cuOpt optimizes one objective per solve. Many real problems have several objectives that pull against each other — cost vs. service level, return vs. risk, makespan vs. overtime, distance vs. vehicle count. A single solve answers "what's optimal for one particular weighting," but it hides the tradeoff the user actually needs to see.

This skill turns a sequence of single-objective cuOpt solves into a Pareto frontier — the set of solutions where you can't improve one objective without giving up another — and gives the discipline to read it. It adds no solver features; it orchestrates the LP / MILP / QP solves already covered by the formulation and API skills.

Reach for this workflow when the problem has two or more objectives with no agreed-upon weighting, signalled by language like:

If there is a single clear objective (everything else is a hard constraint), this skill does not apply — formulate and solve once.

A single optimum encodes one implicit weighting of the objectives. Change the weighting and the optimum moves. The frontier is the curve traced by all the non-dominated optima.

A solution A dominates B when A is at least as good on every objective and strictly better on one. Dominated solutions are never worth choosing. The Pareto frontier is exactly the non-dominated set; the user's job is to pick a point on it, and yours is to show them the whole curve plus where the tradeoff is sharpest.

Do not collapse a multi-objective problem to a single weighted number and report its optimum as "the answer" — that silently makes the tradeoff decision for the user. Trace the frontier and let them choose.

Objectives and constraints are interchangeable. A requirement currently treated as fixed — a coverage floor, a fairness cap, a budget — is often a latent objective: its level was assumed, not given. Promoting such a constraint to a parametric ε-constraint and sweeping it reveals a tradeoff you'd otherwise hide, so read a single-objective model's hard constraints as candidate objectives, not just limits — but only when the level was an assumption. A genuinely fixed, non-negotiable limit (a hard budget cap, a regulatory minimum) stays a constraint; don't manufacture a tradeoff that isn't there. Express any promoted quantity linearly so it can serve as an ε-constraint (see ).

An informative frontier needs objectives that genuinely conflict: if they don't pull against each other, it collapses to a single point with nothing to trade off. And each objective has to be formulated correctly, since a wrong form, sense, or scale distorts the tradeoff and shifts where the knee falls. Formulate each one with before sweeping.

Solve each objective on its own first. For k objectives this is k solves. Record, for each, the value of every objective at that optimum:

              f1        f2        f3
min f1   →   f1*       f2(at f1*) f3(at f1*)
min f2   →   ...       f2*        ...
min f3   →   ...       ...        f3*

The diagonal (, , …) is each objective's best achievable value; the off-diagonals give the range each objective spans across the others' optima. This table does double duty:

If any single-objective solve is already infeasible, stop and fix the model before sweeping — the frontier doesn't exist yet.

Combine the objectives into one and sweep the weights:

minimize  w1·f1(x) + w2·f2(x) + ... ,   for a grid of weight vectors w

Cheap and trivial with any solver. Two limitations to respect:

Keep one objective; move the rest to constraints and sweep their right-hand sides:

minimize  f1(x)
subject to  f2(x) ≤ ε2
            f3(x) ≤ ε3
            (original constraints)

Sweep each across the range from the payoff table. Each 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 ) is simplest kept as the objective while you ε-constrain the linear ones. A convex quadratic objective can instead be ε-constrained directly: add it as a quadratic constraint , which cuOpt supports. 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).

Read that dual as the local exchange rate. Where the frontier is smooth, the dual on a swept ε-constraint is its slope — how much the kept objective moves per unit of the bound — at no cost beyond the solve already run; at a kink it gives only a one-sided rate. A zero dual means the bound is slack: the sweep has run past the frontier's edge. This reading needs LP/QP and a linear ε-constraint (MILP optima and problems with quadratic constraints return no duals) — where duals are unavailable, difference adjacent frontier points instead.

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.

frontier = []
for each weight vector (or ε vector) in the grid:
    set the combined objective (or ε right-hand sides)
    solve with cuOpt              # reuse the prior solution as a warm start
    if status is Optimal/Feasible:
        record (objective values, solution)
discard dominated and duplicate points
sort the survivors to form the frontier

Practical notes:

This skill is solver- and interface-agnostic. The per-solve mechanics — building the objective, adding the ε constraints, passing a warm start, reading status — live in the API skills: