LUNA · OPTFIN ACADEMY

What do you want to understand, optimize or prove?

Luna turns your question into a structured research route across OptFin models, methods, evidence and executable tools.

ACADEMIC LABORATORY · US ↔ EUROPE

Optimization without disciplinary borders.We turn complex decisions into explicit models, executable methods and verifiable results.

OptFin.org connects Operations Research, artificial intelligence, computer science, mobility, computational biology, financial decision science and public systems through explicit models, executable tools and reproducible evidence.

Luna

QUBO & Dimensional CollapseCertified structure freezes variables and concentrates search on an active core.

VERIFICATION STATUS

RUNfresh local suite in progress READmethods and claim boundaries CHECKproof requirements SOURCEpublic code and history

Intelligence restructures the problem.

An informed system isolates the active frontier and transforms a difficult global graph into a smaller explicit decision.

A result must be feasible, not poetic.

Every candidate carries a precise status, objective evaluation, constraint replay and review boundary.

Exactness lives locally; impact travels globally.

Exact kernels, decomposition and hybrid search work together across scientific and institutional domains.

THEORY

The OptFin structural hypothesis

The OptFin structural hypothesis holds that a difficult optimization problem often contains a smaller decisive structure: a set of variables, constraints, symmetries, bottlenecks or decomposable interactions that determines most of the global decision. A sufficiently informed system should identify that structure, isolate a bounded subproblem, solve or repair it with an auditable method, and reintegrate the certified result without violating global feasibility. Intelligence is therefore measured by valid bounds, explicit assumptions, reproducible computation and a traceable certificate of what was proved, improved or left unresolved.

STEP 1

Locate the active frontier

Detect the variables, arcs, residues, assignments or decisions that dominate the objective.

STEP 2

Collapse the decisive structure

Rewrite that region as a bounded model with explicit assumptions and guards.

STEP 3

Recover a certified move

Lift the local result into the global instance and replay feasibility.

STEP 4

Drive bounds to convergence

Track certificates, repairs, lower and upper bounds, and honest stopping criteria.

ARCHITECTURE

A research architecture for exactness, repair and proof

Global decision layer

Domain model and instance parser
Feasible incumbent manager
Bound and proof registry
Multidisciplinary experiment portfolio

Critical structure engine

Dynamic programming
Branch-and-bound and cutting planes
Exact enumeration for small kernels
Restricted master and decomposition methods

Recovery and evidence layer

Feasible reconstruction and repair
Hybrid metaheuristics
Status semantics and claim boundaries
Tests, hashes and reproducibility records

45 DEFINED MATHEMATICAL PROBLEMS

DEFINED MATHEMATICAL PROBLEMS

Arc routing2 problems

CARPArc routing

Capacitated Arc Routing Problem

min z = service cost + deadheading cost

var.rₖ ∈ feasible depot route

s.t.every required arc e is serviced exactly once

Σe∈route qₑ ≤ Q

every route starts and ends at the depot

path scanningrelocate/swap local searchadaptive iterated local search

3 constraint / domain clauses · Implemented · release review pending

NEARP / MCGRPArc routing

Node, Edge and Arc Routing Problem

min z = total traversal + service cost

var.rₖ ∈ feasible depot route

s.t.every required node/edge/arc task is serviced once

load(route) ≤ Q

depot continuity and route connectivity

Floyd-Warshall closurepath scanningrelocate/swapadaptive ILS

3 constraint / domain clauses · Research implementation · bounded evidence

Assignment4 problems

APAssignment

Linear Assignment Problem

min z = Σᵢ Σⱼ cᵢⱼ xᵢⱼ

var.xᵢⱼ ∈ {0,1}

s.t.Σⱼ xᵢⱼ = 1 ∀i

Σᵢ xᵢⱼ = 1 ∀j

xᵢⱼ ∈ {0,1}

Carpaneto-TothHungarianrow/column dual bound

3 constraint / domain clauses · Implemented · release review pending

GAP-MRAssignment

Machine Reassignment / Generalized Assignment

min z = ΣₚΣₘ cₚₘxₚₘ + move penalties

var.xₚₘ ∈ {0,1}

s.t.Σₘxₚₘ=1 ∀ process p

Σₚ rₚᵏxₚₘ ≤ Cₘᵏ ∀ machine m, resource k

xₚₘ∈{0,1}

exact branch and boundcapacity-relaxed lower boundgreedy feasible assignment

3 constraint / domain clauses · Implemented · release review pending

QAPAssignment

Quadratic Assignment Problem

min z = Σᵢ Σⱼ fᵢⱼ d[π(i), π(j)]

var.π(i) ∈ Sₙ

s.t.π ∈ Sₙ (a permutation of locations)

Gilmore-Lawler boundHahn-Grantprojection/DNN boundsexact B&Brobust tabu search

1 constraint / domain clauses · Implemented · release review pending

SPPAssignment

Set Partitioning Problem

min z = Σⱼcⱼxⱼ

var.xⱼ ∈ {0,1}

s.t.Σⱼaᵢⱼxⱼ=1 ∀ row i

xⱼ∈{0,1}

column generationbranch and pricedual stabilization

2 constraint / domain clauses · Defined formulation · implementation not claimed

Covering1 problem

SCPCovering

Set Covering Problem

min z = Σⱼ cⱼxⱼ

var.xⱼ ∈ {0,1}

s.t.Σⱼ aᵢⱼxⱼ ≥ 1 ∀ row i

xⱼ ∈ {0,1}

Caprara-Fischetti-Toth Lagrangian boundsubgradientreduced-cost primal heuristic

2 constraint / domain clauses · Implemented · release review pending

Energy systems1 problem

UC/EDEnergy systems

Unit Commitment and Economic Dispatch

min z = ΣₜΣᵤ [Cᵤ(pᵤₜ)+Sᵤvᵤₜ]

var.pᵤₜ ∈ ℝ₊; uᵤₜ ∈ {0,1}; vᵤₜ ∈ {0,1}

s.t.Σᵤpᵤₜ=Dₜ ∀t

Pᵐⁱⁿᵤuᵤₜ≤pᵤₜ≤Pᵐᵃˣᵤuᵤₜ

commitment/startup and temporal constraints

merit-order constructioneconomic dispatchcommit/decommit local search

3 constraint / domain clauses · Research implementation · bounded evidence

Graph optimization5 problems

GCPGraph optimization

Graph Coloring Problem

min z = Σₖyₖ

var.xᵥₖ ∈ {0,1}; yₖ ∈ {0,1}

s.t.Σₖxᵥₖ=1 ∀ vertex v

xᵤₖ+xᵥₖ≤yₖ ∀ edge {u,v}, color k

xᵥₖ,yₖ∈{0,1}

DSATURbranch and cutcolumn generationconstraint programming

3 constraint / domain clauses · Defined formulation · implementation not claimed

MCPGraph optimization

Maximum Clique Problem

max z = Σᵥwᵥxᵥ

var.xᵥ ∈ {0,1}

s.t.xᵤ+xᵥ≤1 ∀ nonedge {u,v}

xᵥ∈{0,1}

branch and boundmaximum independent set reductionsemidefinite bounds

2 constraint / domain clauses · Defined formulation · implementation not claimed

MAX-CUTGraph optimization

Maximum Cut Problem

max z = Σ{u,v}∈E wᵤᵥyᵤᵥ

var.xᵥ ∈ {0,1}; yᵤᵥ ∈ {0,1}

s.t.yᵤᵥ=1 iff xᵤ≠xᵥ

xᵥ,yᵤᵥ∈{0,1}

branch and cutsemidefinite relaxationGoemans-Williamson rounding

2 constraint / domain clauses · Defined formulation · implementation not claimed

MWISGraph optimization

Maximum-weight Independent Set

max z = Σᵥ wᵥxᵥ

var.xᵥ ∈ {0,1}

s.t.xᵤ+xᵥ≤1 ∀ edge {u,v}∈E

xᵥ∈{0,1}

tree-decomposition DPgraph decompositionmemory-bounded B&B

2 constraint / domain clauses · Implemented · release review pending

MSTGraph optimization

Minimum Spanning Tree Problem

min z = Σₑcₑxₑ

var.xₑ ∈ {0,1}

s.t.Σₑxₑ=|V|−1

Σₑ∈E(S)xₑ≤|S|−1 ∀ nonempty S⊂V

xₑ∈{0,1}

KruskalPrimBoruvka

3 constraint / domain clauses · Defined formulation · implementation not claimed

Location3 problems

CFLPLocation

Capacitated Facility Location Problem

min z = Σᵢ fᵢyᵢ + ΣᵢΣⱼ cᵢⱼxᵢⱼ

var.yᵢ ∈ {0,1}; xᵢⱼ ∈ [0,1]

s.t.Σᵢxᵢⱼ=1 ∀ customer j

Σⱼdⱼxᵢⱼ≤Qᵢyᵢ ∀ facility i

xᵢⱼ≥0

yᵢ∈{0,1}

Benders decompositionLagrangian relaxationbranch and cut

4 constraint / domain clauses · Defined formulation · implementation not claimed

PMPLocation

P-Median Problem

min z = ΣᵢΣⱼdᵢcᵢⱼxᵢⱼ

var.yⱼ ∈ {0,1}; xᵢⱼ ∈ {0,1}

s.t.Σⱼyⱼ=p

Σⱼxᵢⱼ=1 ∀ customer i

xᵢⱼ≤yⱼ

xᵢⱼ,yⱼ∈{0,1}

branch and boundLagrangian relaxationswap local search

4 constraint / domain clauses · Defined formulation · implementation not claimed

UFLPLocation

Uncapacitated Facility Location Problem

min z = Σᵢ fᵢyᵢ + ΣᵢΣⱼ cᵢⱼxᵢⱼ

var.yᵢ ∈ {0,1}; xᵢⱼ ∈ {0,1}

s.t.Σᵢxᵢⱼ=1 ∀ client j

xᵢⱼ≤yᵢ

xᵢⱼ,yᵢ∈{0,1}

Fischetti-style Benders decompositionexact enumeration checks

3 constraint / domain clauses · Implemented · release review pending

Network design2 problems

STPNetwork design

Steiner Tree Problem

min z = Σₑcₑxₑ

var.xₑ ∈ {0,1}

s.t.Σₑ∈δ(S)xₑ≥1 for every cut S separating required terminals

xₑ∈{0,1}

Dreyfus-Wagner DPbranch and cutdual ascent

2 constraint / domain clauses · Defined formulation · implementation not claimed

MIN-MLU SRNetwork design

Time-dependent Segment Routing Min-MLU

min z

var.x ∈ valid K-segment SR paths; z ∈ ℝ₊

s.t.loadₑₜ(x) ≤ z·capacityₑ ∀ link e,time t

every demand uses a valid ≤K-segment SR path with official ECMP splitting

temporal path-change budget is respected

VNS upper boundMCF/cut/Lagrangian lower boundscolumn generationbranch and pricetree decomposition

