2nd/10_Wiki/Topics/Domain_Programming/AI_and_ML/Linear-Programming.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

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

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)

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)

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

# 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

# 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

# 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