Decision Matrix
A lookup table for the question "given my objective, my bounds, and my formulation — is the problem well posed, which solution will I get, and can I trust it?" Read the prose reasoning behind every entry in Bounds, Branches, and Feasibility.
- Dataset authors: find your intended objective and confirm the instance lands in a green row before shipping.
- Validators: this is the map of where your formulation should agree with the benchmark and where it is expected to diverge. Divergence inside a "✗ not physical" cell is not a bug in your code.
Objective × bounds × formulation
Legend: ✓ = well behaved / exact / physical; ⚠ = conditional, verify; ✗ = ill posed / inexact / non-physical.
| Objective family | Bounded? (no op. limits) | Bounded? (full op. limits) | Branch favoured | Relaxation exactness (SOC/SDP) | Recommended model |
|---|---|---|---|---|---|
| Min generation cost (non-neg. coeffs) | ⚠ coercive only if strongly convex (e.g. strictly convex quadratic) | ✓ | High-voltage | ✓ under standard conditions | Relaxation, verify ex post |
| Min losses / min slack power | ✓ (coercive) | ✓ | High-voltage | ✓ | Relaxation |
| Feasibility (constant objective) | ✗ vacuous: every profile feasible, iterates drift (not "unbounded" — a constant objective can't be) | ✓ | Undetermined among feasible pts | ⚠ one-sided certificate only | Nonconvex or verified relaxation |
| Min import from grid | ✓ | ✓ | High-voltage | ⚠ with dispatchable DER (drives local injection up → upper v bounds) | Relaxation, watch upper bounds |
| Max Σ voltage magnitude | ⚠ unbounded above without v̄ | ✓ | High-voltage | ⚠ fails when upper v bounds bind | Relaxation, watch upper bounds |
| Max PV / DER injection | ✗ | ⚠ bounded but boundary-seeking | Drives to upper v bound | ✗ typically inexact | Nonconvex; see list |
| Max load served / loadability | ✗ | ⚠ drives to the nose | Toward low-voltage / nose | ✗ inexact near collapse | Nonconvex + continuation |
| Max transfer / ATC / interface flow | ✗ | ⚠ boundary-seeking | Maximizes current | ✗ | Nonconvex |
| Max export / feed-in | ✗ | ⚠ | Reverse-flow current max | ✗ | Nonconvex |
| Max storage throughput / arbitrage volume | ✗ | ⚠ | Cycling-current max | ✗ (plus SoC-constraint inexactness) | Nonconvex |
| Max revenue / profit | ✗ | ⚠ | More generation & flow | ✗ | Nonconvex |
| Min cost with a negative coefficient | ✗ | ⚠ | Inverted → toward low-voltage | ✗ data-induced | Audit signs; nonconvex if real |
"Bounded," "branch," and "exact" are three readings of one underlying property (see §5). A row that is ✗ on exactness is almost always boundary-seeking and low-voltage-leaning too. The green family at the top — objectives non-decreasing in generation/losses — is green in every column for the same reason.
The "branch favoured" column records which branch an objective pulls toward, not which branch a local solver will return. Confirm the sheet a solve actually landed on with the diagnostics; a green row is a reason to expect an operational answer, not evidence that you got one.
Formulation cross-cut
The table's exactness column assumes a single-phase, radial-or-mild-mesh setting. Adjust as follows:
- Linear approximations (LinDistFlow, network/DC): always bounded and convex given bounds; never "inexact" in the relaxation sense, but they cannot represent collapse or branch multiplicity at all. Use for screening, not for any question on this page.
- Convex relaxations (SOC-BFM/BIM, SDP): the exactness column applies. On meshed networks the radial guarantees weaken; on multiphase / unbalanced networks treat all ✓ as ⚠ and verify (Bernstein et al., 2018).
- Nonconvex AC: no exactness question, but every ✗/⚠ row can still converge to a low-voltage or locally optimal solution depending on the start point. Use a high-voltage (flat or warm) start and check the diagnostics.
See also: Bounds, Branches, and Feasibility · Objectives that imply loss maximization · Diagnostics & validation · Trusting the solver · References