3 constraint / domain clauses · Research implementation · bounded evidence

Network flow5 problems

MAX-FLOWNetwork flow

Maximum Flow Problem

max z = Σ(s,v)∈E xₛᵥ − Σ(v,s)∈E xᵥₛ

var.xₑ ∈ [0,uₑ]

s.t.Σ(u,v)∈E xᵤᵥ−Σ(v,w)∈E xᵥw=0 ∀ v∉{s,t}

0≤xₑ≤uₑ

Edmonds-KarpDinicpush-relabel

2 constraint / domain clauses · Defined formulation · implementation not claimed

MCFNetwork flow

Minimum Cost Flow Problem

min z = Σₑcₑxₑ

var.xₑ ∈ [0,uₑ]

s.t.Σₑ∈δ⁺(v)xₑ−Σₑ∈δ⁻(v)xₑ=bᵥ ∀ vertex v

0≤xₑ≤uₑ

network simplexsuccessive shortest pathcost scaling

2 constraint / domain clauses · Defined formulation · implementation not claimed

MCNFNetwork flow

Multicommodity Network Flow Problem

min z = ΣₖΣₑcₑₖxₑₖ

var.xₑₖ ∈ ℝ₊

s.t.flow balance holds for every commodity k and vertex v

Σₖxₑₖ≤uₑ ∀ arc e

xₑₖ≥0

linear programmingLagrangian decompositioncolumn generation

3 constraint / domain clauses · Defined formulation · implementation not claimed

SPNetwork flow

Shortest Path Problem

min z = Σₑcₑxₑ

var.xₑ ∈ {0,1}

s.t.flow balance is 1 at source, −1 at destination and 0 elsewhere

xₑ∈{0,1}

DijkstraBellman-FordA-star

2 constraint / domain clauses · Defined formulation · implementation not claimed

TPNetwork flow

Transportation Problem

min z = ΣᵢΣⱼcᵢⱼxᵢⱼ

var.xᵢⱼ ∈ ℝ₊

s.t.Σⱼxᵢⱼ=sᵢ ∀ supply node i

Σᵢxᵢⱼ=dⱼ ∀ demand node j

xᵢⱼ≥0

transportation simplexnetwork simplexcost scaling

3 constraint / domain clauses · Defined formulation · implementation not claimed

Packing9 problems

BPPCPacking

Bin Packing with Conflicts

min z = Σᵦ yᵦ

var.xᵢᵦ ∈ {0,1}; yᵦ ∈ {0,1}

s.t.assignment and capacity constraints

xᵢᵦ + xⱼᵦ ≤ 1 ∀ conflict {i,j}, bin b

xᵢᵦ,yᵦ ∈ {0,1}

conflict-aware pricingbranch and price

3 constraint / domain clauses · Implemented · release review pending

BPP-PPacking

Bin Packing with Precedence Constraints

min z = Σᵦ yᵦ

var.xᵢᵦ ∈ {0,1}; yᵦ ∈ {0,1}; position(i) ∈ ordered index

s.t.assignment and bin-capacity constraints

position(i) ≤ position(j) ∀ precedence i ≺ j

xᵢᵦ,yᵦ ∈ {0,1}

topological B&Bmaterial/CPM/flow boundsantichain branch-and-price bound

3 constraint / domain clauses · Implemented · release review pending

CSPPacking

Cutting Stock Problem

min z = Σₚ yₚ

var.yₚ ∈ ℤ₊

s.t.Σₚ aᵢₚ yₚ ≥ dᵢ ∀ item type i

yₚ ∈ ℤ₊

each pattern p satisfies Σᵢ wᵢaᵢₚ ≤ C

Gilmore-Gomory column generationdemand expansionbranch and price

3 constraint / domain clauses · Implemented · release review pending

BPPPacking

One-dimensional Bin Packing Problem

min z = Σᵦ yᵦ

var.xᵢᵦ ∈ {0,1}; yᵦ ∈ {0,1}

s.t.Σᵦ xᵢᵦ = 1 ∀ item i

Σᵢ wᵢxᵢᵦ ≤ C yᵦ ∀ bin b

xᵢᵦ,yᵦ ∈ {0,1}

FFDcolumn-generation LPRyan-Foster branch and pricesubset-row cuts

3 constraint / domain clauses · Implemented · release review pending

QKPPacking

Quadratic Knapsack Problem

max z = Σᵢ pᵢxᵢ + Σi<j pᵢⱼxᵢxⱼ

var.xᵢ ∈ {0,1}

s.t.Σᵢ wᵢxᵢ ≤ C

xᵢ ∈ {0,1}

Caprara-Pisinger-Toth surrogate boundexact branch and bound

2 constraint / domain clauses · Implemented · release review pending

SPP-KPacking

Set Packing Problem

max z = Σⱼpⱼxⱼ

var.xⱼ ∈ {0,1}

s.t.Σⱼaᵢⱼxⱼ≤1 ∀ row i

xⱼ∈{0,1}

branch and cutLagrangian relaxationconflict-graph methods

2 constraint / domain clauses · Defined formulation · implementation not claimed

2D-GKPPacking

Two-dimensional Guillotine Knapsack

max z = Σᵢ pᵢ xᵢ

var.xᵢ ∈ {0,1}; (uᵢ,vᵢ) ∈ placement coordinates

s.t.selected rectangles fit in W×H

placements do not overlap

every recursive cut is guillotine

xᵢ ∈ {0,1}

Caprara-Monaci exact dynamic programming

4 constraint / domain clauses · Implemented · release review pending

2D-VPPPacking

Two-dimensional Vector Packing

min z = Σᵦ yᵦ

var.xᵢᵦ ∈ {0,1}; yᵦ ∈ {0,1}

s.t.Σᵢ wᵢᵏxᵢᵦ ≤ Cᵏyᵦ ∀ bin b, dimension k∈{1,2}

Σᵦxᵢᵦ=1

