Problems catalog

Choose a problem by its formulation, encoding and dimensions, then check the algorithm compatibility matrix. This catalog describes the 136 registered problems in VAMOS 1.1.0. The named selection helpers are stable public APIs. A synthetic or engineering benchmark is a test model, not a validated model of your application.

Choose a starting point

What you want to exercise	Start with	Why
A first continuous run and a visible two-objective front	zdt1	Inexpensive evaluation, 30 variables, packaged 2D reference sample
Three or more objectives	dtlz2	Scalable objectives and a standard dimension rule
More varied geometry and difficulty	wfg1–wfg9, zcat1–zcat20	Configurable objective count; record the family parameters
Hundreds of variables	lsmop1–lsmop9	300 variables by default; group structure matters
Explicit constraint handling	c1dtlz1, mw1	Emit inequality constraints separately from objectives
Binary, integer or mixed operators	zdt5, int_alloc, mixed_design	Small, deterministic examples with explicit encodings
Permutation operators	tsp6, then kroa100	Closed tours with two specified route objectives
Application-inspired test functions	RE and RWA tables below	Fixed engineering formulations with different encodings and objective counts
Expensive model evaluations	fs_real, ml_tuning	Train a classifier for each candidate; require scikit-learn

Use optimize("zdt1", ...) in Python or --problem zdt1 in the CLI. Inspect the installed version through the public facade:

from vamos import available_problem_names, make_problem_selection

print(sorted(available_problem_names()))
selection = make_problem_selection("dtlz2", n_obj=3)
problem = selection.instantiate()
print(problem.n_var, problem.n_obj, problem.encoding)  # 12 3 real
print(problem.xl, problem.xu)

The tables report the instantiated default dimensions. n is the number of decision variables, M the number of objectives, and G the number of explicit inequality outputs (G <= 0 is feasible). Zero G means no separate inequality outputs; bounds and encoding rules still apply. All objectives returned in F are minimized, including sign-flipped maximization objectives.

Reference sample means a CSV shipped with the package, with its objective dimension shown. It does not certify an exact Pareto front or guarantee suitability after changing the formulation. See Reference fronts before computing indicators.

Continuous test suites

These families use real-valued variables (real or continuous in the metadata) and need only the base installation, including WFG. Source links point to the corresponding implementation or family registration in the 1.0.1 release.

Registered keys	Default n	Default M	G	Default parameters / dimension rule	Source	Reference sample
zdt1, zdt2, zdt3	30	2	0	Variables in [0, 1]; fixed M	ZDT	2D, each
zdt4	10	2	0	First variable [0, 1], remainder [-5, 5]; fixed M	ZDT	2D
zdt6	10	2	0	Variables in [0, 1]; fixed M	ZDT	2D
dtlz1	7	3	0	n = M + 4 (k = 5) when n is omitted	DTLZ	3D
dtlz2–dtlz6	12	3	0	n = M + 9 (k = 10); DTLZ4 uses alpha = 100	DTLZ	3D, each
dtlz7	22	3	0	n = M + 19 (k = 20)	DTLZ	3D
wfg1–wfg9	24	3	0	k = 4, l = 20; bounds [0, 2i] for variable i	WFG	2D only, each
lz09_f1, lz09_f7, lz09_f8	10	2	0	Fixed M; family-specific linkage	Li–Zhang LZ09	None packaged
lz09_f2–lz09_f5, lz09_f9	30	2	0	Fixed M; family-specific linkage	Li–Zhang LZ09	None packaged
lz09_f6	10	3	0	Fixed M	Li–Zhang LZ09	None packaged
cec2009_uf1–cec2009_uf7	30	2	0	Fixed M; UF5: N=10, UF6: N=2, both epsilon=0.1	CEC2009	2D, each
cec2009_uf8–cec2009_uf10	30	3	0	Fixed M; UF9: epsilon=0.1	CEC2009	3D, each
lsmop1–lsmop9	300	3	0	nk = 5 subcomponents per group	LSMOP	2D only, each
zcat1–zcat20	30	2	0	level = 1; complicated Pareto set, bias and imbalance all False	ZCAT	2D, 3D, 4D and 6D, each

DTLZ, WFG, LSMOP and ZCAT allow changing n_obj in named selection. For DTLZ, omitting n_var recalculates the standard dimension from M; setting a different n is allowed but changes the experimental setting. For WFG, k = 4 at M = 2, otherwise k = 2(M - 1), and l = n - k; WFG2/3 require an even distance-variable count. LSMOP needs enough variables for all groups to remain nonempty. Family constructor checks still apply to dimension overrides.

