Multi-Objective Exploration

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.

When this applies

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

• "balance X and Y", "trade off", "as cheap as possible without hurting service"
• "minimize cost and maximize coverage", "I want options, not one answer"
• any objective the user is willing to relax in exchange for another

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

Core idea — one solve is one point on a curve

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 cuopt-numerical-optimization-formulation).

Step 1 — define the objectives

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 cuopt-numerical-optimization-formulation before sweeping.

Step 2 — build a payoff table (anchor each objective)

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 (f1*, f2*, …) 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:

• It sets the sweep bounds for the ε-constraint method (the feasible range of each constrained objective).
• It supplies the scales for normalization — objectives in dollars, percent, and hours can't be weighted meaningfully until divided by their ranges.

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

Step 3 — choose a scalarization

Weighted sum

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:

• It only finds points on the convex hull of the frontier. Concave (non-convex) regions of the frontier are unreachable no matter how you choose weights, and for MILP the reachable points can be sparse with large gaps. A frontier that looks suspiciously linear or has only a few clustered points is the symptom.
• Weights are not priorities until the objectives are normalized. Divide each f_k by its payoff-table range first; otherwise the largest-magnitude objective dominates regardless of intent.

ε-constraint (preferred for a complete frontier)

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 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, 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 f1 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 usually means the bound is slack — the sweep has run past the frontier's edge (one-way: a slack bound always shows a zero dual, but under degeneracy a binding bound can too). 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.

Step 4 — sweep, collect, and filter

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:

• Warm-start LP sweeps. For an LP frontier, carry the previous solve's PDLP warmstart data into the next to cut solve time. Per cuOpt this is LP-only: a MILP solve doesn't take a PDLP warmstart (you can optionally seed a MIP start instead). See cuopt-numerical-optimization-api for the calls.
• Cap each MILP solve. Set a per-solve time limit on MILP sweeps (see cuopt-numerical-optimization-api) — 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. On MILP, 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 — complete the frontier: measure and fill what the sweep missed

A weighted-sum sweep returns only supported points (Step 3's convex-hull limitation); on MILP frontiers, non-supported points — the ones no weighted-sum weighting returns — often make up much of the non-dominated set. A coarse ε-constraint grid leaves gaps the same way: any finite sweep can miss regions. Before presenting a swept frontier, measure the likely miss and decide whether to fill.

Measure the miss

Sort the swept points by one objective. For each adjacent pair, form the rectangle (in general, the box) between them in objective space; flag any box much larger than the median adjacent box (3× is a reasonable bar) or covering a large share of the frontier's spanned area — a sweep that returned only a handful of points is all gaps, so no box stands out from the median. Large boxes have two causes — non-supported regions (weighted sum cannot reach them, common under fixed-charge structure) and weight clustering (a finite grid re-discovering the same corners, even on a nearly convex frontier). The fill step treats both the same.

If all boxes are small and even, the sweep is likely adequate — say so and stop.

Fill the largest gaps first

For each flagged box, solve one ε-constraint subproblem targeted inside it: optimize one objective with the other bounded at the box midpoint (bi-objective; with more objectives, sort by each objective in turn and place one target per flagged box instead of recursing). Only certified Optimal results settle or steer anything here — a time-limited incumbent is kept as a point (tagged, below) but proves nothing about the gap. A new certified point that survives Step 4's dominance filter means the gap was real (an ε solve can return a weakly optimal point) — bisect: two more targets inside the two sub-boxes it creates. A certified endpoint coming back clears just the probed side of the bound; certifying the whole box as a true discontinuity also needs a known objective step size — all-integer objective coefficients over integer variables give one — to place the bound just inside the far endpoint and match its certified optimum. Without that step size, report the box as a candidate gap, not a proven discontinuity. Stop on a solve budget, or when the remaining boxes fall below the flag bar.

Warm-start each solve (cheap insurance)

Consecutive fill solves differ by one bound, so seed each with its neighbor as a MIP start (Step 4's warm-start note) — one line, and it never changes what is optimal. Expect unchanged solve times; the value is insurance on hard subproblems.

Degrade gracefully, never silently

If a subproblem hits its time limit with a feasible incumbent (FeasibleFound), keep the point — it is feasible, and the solve's reported gap bounds its suboptimality — but record it as approximate. The time-capped solve is the primary fallback: it returns both an incumbent and a bound. Heuristics-only mode (mip_heuristics_only) drops the proof work and returns feasible points with no gap bound — use it when feasible points are all you need, and tag everything it returns approximate.

Report with provenance

Every presented point carries one of two tags:

• exact — Optimal at your gap setting, i.e. optimal to that gap (Step 4);
• approximate — time-limited incumbent (quote its reported gap) or heuristics-only result (no bound exists; say so).

State the counts with the frontier ("14 points, 11 exact, 3 approximate near the low-cost end, worst gap 2.4%"). Never present a mixed frontier as uniformly optimal.

Step 6 — 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. 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 (return to Step 5 and fill the gaps) 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. 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

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:

• cuopt-numerical-optimization-api — LP, MILP, QP solves (Python, C, CLI).
• cuopt-routing-api-python — the same frontier workflow applies to routing tradeoffs (distance vs. vehicles vs. time).