xᵢᵦ,yᵦ∈{0,1}

Caprara-Toth lower boundsFFDexact feasibility B&B

3 constraint / domain clauses · Implemented · release review pending

0-1 KPPacking

Zero-one Knapsack Problem

max z = Σᵢ pᵢ xᵢ

var.xᵢ ∈ {0,1}

s.t.Σᵢ wᵢ xᵢ ≤ C

xᵢ ∈ {0,1}

exact dynamic programmingbranch and bound

2 constraint / domain clauses · Implemented · release review pending

Production planning1 problem

CLSPProduction planning

Capacitated Lot Sizing Problem

min z = Σₜ(cₜxₜ+hₜIₜ+fₜyₜ)

var.xₜ ∈ ℝ₊; Iₜ ∈ ℝ₊; yₜ ∈ {0,1}

s.t.Iₜ₋₁+xₜ=dₜ+Iₜ ∀ period t

xₜ≤Qₜyₜ

xₜ,Iₜ≥0

yₜ∈{0,1}

Wagner-Whitin DP for uncapacitated casesbranch and cutDantzig-Wolfe decomposition

4 constraint / domain clauses · Defined formulation · implementation not claimed

Routing5 problems

ATSPRouting

Asymmetric Travelling Salesman Problem

min z = Σᵢ Σⱼ cᵢⱼ xᵢⱼ

var.xᵢⱼ ∈ {0,1}

s.t.Σⱼ xᵢⱼ = 1 ∀i

Σᵢ xᵢⱼ = 1 ∀j

Σᵢ∈S Σⱼ∉S xᵢⱼ ≥ 1 ∀ ∅ ⊂ S ⊂ V

xᵢⱼ ∈ {0,1}

Held-Karp DPCarpaneto-Toth assignment relaxationnearest neighbourDynaBranch

4 constraint / domain clauses · Implemented · release review pending

CVRPRouting

Capacitated Vehicle Routing Problem

min z = Σᵣ cost(r) yᵣ

var.yᵣ ∈ {0,1}

s.t.Σᵣ aᵢᵣ yᵣ = 1 ∀ customer i

Σᵢ qᵢ aᵢᵣ ≤ Q ∀ route r

yᵣ ∈ {0,1}

exact partition + route DPedge lower boundClarke-Wright savings2-opt

3 constraint / domain clauses · Implemented · release review pending

OPRouting

Orienteering Problem

max z = Σᵢpᵢyᵢ

var.yᵢ ∈ {0,1}; xᵢⱼ ∈ {0,1}

s.t.selected arcs form one source-to-destination route

Σ(i,j)∈A tᵢⱼxᵢⱼ≤T

yᵢ,xᵢⱼ∈{0,1}

dynamic programmingbranch and cutlarge neighborhood search

3 constraint / domain clauses · Defined formulation · implementation not claimed

TSPRouting

Symmetric Travelling Salesman Problem

min z = Σ(i,j)∈E cᵢⱼ xᵢⱼ

var.xᵢⱼ ∈ {0,1}

s.t.Σⱼ xᵢⱼ = 2 ∀i

Σ(i,j)∈δ(S) xᵢⱼ ≥ 2 ∀∅≠S⊂V

xᵢⱼ ∈ {0,1}

Held-Karp DPone-tree lower boundnearest neighbour2-optDynaSearch

3 constraint / domain clauses · Public + internal methods

VRPTWRouting

Vehicle Routing Problem with Time Windows

min z = ΣᵢΣⱼcᵢⱼxᵢⱼ

var.xᵢⱼ ∈ {0,1}; tᵢ ∈ [aᵢ,bᵢ]

s.t.every customer has one incoming and one outgoing route arc

Σᵢqᵢyᵢₖ≤Q ∀ vehicle k

aᵢ≤tᵢ≤bᵢ

time propagation and depot continuity hold

branch and pricelabel-setting pricingadaptive large neighborhood search

4 constraint / domain clauses · Defined formulation · implementation not claimed

Scheduling6 problems

FSPScheduling

Flow Shop Scheduling Problem

min z = Cₘₐₓ

var.π ∈ Sₙ; Cⱼₘ ∈ ℝ₊

s.t.every job visits machines in the same order

no machine processes two jobs simultaneously

π∈Sₙ

Cₘₐₓ≥Cⱼₘ ∀ job j

Johnson rule for two machinesbranch and boundNEH heuristiciterated greedy

4 constraint / domain clauses · Defined formulation · implementation not claimed

JSPScheduling

Job Shop Scheduling Problem

min z = Cₘₐₓ

var.Sᵤ ∈ ℝ₊; yᵤᵥ ∈ {0,1}

s.t.Sᵥ≥Sᵤ+pᵤ for every technological precedence u≺v

operations sharing a machine do not overlap

Cₘₐₓ≥Sᵤ+pᵤ ∀ operation u

disjunctive-graph B&Bconstraint programmingshifting bottlenecktabu search

3 constraint / domain clauses · Defined formulation · implementation not claimed

RCSPScheduling

Railway Crew Scheduling

min z = Σₚ cₚyₚ

var.yₚ ∈ {0,1}

s.t.Σₚ aₜₚyₚ ≥ 1 ∀ trip t

yₚ∈{0,1}

every pairing p satisfies duty and connection rules

pairing generationset-covering Lagrangian method

3 constraint / domain clauses · Implemented · release review pending

RCPSPScheduling

Resource-Constrained Project Scheduling Problem

min z = Cₘₐₓ

var.xⱼₜ ∈ {0,1}; Cₘₐₓ ∈ ℝ₊

s.t.each activity starts once

precedence relations are respected

renewable resource use never exceeds capacity in any period

start variables are binary

time-indexed MILPbranch and boundserial schedule generationconstraint programming

