---
id: wiki-2026-0508-linear-programming
title: Linear Programming
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [LP, Linear Optimization, Simplex]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [optimization, lp, simplex, scipy, pulp, or-tools]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack: { language: Python, framework: scipy/PuLP/OR-Tools }
---

# Linear Programming

## 매 한 줄
> **"매 LP는 선형 목적함수 + 선형 제약 → 최적해는 vertex에 있다"**. Simplex가 vertex 사이를 움직이고, interior point는 내부를 가로지른다.

## 매 핵심
### 매 표준형
- minimize cᵀx s.t. Ax ≤ b, x ≥ 0
- Feasible region = polytope (convex). 최적해는 항상 corner(꼭짓점) 또는 edge.
- LP relaxation → IP/MIP의 lower bound.

### 매 알고리즘 비교
- **Simplex**: vertex hopping. exponential worst case지만 실제는 빠름.
- **Interior point (Karmarkar)**: polynomial time. 큰 LP에 유리.
- **Dual simplex**: 제약 추가 후 warm start에 강함 (branch-and-bound 내부).

### 매 응용
1. Resource allocation, transportation, assignment
2. Diet/blending, production scheduling
3. Network flow (max-flow, min-cost-flow)
4. ML: SVM (QP지만 LP-like), L1 regression, portfolio optimization

## 💻 패턴
### scipy.optimize.linprog
```python
from scipy.optimize import linprog
# minimize -x0 - 2x1 s.t. x0+x1<=4, x0+3x1<=6, x>=0
res = linprog(c=[-1, -2], A_ub=[[1,1],[1,3]], b_ub=[4,6],
              bounds=[(0, None), (0, None)], method="highs")
print(res.x, res.fun)  # [3, 1] -> -5
```

### PuLP (선언적)
```python
from pulp import LpProblem, LpVariable, LpMaximize, lpSum
prob = LpProblem("prod", LpMaximize)
x = LpVariable.dicts("x", ["A", "B"], lowBound=0)
prob += 40*x["A"] + 30*x["B"]               # objective
prob += 2*x["A"] + x["B"] <= 100             # labor
prob += x["A"] + x["B"] <= 80                # material
prob.solve()
```

### OR-Tools (production)
```python
from ortools.linear_solver import pywraplp
solver = pywraplp.Solver.CreateSolver("GLOP")  # LP. "CBC" for MIP
x = solver.NumVar(0, solver.infinity(), "x")
y = solver.NumVar(0, solver.infinity(), "y")
solver.Add(x + 2*y <= 14); solver.Add(3*x - y >= 0)
solver.Maximize(3*x + 4*y)
solver.Solve()
```

### Integer/Mixed-Integer LP
```python
# PuLP with integer variables
x = LpVariable("x", lowBound=0, cat="Integer")
y = LpVariable("y", lowBound=0, upBound=1, cat="Binary")
# branch-and-bound: LP relaxation → branch on fractional vars
```

### Transportation problem
```python
# min sum c_ij x_ij s.t. sum_j x_ij = supply_i, sum_i x_ij = demand_j
costs = [[8, 6, 10], [9, 12, 13], [14, 9, 16]]
supply = [20, 30, 25]; demand = [10, 35, 30]
# flatten to linprog
```

### Sensitivity / dual
```python
# scipy returns marginals via res.ineqlin.marginals (HiGHS)
# Dual variable = shadow price of constraint
```

## 매 결정 기준
| 상황 | Tool |
|---|---|
| 빠른 prototype | scipy.linprog (HiGHS) |
| 선언적/큰 모델 | PuLP |
| Production / MIP | OR-Tools, Gurobi, CPLEX |
| 정수 변수 多 | CBC, Gurobi (commercial) |
| Network flow | NetworkX, OR-Tools min_cost_flow |

**기본값**: scipy HiGHS → 부족하면 PuLP+CBC → 상용 Gurobi.

## 🔗 Graph
- 부모: [[Optimization]]
- 변형: [[Integer-Programming]]
- Adjacent: [[SVM]]

## 🤖 LLM 활용
**언제**: LP 모델링 (변수/제약 도출), code 생성, 결과 해석.
**언제 X**: 대규모 commercial solver tuning, numerical stability 진단은 전문가.

## ❌ 안티패턴
- 비선형 제약을 LP로 모델링 (→ NLP/QP 필요)
- 정수 변수에 LP 그대로 적용 (반올림 ≠ 최적)
- 큰 dense A matrix를 sparse 변환 없이 사용
- Unbounded/infeasible 모델 진단 없이 결과 신뢰

## 🧪 검증 / 중복
- Verified (Bertsimas Tsitsiklis "Intro to LO", scipy/PuLP/OR-Tools docs). 신뢰도 A.
- 중복: 없음 (LP는 독립 토픽).

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — 매 prefix, scipy/PuLP/OR-Tools 패턴 |