variable-aggregation

name

variable-aggregation

description

Use when reducing a discopt nonlinear optimization model with variable aggregation (reduced-space presolve) — substituting variables defined by equality constraints to shrink the model, improve interior-point convergence reliability, or reproduce Naik et al. (arXiv:2502.13869). Covers method choice (ld1/ecd2/ld2/d2/gr/lm), the structure-preserving-vs-Hessian-cost tradeoff, numerical guards, primal/dual recovery, implicit elimination of cyclic blocks, structural diagnostics, and the study/benchmark harnesses. Triggers on the discopt.aggregation package, "aggregate variables", "reduced-space formulation", "presolve substitution", or eliminating intermediates from an NLP/DAE model.

Variable aggregation (discopt.aggregation)

Overview

Variable aggregation substitutes a variable that is defined by an equality constraint into the rest of a model, eliminating both a variable and a constraint to produce a smaller reduced-space formulation, then recovers the eliminated variables from the reduced solution. It is a presolve reduction for nonlinear programs, implementing Naik, Biegler, Bent & Parker, Variable aggregation for nonlinear optimization problems (arXiv:2502.13869), as the discopt.aggregation plugin.

import discopt.modeling as dm
from discopt.aggregation import aggregate, solve

m = dm.Model("ex")
x = m.continuous("x", lb=-5, ub=5)
y = m.continuous("y", lb=-5, ub=5)
m.subject_to(y - (x + 1) == 0.0)      # y is DEFINED by x
m.minimize(x**2 + y**2)

res = solve(m, method="d2")            # aggregate -> solve -> recover
print(res.status, res.objective)       # optimal 0.5
print(res.x)                           # {'x': -0.5, 'y': 0.5}  (y recovered)

Core value: a smaller, often better-conditioned problem that interior-point solvers converge on more reliably — if you pick a method that doesn't inflate the Hessian. The whole skill is about that "if".

The one thing to get right: structure-preserving vs approximate-maximum

This is the central tradeoff and the source of every pitfall.

Family	Methods	Eliminates	Effect	Use when
Structure-preserving	ld1, ecd2, ld2, d2	≤ ~60% (only ≤2-variable defining eqs)	keeps per-constraint density ~flat; per-iteration cost stays low	default choice; d2 is the recommended balance
Approximate-maximum	gr, lm	70–90% (any linearly-defined var)	substitutes multi-variable / nonlinear expressions everywhere → denser, more nonlinear constraints; Hessian evaluation can become the bottleneck	you want maximum size reduction AND have checked the reduced model isn't expression-blown-up (use order="min_fill")

Rule of thumb: start with method="d2" (the paper's recommendation and the default). Only reach for gr/lm when you specifically need aggressive elimination, and then pass order="min_fill" and/or max_def_nodes= to bound the substitution blow-up. See references/methods.md and references/troubleshooting.md.

Two entry points

• aggregate(model, method="d2", ...) -> AggregationResult — build the reduced model + recovery map. Inspect .reduced_model, .eliminated, .n_passes, .recover_full(kept_values).
• solve(model, method="d2", ...) -> AggregatedSolveResult — aggregate, solve the reduced model, recover the full-space solution in one call. Returns .status, .objective, .x (full-space, original shapes), .duals (if recover_duals=True).

Both accept the same aggregation options (below). solve forwards extra kwargs to the reduced model's .solve().

Method selection (quick)

ld1   fixed variables  y = a                        (most conservative)
ecd2  equal-coeff 2-var y = x + a                    structure-preserving
ld2   linear 2-var     y = a*x + b                   structure-preserving
d2    degree-2         y = f(x)   (≤2 vars)          DEFAULT, recommended
d<k>  degree-k         y = f(x1..xk) (≤k vars)       d3, d4, ... (intermediate)
gr    greedy           y = f(w,x,…) (y linear)       approximate-maximum
lm    linear-matching  y = f(w,x,…) (y linear)       approximate-maximum

ld1/ecd2/ld2/d2/d<k> run fixed-variable elimination first, then the degree-bounded pass, recursively to a fixed point. gr/lm are applied once. Integer/binary variables are not eliminated by default; opt in with integer_aggregation=True to also eliminate a scalar integer/binary variable when its integrality is implied (see the integer_aggregation row below and references/troubleshooting.md). Full decision guidance: references/methods.md.

Options reference (when to use each)

All are keyword args to aggregate/solve. Defaults are safe no-ops unless noted.