Explicitly constrained suites

All entries in this table are real-valued. They emit G alongside F; choose an algorithm with the required constraint support in the matrix.

Registered keys	Default n	Default M	G	Default parameters / objective count	Source	Reference sample
cec2009_cf1	30	2	1	Fixed M and constraint function	CEC2009	None packaged
c1dtlz1	12	3	1	Configurable M; n remains 12 unless supplied	C-DTLZ	2D only
c1dtlz3	12	3	1	Default radius r = 9 at M = 3; varies with M	C-DTLZ	2D only
c2dtlz2	12	3	1	Default radius r = 0.4 at M = 3; varies with M	C-DTLZ	2D only
c3dtlz1	12	3	M	Configurable M	C-DTLZ	None packaged
dc1dtlz1, dc1dtlz3	12	3	1	Configurable M; a = 5, b = 0.95	DC-DTLZ	2D only, each
dc2dtlz1, dc2dtlz3	12	3	2	Configurable M; a = 3, b = 0.9	DC-DTLZ	2D only, each
dc3dtlz1, dc3dtlz3	12	3	M	Configurable M; a = 5, b = 0.5	DC-DTLZ	None packaged
mw1, mw2, mw6, mw9	15	2	1	Actual M fixed at 2	MW	2D for MW1/2/6; none for MW9
mw3, mw7, mw12, mw13	15	2	2	Actual M fixed at 2	MW	2D for MW3/7; none for MW12/13
mw5, mw10	15	2	3	Actual M fixed at 2	MW	2D for MW5; none for MW10
mw11	15	2	4	Actual M fixed at 2	MW	None packaged
mw4, mw8, mw14	15	3	1	Configurable M	MW	None packaged

In 1.1.0, the registry accepts an objective override for every MW key, but only MW4, MW8 and MW14 actually change their objective count. Use the instantiated problem's n_obj and keep the other MW cases at two objectives.

In 1.0.2, cec2009_cf1 did not declare its constraint, so algorithms ran it as an unconstrained problem; it also stored the CEC2009 constraint c(x) >= 0 without changing its sign and searched [-1, 1] for variables 2 to n. VAMOS 1.1.0: CF1 declares its constraint, stores G = -c(x) so that G <= 0 is feasible, and uses the CEC2009 domain [0, 1]^n.

Binary, integer, mixed and permutation examples

These examples are useful for checking representations and operators. The synthetic instances have fixed generation seeds, independent of the optimizer's seed. Changing n constructs a different instance; it does not merely increase the search budget.

Key	Encoding	Default n / M	G	Default parameters	Source	Reference sample
zdt5	Binary	80 / 2	0	30-bit first block, ten 5-bit tail blocks	ZDT5	2D, 31 points for default n
bin_feat	Binary	50 / 2	0	Instance seed 12345	VAMOS synthetic	None packaged
bin_knapsack	Binary	50 / 2	0	Capacity ratio 0.4; instance seed 2023	VAMOS synthetic	None packaged
bin_qubo	Binary	30 / 2	0	Symmetric Q; instance seed 7	VAMOS synthetic	None packaged
int_alloc	Integer	20 / 2	0	Each allocation in 0..10; instance seed 321	VAMOS synthetic	None packaged
int_jobs	Integer	30 / 2	0	Five job labels 0..4; instance seed 99	VAMOS synthetic	None packaged
mixed_design	Mixed	9 / 2	0	Three real, three integer, three categorical; instance seed 123	VAMOS synthetic	None packaged
tsp6	Permutation	6 / 2	0	Six equally spaced cities on the unit circle	VAMOS TSP	None packaged
kroa100, krob100, kroc100, krod100, kroe100	Permutation	100 / 2	0	Fixed packaged coordinates from the respective TSPLIB instance	TSP formulation, coordinates	None packaged

What the objectives actually mean

For binary x, each bit selects one item. Every expression below is minimized.