4 constraint / domain clauses · Defined formulation · implementation not claimed

1||ΣwⱼTⱼScheduling

Single-machine Total Weighted Tardiness

min z = Σⱼ wⱼTⱼ

var.Cⱼ ∈ ℝ₊; Tⱼ ∈ ℝ₊

s.t.Tⱼ = max(Cⱼ − dⱼ, 0)

one nonpreemptive job at a time on one machine

DynaSearch adjacent interchange DPbranch-on-last-job B&BDynaBranch

2 constraint / domain clauses · Implemented · release review pending

TTPScheduling

Train Timetabling Problem

max z = ΣₜΣᵣ pₜᵣxₜᵣ

var.xₜᵣ ∈ {0,1}

s.t.Σᵣxₜᵣ ≤ 1 ∀ train t

xₜᵣ + xᵤₛ ≤ 1 for every conflicting timetable pair

xₜᵣ∈{0,1}

Caprara-Fischetti-Toth Lagrangian relaxationdrop-and-add searchexact B&B

3 constraint / domain clauses · Implemented · release review pending

Synchronized routing1 problem

TSP-D / FSTSP kernelSynchronized routing

One-truck / one-drone routing kernel

min z = Cₘₐₓ

var.x ∈ mixed binary and continuous schedule

s.t.each customer is served exactly once by truck or drone

launch precedes rendezvous

drone flight time ≤ endurance

truck and drone synchronize at every rendezvous

exact synchronization-state DProute-first/split-secondweighted-interval DPset-packing B&BVNDfixed-tour QUBO

4 constraint / domain clauses · Research implementation · bounded evidence

No mathematical problem matches that search.

20 MATHEMATICAL MODEL FAMILIES

Select any diagram to load the model and its representative executable algorithm directly in this page.

Optimization, decomposition and proof, drawn clearly.

SELECTED MODEL · EXPLAINED IN PLACE

Linear Programming

Objective contours meet a convex feasible polytope.

Representative executable algorithm

Hungarian assignment

Problem. Minimum- or maximum-cost one-to-one row/column assignment.

Core idea. Dual potentials expose zero reduced-cost augmenting assignments.

Normalize the cost matrix.
Update potentials while finding augmenting paths.
Recover the primal matching and objective.

Certificate. Primal assignment and dual reduced-cost conditions.

Complexity. O(n³) after padding

Limits. Additional side constraints need another model.

from optfin_orlab import HungarianAssignmentSolver

solver = HungarianAssignmentSolver()
result = solver.solve(...)

Open full guide View source Open executable test

MULTIDISCIPLINARY SCIENCE

Operations Research is the mathematical core; each program connects it to a different scientific or institutional domain.

Multidisciplinary research programs

OPERATIONS RESEARCH

Exact optimization and industrial systems

Branch-and-bound, cutting planes, decomposition, routing, packing, scheduling and certified large-scale benchmarks.

Exact algorithmsMathematical programmingROADEF benchmarks

ARTIFICIAL INTELLIGENCE

AI structures for decisions with proof

Agents identify decisive substructures, call explicit optimization models and return feasible decisions with traceable bounds.

Agent systemsNeuro-symbolic workflowsDecision APIs

MOBILITY AND AUTONOMOUS SYSTEMS

Truck, drone and multimodal logistics

Clean routing kernels, synchronized decisions and heterogeneous travel models before operational constraints are layered in.

RoutingDynamic programmingHybrid exact methods

COMPUTATIONAL BIOLOGY

Molecular and biomedical optimization

A research agenda for exact subgraph refinement, candidate selection and auditable optimization in biological design.

Graph optimizationCombinatorial designScientific validation

FINANCIAL DECISION SCIENCE

Optimization and business analytics

Transparent allocation, scenario analysis, risk-aware planning and decision support built from explicit objectives and constraints.

Business analyticsRobust scenariosExplainable decisions

PUBLIC SYSTEMS

Audit, grants and evidence governance

Reproducibility, software assurance and government-project readiness connect scientific work to accountable institutions.

Independent auditGrant readinessEvidence systems

DIRECTOR IN PRACTICE · DESIGN DISTRICT

Jose Diaz directs OptFin.org from the intersection of optimization, business analysis and applied decision-making. This editorial series places the person, the city and the work in the same frame.

Research has a human point of view.

Jose Diaz in a navy jacket beside geometric Design District architecture. — Jose Diaz · institutional portrait

RUN · READ · VERIFY

Theory is linked to runnable software, working papers, research notes and machine-readable evidence.

Academic tools

V1.3 TOOL

V3 Dimensional Collapse Simulator

Explore how an active search domain shrinks while bounds, frozen variables and parallel exact work evolve.

Launch simulator

PUBLICATION

Working Papers

Formal research tracks for structural optimization, industrial benchmarks, molecular refinement and proof standards.

Read working papers

RESEARCH NOTES

Optimization Blog

Professional notes explaining exactness, feasibility, dimensional collapse and hybrid optimization.

Read research notes

OPEN SOURCE

OR-Lab Python Library

Reviewed algorithm implementations, tests, reproducible benchmarks, citation metadata and public CI.