Option	Default	Use when
method	"d2"	pick the family (above)
recursive	True	False = exactly one pass of the named strategy
pivot_tol	1e-6	relative pivot guard; 0.0 = exact paper behavior
prune_bounds	True	omit box→inequality constraints proven redundant by interval propagation (keeps bounded-variable elimination a net win)
scalarize	True	rewrite array variables to scalars so aggregation reaches discretized/DAE models (no-op on scalar models)
max_fill	None	cap per-elimination Jacobian fill; drops high-fan-out eliminations
max_def_nodes	None	cap the substituted expression's node count — use with gr/lm to bound Hessian blow-up
max_condition	None	cap worst-case recovery-error amplification (drops ill-conditioned chains a large coefficient or tiny pivot would create)
order	"index"	"min_fill" = fill-aware greedy; shrinks gr/lm expression graphs ~40× for the same eliminations
tearing	"greedy"	"exact" = maximum-acyclic-subset tear of cyclic blocks (eliminates ≥ greedy per block)
decomposable	False	local-search the lm matching for a more decomposable (higher Theorem-2 bound) structure
preserve_target_linearity	False

Deep dive with recipes: references/options.md.

Safety and correctness (built in)

• Infeasibility is surfaced, not hidden. If a fixed-variable elimination's value violates its bounds, or a fully-determined system is inconsistent, solve() returns status="infeasible" (with a warning) instead of a bogus optimal.
• No dangling references. Variables inside opaque nodes (CustomCall, matrix multiply) are correctly tracked and substituted; genuinely unanalyzable nodes raise rather than silently corrupt the model.
• Deterministic. Results are reproducible across PYTHONHASHSEED (sorted graph iteration), so a run is repeatable.
• Guards are sound. max_fill, max_def_nodes, max_condition only ever drop eliminations; the remaining set stays validly lower-triangular.

When aggregation helps — and when it hurts

Helps: convergence reliability (the paper's headline — more parameter instances converge), and solve time when the reduced KKT system factorizes faster. Best on models with many linearly-defined intermediates (DAE discretizations, flowsheets, recycle loops).

Hurts: gr/lm on a model where the eliminated intermediates feed a nonlinear objective/constraint — substitution compounds the expression graph (worst case degree-2^n), and Hessian evaluation dominates. A chain of 2-variable-eliminable constraints is the classic trap. Mitigations, in order: order="min_fill" → max_def_nodes= → use d2 instead → keep the block implicit (see below).

Beyond primal recovery

• Dual recovery — solve(..., recover_duals=True).duals gives full-space Lagrange multipliers (surviving + eliminated defining equalities), validated by a KKT stationarity self-check. Reports available=False with a reason for out-of-scope cases (maximize objective, active eliminated-variable bound) rather than a wrong answer. See references/recovery-and-implicit.md.
• Implicit elimination — eliminate_implicit(model) eliminates cyclic (irreducible) blocks that explicit substitution can't, via a differentiable inner solve (Model.implicit), keeping the reduced model AD-differentiable. This is the reduced-space move for recycle loops / index-1 DAEs.
• Structural diagnostics — structural_diagnosis(model) runs a Dulmage–Mendelsohn decomposition to flag over/under-determined subsystems before eliminating (the square block is exactly what aggregation targets).
• Schur/ordering — kkt_schur_indices(model) and block_triangular_ordering(model) emit the reducible block / permutation for a structured-KKT (Schur-complement) linear solve.

Study & reproduction harnesses (paper Tables 4/6/7)

• compare_methods(model) → Table-4 structural comparison (vars, cons, elim, NNZ/con, Hessian coupling) across all methods.
• benchmark_model(model) / callback_breakdown(model, method) → runtime and per-callback (func/grad/jac/hess) cost breakdown (Table-6 analogue).
• reliability_sweep(problem, methods, points) + virtual_best(results) + sweep_to_csv(...) → convergence-reliability grids (Figs 9–12, Table 7).
• PROBLEMS registry of parameterized test problems (reaction_diffusion, unit_selection, gas_pipeline, recycle_loop, cstr_dynamic, inventory_chain). Details and the local-vs-global-solve caveat: references/study-tools.md.

Task routing

• Which method / when → references/methods.md
• Every option, with recipes → references/options.md
• gr/lm is slow, hangs, or densifies; infeasibility; recovery failed → references/troubleshooting.md
• Dual recovery, implicit cyclic-block elimination, Schur/ordering → references/recovery-and-implicit.md
• Reproducing the paper's structural/runtime/reliability study → references/study-tools.md
• Runnable end-to-end scripts → examples/ (and python -m discopt.aggregation.examples for the in-tree worked examples, one per feature)

Installation / sanity check

pip install discopt-aggregation          # or: uv pip install -e .  (in the repo)
python -c "from discopt.aggregation import aggregate, solve; print('ok')"

discopt.aggregation (requires discopt >= 0.6) is a PEP-420 namespace subpackage that merges into the installed discopt package at import time.

This skill itself ships inside the wheel. Claude Code does not auto-discover skills from site-packages, so after a pip install run the bundled installer once:

discopt-aggregation-skill install         # -> ./.claude/skills/ (project)
discopt-aggregation-skill install --user  # -> ~/.claude/skills/ (all projects)