Key	First objective	Second objective	Interpretation
bin_feat	Negative sum of selected utilities	Sum of selected feature costs	Synthetic utility/cost trade-off; no classifier is trained
bin_knapsack	Absolute difference between selected weight and target capacity	Negative selected value	Capacity deviation is an objective; overweight selections are allowed
bin_qubo	xᵀQx + bᵀx	Negative number of selected bits	A two-objective synthetic QUBO extension
int_alloc	Total allocation cost	Negative sum of weighted square-root returns	Diminishing returns; no global budget constraint
int_jobs	Weighted mismatches against preferred job labels	Largest job-type count divided by n	Encourages a balanced use of labels; does not model machines or makespan
mixed_design	Squared real/integer target errors plus category mismatches	Integer and category costs	Synthetic mixed-variable design trade-off
All TSP keys	Sum of edge lengths in the closed tour	Maximum edge length in that tour	Total distance versus longest leg

TSP decisions must contain every city exactly once. Both objectives use the same coordinates and unrounded Euclidean distances, including the closing edge. A kroa100 run therefore uses TSPLIB coordinates in VAMOS's two-objective formulation; published single-objective TSPLIB tour costs with integer distance rounding are not directly comparable. The named tsp6 uses circle coordinates; constructing TSP() without arguments uses a separate asymmetric six-city toy.

ZDT5 requires n >= 35 and (n - 30) divisible by 5. Changing its number of tail blocks changes its front, so the packaged 80-bit reference is not universal. The TSPLIB factories have fixed dimensions; supplying n_var does not resize their coordinate files.

Application-inspired benchmarks

RE: Tanabe–Ishibuchi engineering suite

These are fixed formulations: keep their dimensions as listed. G = 0 for all RE entries. Several include the sum of constraint violations as an additional objective; that trade-off differs from enforcing feasibility through G. The default bounds and discrete choices are fixed by the linked implementation.

Key / formulation	Encoding	n / M	Violation objective	Default parameters	Source	Reference sample
re21 — four-bar truss	Real	4 / 2	No	Fixed design bounds	RE2	2D
re22 — reinforced concrete beam	Mixed	3 / 2	f₂	Discrete reinforcement-area choices	RE2	None packaged
re23 — pressure vessel	Mixed	4 / 2	f₂	Thickness step 0.0625; two integer counts	RE2	None packaged
re24 — hatch cover	Real	2 / 2	f₂	Bounds [0.5, 4] to [4, 50]	RE2	2D
re25 — coil compression spring	Mixed	3 / 2	f₂	Integer coil count, discrete wire diameter	RE2	None packaged
re31 — two-bar truss	Real	3 / 3	f₃	Fixed design bounds	RE3a	3D
re32 — welded beam	Real	4 / 3	f₃	Fixed design bounds	RE3a	3D
re33 — disc brake	Real	4 / 3	f₃	Fixed design bounds	RE3a	3D
re34 — vehicle crashworthiness	Real	5 / 3	No	Fixed response surfaces	RE3a	3D
re35 — speed reducer	Mixed	7 / 3	f₃	One integer gear-teeth variable	RE3b	None packaged
re36 — gear train	Integer	4 / 3	f₃	Integer tooth counts 12..60	RE3b	None packaged
re37 — rocket injector	Real	4 / 3	No	Fixed response surfaces	RE3b	3D
re41 — car side impact	Real	7 / 4	f₄	Fixed response surfaces	RE4	4D
re42 — conceptual marine design	Real	6 / 4	f₄	Fixed design bounds	RE4	4D
re61 — water resource planning	Real	3 / 6	f₆	Fixed design bounds	RE6/9	6D
re91 — car cab	Real	7 / 9	No separate violation objective	Deterministic mean-value evaluation (stochastic=False)	RE6/9	9D

Version 1.1.0 has stale descriptive labels for several RE registry entries. In particular, the re24 selection metadata reports four variables while its factory creates the two-variable hatch-cover formulation. The table follows the implementation; inspect problem.n_var after instantiation and use make_problem_selection("re24", n_var=2) to keep this selection's dimension metadata aligned. Keep RE91's deterministic setting explicit when comparing it with stochastic formulations.

RWA: Zapotecas-Martínez and collaborators

All RWA entries use real variables, fixed dimensions and bounds, and zero separate G outputs. They evaluate application response functions. The implementation converts maximization goals to negative objective values.