Open GitHub library

EVIDENCE LAB

Reproducibility Console

Exact commands, observed test output, benchmark metrics and repetition hashes in one review surface.

Inspect evidence

DISTRIBUTED RESEARCH

Hadoop / Spark Experiment Planner

Deterministic partitioning and result aggregation for larger experiment campaigns without changing solver semantics.

Review architecture

32 EXACT, HEURISTIC AND METAHEURISTIC METHODS

Algorithm ledger

Public · locally testedGraph algorithms

Dijkstra shortest path

Dijkstra

label settingbinary heap

Evidence boundary. Registered-edge path and settled nonnegative labels.

API: ExactGraphOracle.shortest_path
Source: src/optfin_orlab/graphs.py
Complexity: O((|V| + |E|) log |V|)

Implemented · release review pendingToth lineage

Carpaneto-Toth primal-dual assignment

Carpaneto · Martello · Toth · Dell'Amico

row/column reductionshortest augmenting pathdual bound

Evidence boundary. Primal assignment agrees with brute force and Hungarian checks in the registered stress suite.

API: AP.solve_carpaneto_toth
Source: source/orlab/problems/assignment.py
Complexity: Polynomial assignment method; public-release review pending

Public · locally testedExact combinatorial optimization

Hungarian / Kuhn-Munkres assignment

Kuhn · Munkres

dual potentialsaugmenting paths

Evidence boundary. Primal assignment and dual reduced-cost conditions.

API: HungarianAssignmentSolver.solve
Source: src/optfin_orlab/exact.py
Complexity: O(n³)

Public · locally testedDynamic programming

Held-Karp TSP and ATSP dynamic programming

Held · Karp

subset DPtour reconstruction

Evidence boundary. Complete subset recurrence and minimum depot return.

API: TSP.solve / ATSP.solve
Source: src/optfin_orlab/tsp.py
Complexity: O(2ⁿn²) time; O(2ⁿn) memory

Public · locally testedExact methods

Branch and bound

Land · Doig

LP boundsincumbent pruningbinary branching

Evidence boundary. Feasible incumbent plus exhausted or bounded tree.

API: BranchAndBound.solve
Source: src/optfin_orlab/branch_and_bound.py
Complexity: Exponential worst case

Public · locally testedExact methods

Branch and cut

Polyhedral integer optimization

zero-half cutsMIR cutslifted-cover cuts

Evidence boundary. Incumbent, valid-cut ledger and bounded search tree.

API: BranchAndBound.solve(use_cuts=True)
Source: src/optfin_orlab/branch_and_bound.py
Complexity: Problem-dependent; exponential worst case

Implemented · release review pendingDecomposition

Branch and price for bin packing

Dantzig-Wolfe · Gilmore-Gomory

restricted masterpricingbranching

Evidence boundary. Column-master bound and branching ledger within the implemented model.

API: BranchAndPriceBpp
Source: source/orlab/pipeline/branch_and_price_bpp.py
Complexity: Pricing- and tree-dependent

Implemented · release review pendingDecomposition

Set-partitioning column generation

Dantzig-Wolfe

restricted masterreduced-cost pricingcolumn pool

Evidence boundary. Dual bound only when pricing exhaustion is established.

API: SetPartitionColumnGeneration
Source: source/orlab/pipeline/setpart_column_generation.py
Complexity: Master iterations × pricing cost

Implemented · release review pendingDecomposition

Fischetti-style Benders decomposition

Benders · Fischetti

master problemfeasibility cutsoptimality cuts

Evidence boundary. Master/subproblem cut ledger under the registered assumptions.

API: BendersDecomposition
Source: source/orlab/exact_masters/fischetti_benders.py
Complexity: Problem- and cut-dependent

Implemented · release review pendingMatheuristics

Fischetti-Lodi local branching

Fischetti · Lodi

Hamming-ball neighborhoodre-centeringexact subproblem

Evidence boundary. Fast-incumbent evidence; exactness requires complete outer search.

API: LocalBranching.solve
Source: source/orlab/exact_masters/fischetti_lodi_local_branching.py
Complexity: Neighborhood and MIP dependent

Implemented · release review pendingMatheuristics

Fischetti feasibility pump

Fischetti · Glover · Lodi

LP roundingdistance minimizationcycle perturbation

Evidence boundary. Feasibility only; no optimality claim.

API: FeasibilityPump.solve
Source: source/orlab/exact_masters/fischetti_feasibility_pump.py
Complexity: Iterations × relaxation cost

Implemented · release review pendingLocal search operators

DynaSearch with exact exponential-neighborhood DP

Congram · Potts · van de Velde

compound 2-optindependent movesdynamic programming

Evidence boundary. Each pass is at least as strong as its best single 2-opt move in registered checks.

API: DynaSearch.solve
Source: source/orlab/dynasearch.py
Complexity: O(n²) per DP-searched pass

Public · locally testedLocal search operators

2-opt segment reversal

Croes

edge exchangebest improvement

Evidence boundary. Feasible tour and recomputed objective after every accepted move.

API: TSP.heuristic / VRP._two_opt
Source: src/optfin_orlab/tsp.py
Complexity: O(n²) candidates per pass

Research implementation · bounded evidenceLocal search operators

Relocate and swap neighborhoods

General routing neighborhoods

relocateswapexact re-evaluation

Evidence boundary. Accepted moves are replayed for feasibility in their owning model.

API: TruckDroneProblem.improve_tour_vnd / CARP.improve
Source: source/orlab/problems/truck_drone.py
Complexity: Neighborhood- and evaluator-dependent

Implemented · release review pendingConstructive heuristics

Clarke-Wright savings + 2-opt

Clarke · Wright

parallel savingscapacity-feasible merge2-opt

Evidence boundary. Capacity-feasible routes with recomputed cost.

