user's preference, not a rule. At the knee the slope is two-sided — the dual just below differs from just above — so quote the exchange rate there as a range, not one number.
 - **Treat dominated or gappy output as a diagnostic.** If dominated points survive filtering, or the frontier is implausibly sparse or perfectly linear, suspect the sweep or the model — most often weighted-sum hiding a concave region (switch to ε-constraint) or a normalization mistake.
-- **State the weighting/ε you used.** Every reported point is conditional on its scalarization. Make that explicit so a single solve is never mistaken for "the" optimum.
+- **State the weighting/ε you used.** Every reported point is conditional on its scalarization. Make that explicit so a single solve is never mistaken for "the" optimum. On LP/QP, the ε-constraint duals are the *implicit weights* at that point — the effective price the solution puts on each constrained objective, and the weights a weighted-sum solve would need to reproduce that tradeoff. Reporting them makes the accepted tradeoff ratio explicit.
 
 ## Interfaces
 

skills/cuopt-multi-objective-exploration/evals/evals.json

@@ -24,7 +24,22 @@
       "Builds a payoff table (each objective alone) for ranges/normalization",
       "Traces the Pareto frontier via repeated single-objective cuOpt solves; prefers epsilon-constraint over weighted-sum for completeness on a non-convex/MILP problem",
       "Filters dominated and duplicate portfolios",
-      "Reports the tradeoff and flags the knee, leaving the final pick to the lead"
+      "Reports the tradeoff and flags the knee, leaving the final pick to the lead",
+      "Since supplier selection is MILP (no duals), estimates the cost/resilience exchange rate by differencing adjacent frontier points, not from constraint duals"
+    ]
+  },
+  {
+    "id": "multiobj-explore-eval-004-dual-exchange-rate",
+    "question": "An analyst is building a risk-return efficient frontier across a set of assets: minimize portfolio variance and maximize expected return, with no fixed risk appetite. Using cuOpt, how would you trace the frontier, and at each point how would you tell the investor the rate at which they are paying variance for extra return without spending additional solves?",
+    "expected_skill": "cuopt-multi-objective-exploration",
+    "expected_script": null,
+    "ground_truth": "The agent recognizes a two-objective tradeoff (variance vs return) with no fixed weighting and traces the frontier by epsilon-constraint: it keeps the quadratic variance as the objective and sweeps a linear return floor (return >= epsilon), each value a single convex QP solve. At each solved point it reads the dual value on the binding (linear) return-floor constraint as the marginal variance per unit of required return -- the local exchange rate, i.e. the slope (tangent) of the frontier at that point -- taken directly off the solve at no extra cost rather than by differencing two adjacent points, and accurate to the solve's optimality tolerance. It notes the dual reading applies to continuous LP/QP solutions, not MILP, and that the swept constraint must be linear since any quadratic constraint makes cuOpt return no duals for the whole solve. It filters dominated points, flags the knee, and leaves the risk appetite to the investor, deferring per-solve mechanics to the api-* skills and formulation to cuopt-numerical-optimization-formulation.",
+    "expected_behavior": [
+      "Frames variance vs return as two competing objectives with no agreed weighting",
+      "Traces the frontier via epsilon-constraint, keeping the quadratic variance as the objective and sweeping a linear return floor (each a single QP solve)",
+      "Reads the swept (linear) constraint's dual as the local exchange rate (marginal variance per unit return) and slope of the frontier, taken straight off each solve rather than by differencing",
+      "Notes the dual / exchange-rate reading applies to continuous LP/QP, not MILP, and is accurate to the solve tolerance",
+      "Filters dominated points, flags the knee, and leaves the final risk appetite to the investor"
     ]
   },
   {