Key / application	n / M	Default parameters	Source	Reference sample
rwa1 — honeycomb heat sink	5 / 2	Fixed response surface; maximize Nu, minimize friction factor	RWA1–4	2D
rwa2 — vehicle crashworthiness	5 / 3	Fixed response surface	RWA1–4	3D
rwa3 — synthesis gas production	3 / 3	Maximize CH₄ and CO; minimize H₂/CO	RWA1–4	3D
rwa4 — wire electrical discharge machining	5 / 3	Maximize cutting rate; minimize roughness and dimensional deviation	RWA1–4	3D
rwa5 — thermal storage	9 / 3	Fixed packed-bed response surface	RWA5	3D
rwa6 — steel milling	4 / 3	z = 1, d = 1; maximize removal rate	RWA6	3D
rwa7 — rocket injector	4 / 3	Three response objectives	RWA7–8	3D
rwa8 — rocket injector	4 / 4	Four response objectives	RWA7–8	4D
rwa9 — ultra-wideband antenna	10 / 5	Fixed passband/stopband response functions	RWA9–10	5D
rwa10 — repellent fabric	3 / 7	Fixed fabric performance response functions	RWA9–10	7D

Learning and mixed engineering examples

Install python -m pip install "vamos-optimization[examples]==1.1.0" for the two classifier examples. The base package is sufficient for welded_beam. See Installation for environments and extras.

Key	Encoding	Default n / M	G	Defaults and objective meaning	Source	Reference sample
fs_real	Binary	30 / 2	0	Breast-cancer data; stratified 70/30 split, random_state=0; logistic-regression validation error vs selected-feature count	Feature selection	None packaged
ml_tuning	Mixed	4 / 2	0	Same dataset/split; SVC validation error vs support-vector count plus 0.1 × degree	SVM tuning	None packaged
welded_beam	Mixed	6 / 2	4	Four real dimensions, one material category, one stiffener count; fabrication cost vs deflection	Engineering example	None packaged

fs_real penalizes the empty mask with error 1 and feature-count objective n. ml_tuning searches C in [10⁻², 10²], gamma in [10⁻³, 10¹], kernels rbf/linear/poly and degrees 2–5. These are validation objectives, not unbiased held-out test estimates. Both public classes also accept the wine dataset, which changes fs_real to 13 decision variables; named registry selection uses the defaults above.

VAMOS 1.1.0 constraint metadata: WeldedBeamDesignProblem declares its four constraint columns, so registry-created problems enforce them. In 1.0.2, an explicit problem.n_constraints = 4 was required; that assignment remains harmless. Follow the engineering lesson and inspect the returned G before selecting a design.

welded_beam is VAMOS's six-variable mixed design example; it is a different formulation from the four-variable, three-objective re32. Its four emitted G columns are shear stress, normal stress, deflection and the h <= b condition. Its formulas are not certified as the canonical four-variable welded-beam benchmark. Likewise, cec2009_uf3 currently uses [-1, 1] for variables 2 to n, so its domain needs independent reference validation before use as canonical CEC2009 UF3. See Known limitations.

Reference fronts

The packaged CSV directory contains finite reference samples. Check the CSV's number of columns, objective definitions, signs, scaling and parameter settings against the run. A file's existence alone is insufficient evidence of compatibility or a certified optimum.

• ZDT fronts have two columns; ZDT5's sample is tied to its default 80 bits.
• DTLZ fronts have three columns. Changing M needs a corresponding reference.
• WFG, LSMOP and the listed C-/DC-DTLZ samples have two columns, even though their registered default is three objectives. Supply a matching reference for default runs; do not use those two-column files for three-objective indicators.
• ZCAT supplies dimension-specific samples at M = 2, 3, 4 and 6. Other objective counts need a supplied reference; 1.1.0's automatic lookup can otherwise fall back to a 2D file.
• RE/RWA samples have the dimensions shown above. Their presence does not imply that every sample point is a certified global Pareto optimum.
• Families marked “None packaged” require a suitable external reference or a documented empirical reference construction when using reference-front metrics.

The CLI accepts --hv-reference-front path/to/front.csv with --hv-threshold; use one column per objective and validate the selected file before launching a study. A reference front is a set of objective vectors; a hypervolume reference point is one vector bounding the evaluated region. See Analysis for indicators and comparison guidance.

CLI presets

Preset	Members
families	zdt1, dtlz2, wfg4, tsp6 (includes different encodings)
zdt	ZDT1/2/3/4/6; binary ZDT5 is selected separately
dtlz, wfg, zcat, lz, lsmop	The corresponding suites above
cec	CEC2009 UF1–10 and CF1
constrained_many	The 10 C-/DC-DTLZ and 14 MW cases
tsp	tsp6
tsplib	kroa100, krob100, kroc100, krod100, kroe100
real_world	ml_tuning, welded_beam, fs_real; this preset does not include RE/RWA

Use --problem-set <preset>. A shared dimension override must make sense for every member; prefer explicit problem selections for studies that mix fixed and scalable formulations. For a new objective function, follow Build your own problem.

For verified application walkthroughs, see feature selection, model hyperparameters, and decision representations.