API: VRP.heuristic
Source: source/orlab/problems/vrp.py
Complexity: O(n² log n) plus local improvement

Implemented · release review pendingConstructive heuristics

CARP path scanning

Arc-routing construction

path scanningcapacity splitdeadheading shortest paths

Evidence boundary. Every required task is serviced within vehicle capacity.

API: CARP.construct
Source: source/orlab/problems/carp.py
Complexity: Problem- and shortest-path dependent

Implemented · release review pendingGeneral metaheuristics

POPMUSIC partial optimization

Taillard · Voss

related-part selectionexact/near-exact subsolverreinsertion

Evidence boundary. Monotone accepted improvements; global optimality is not implied.

API: POPMUSIC.optimize
Source: source/orlab/popmusic.py
Complexity: Rounds × selected-subproblem cost

Research implementation · bounded evidenceGeneral metaheuristics

Variable Neighborhood Search

Mladenović · Hansen

shakingneighborhood changelocal descent

Evidence boundary. Checker-valid feasible upper bound when the official evaluator accepts the solution.

API: RoadefVnsUpperBound.solve
Source: source/orlab/roadef/roadef_vns_ub.py
Complexity: Time-budgeted

Research implementation · bounded evidenceGeneral metaheuristics

Adaptive Large Neighborhood Search controller

Ropke · Pisinger

destroy/repair operatorsadaptive scoresacceptance gate

Evidence boundary. Operator ledger and feasible incumbent; no global proof.

API: AdaptiveOperatorController
Source: source/orlab/solvers/roadef_alns_controller.py
Complexity: Iterations × operator/evaluation cost

Implemented · release review pendingGeneral metaheuristics

Taillard robust tabu search for QAP

Taillard

swap neighborhoodtabu tenureaspiration

Evidence boundary. Feasible assignment and recomputed QAP objective; no global proof.

API: QAP.tabu_search
Source: source/orlab/problems/qap.py
Complexity: Iterations × O(n²) swap neighborhood

Implemented · release review pendingGeneral metaheuristics

Adaptive iterated local search for CARP/NEARP

Iterated local search · arc routing

destroygreedy repairadaptive operator choicelocal descent

Evidence boundary. Feasible required-task service and incumbent trace.

API: CARP.solve
Source: source/orlab/problems/carp.py
Complexity: Iterations × repair/local-search cost

Public · locally testedGeneral metaheuristics

Seeded classical simulated annealing

Kirkpatrick · Gelatt · Vecchi

QUBO energytemperature schedulefeasibility decode

Evidence boundary. Energy and decoded feasibility only; no quantum or optimality claim.

API: TruckDroneProblem.solve_qubo_annealing
Source: src/optfin_orlab/truck_drone.py
Complexity: Restarts × sweeps × interactions

Implemented · release review pendingHybrid exact/metaheuristic

DynaBranch

OptFin OR-Lab synthesis

DynaSearch warm startCarpaneto-Toth boundsubtour branching

Evidence boundary. Exact branch-and-bound retains the proof boundary; the heuristic supplies only the incumbent.

API: DynaBranch.solve
Source: source/orlab/dynabranch.py
Complexity: Exponential worst case with DP warm start

Implemented · release review pendingHybrid exact/metaheuristic

Corridor Method with exact window DP

Sniedovich · Voss

moving corridorexact window DPiterated improvement

Evidence boundary. Each corridor is solved exactly; the overall result remains heuristic.

API: CorridorMethod.solve
Source: source/orlab/corridor_method.py
Complexity: Windows × subset-DP cost

Research implementation · bounded evidenceHybrid exact/metaheuristic

Residual VNS + column generation

OptFin ROADEF research

residual VNScolumn extractionrestricted mastercompact polish

Evidence boundary. Separate UB feasibility and master-bound evidence; closure only when all proof gates pass.

API: ResidualVnsColumnGeneration.run
Source: source/orlab/solvers/roadef_residual_vns_column_generation.py
Complexity: Campaign-budgeted

Research implementation · bounded evidenceLocal search operators

Truck-drone best-improvement VND

VND routing

2-optrelocateexact split evaluation

Evidence boundary. Every candidate is evaluated through exact fixed-tour split; independent W03 remains separate.

API: TruckDroneProblem.improve_tour_vnd
Source: source/orlab/problems/truck_drone.py
Complexity: Rounds × neighborhoods × split-DP cost

Implemented · release review pendingToth lineage

Set-covering Lagrangian relaxation

Caprara · Fischetti · Toth

Lagrangian relaxationsubgradientreduced-cost primal heuristic

Evidence boundary. Dual lower bound and feasible primal upper bound are reported separately.

API: SetCoveringLagrangian.solve
Source: source/orlab/caprara/set_covering_lagrangian.py
Complexity: Iterations × decomposed covering evaluation

Implemented · release review pendingToth lineage

Train timetabling Lagrangian + branch and bound

Caprara · Fischetti · Toth

conflict relaxationsubgradientdrop-and-add searchbranch and bound

Evidence boundary. Dual bound, feasible timetable and exact-tree closure on registered instances.

API: TrainTimetabling.solve
Source: source/orlab/caprara/train_timetabling_lagrangian.py
Complexity: Timetable choices and tree dependent

Implemented · release review pendingToth lineage

Railway crew set-covering method

Caprara · Fischetti · Toth · Vigo · Guida

legal pairing generationset coveringLagrangian bound

Evidence boundary. Set-covering LB/UB evidence within the generated pairing universe.

API: RailwayCrewScheduling.solve
Source: source/orlab/caprara/railway_crew_scheduling.py
Complexity: Pairing generation and covering iterations

Implemented · release review pendingToth lineage

Quadratic knapsack exact branch and bound

Caprara · Pisinger · Toth

linear-surrogate upper boundbranch and bound

Evidence boundary. LB=UB tree closure on registered small instances.

API: QuadraticKnapsack.solve
Source: source/orlab/caprara/quadratic_knapsack.py
Complexity: Exponential worst case

Implemented · release review pendingToth lineage

Two-dimensional vector packing

Caprara · Toth

dimension boundscombined lower boundFFD upper boundexact feasibility B&B

Evidence boundary. Lower/upper bounds and exact bin-feasibility closure on registered instances.

API: VectorPacking2D.solve
Source: source/orlab/caprara/vector_packing_2d.py
Complexity: Exponential exact phase

Research implementation · bounded evidenceQuantum-ready classical baselines

Truck-drone fixed-tour QUBO

Binary set packing · quantum-ready modeling

linear savingsquadratic conflict penaltiesclassical annealing

Evidence boundary. Feasible decoded sortie set only; zero quantum-advantage claim.

API: TruckDroneProblem.build_qubo / solve_qubo_annealing
Source: source/orlab/problems/truck_drone.py
Complexity: Candidate interactions plus annealing budget

No method matches that search.

PROOF STANDARD

The lab standard for a real result

A result becomes scientific evidence only when the model, instance, objective, feasibility, bounds, runtime, stopping reason and checker record remain together.

Field	Requirement
Feasibility	Independent checker or replayable constraints
Objective	Explicit value with reproducible evaluation
Bound status	Lower bound, upper bound and certified gap
Trace	Solver path, fallback path, hashes and stopping reason

EXECUTABLE EVIDENCE

Use the exact commands below, inspect their terminal output, then compare the two deterministic benchmark runs.

Reproduced locally · what this evidence state means

The latest canonical local suite passed. Independent auditor replay and the standalone package collection remain separate evidence gates.

Reproducibility before rhetoric

CANONICAL SUITE

Canonical Suite

governance, component contract, folder index, builder, graph core and truck-drone canonical tests

47passed

Command

$ python -m pytest -q tests/optfin/orlab/test_public_repository_builder.py tests/optfin/test_optfin_orlab_graph_core.py source/orlab/tests/test_truck_drone.py tests/optfin/test_solvida_ui_component_rule0_core.py tests/optfin/test_folder_index_governance.py

Environment: local Windows development lane
Result: 47 passed

Observed output · show exact terminal evidence

...............................................                  [100%]
47 passed in 47.05s

PUBLIC PACKAGE

Public Package

standalone public package

BLOCKEDpassed

Command

$ cd output/orlab/public/optfin-orlab && python -m pytest -q --basetemp=.test-tmp tests

Environment: local Windows development lane with public src only
Result: BLOCKED

Observed output · show exact terminal evidence

Collection blocked
Missing optfin_orlab.distributed import in the projected package

BASH

Reproduce locally

Python 3.11+ · public package only

$ git clone https://github.com/jose-diaz-fl/optfin-orlab.git
$ cd optfin-orlab
$ python -m pip install -e ".[test]"
$ python -m pytest -q
$ python benchmarks/run_smoke.py --output benchmark.json

2× HASH MATCH

Deterministic smoke benchmark

A minimal four-algorithm smoke sample is executed twice from the same public release inputs. It is not the algorithm catalog.

Command

$ python benchmarks/run_smoke.py --output benchmark.json

Algorithm	Measured quantity	Result	Check
Dijkstra	Shortest-path distance	4.0	PASS
Exact assignment	Minimum assignment objective	5.0	PASS
Branch-and-cut	Binary objective	20.0	PASS
Truck + drone	Makespan objective	20.0	PASS

OPEN RELEASE

Code, tests, documentation and approved evidence

Apache-2.0 release. Every consumer uses equal public terms and cites OptFin.org / OR-Lab.

FIELD WORK · REAL ESTATE EDUCATION

A professional real estate education seminar documented through presentation, discussion, questions and shared experience in Florida.

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2022-02-08Florida, United States

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Future OptFin.org encounters will connect multidisciplinary research, real estate education and practical decision science. Dates will be published only after they are confirmed.

PARTNERS & EMERGING INSTITUTIONS

A focused Florida-Italy academic connection for Operations Research, optimization and applied research.

US-Europe Academic Network

HOME INSTITUTE

OptFin.org · Florida

Independent academic and applied-research program directed by Jose Diaz, with OR-Lab as its open scientific software laboratory.

Active multidisciplinary research institute

ACADEMIC CONNECTION

ICOOR.it · Italy

Interuniversity Consortium for Optimization and Operation Research in Italy. Its current official Assembly represents ten Italian universities collaborating in optimization and Operations Research.

Universities represented in ICOOR's official Assembly

University of Bologna
Polytechnic University of Bari
University of Cagliari
University of Camerino
Polytechnic University of Milan
University of Modena and Reggio Emilia
University of Pisa
University of Trieste
Polytechnic University of Turin
University of Bergamo

ICOOR official organization and Assembly ICOOR official members page

Director-verified continuing academic affiliation

LEADERSHIP

Institute leadership

DIRECTOR

Jose Diaz

Operations Research · Optimization · Business Analysis

Profile and responsibilities

COMPUTER ENGINEER

Ivet Delgado-Diaz

Audits · Government Grants · Research Reproducibility

Profile and responsibilities