55 lines
1.8 KiB
Python
55 lines
1.8 KiB
Python
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#!/usr/bin/env python3
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# QP_Grundlagen.py
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"""
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Kapitel QP/NLP: Die drei Faelle aus der Konvexitaets-Tabelle (Abschnitt
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'Das quadratische Programm') an
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einem Mini-QP demonstriert: P positiv definit, P (singulaer) semidefinit,
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P mit negativem Eigenwert.
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"""
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import numpy as np
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import cvxpy as cp
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def loese_qp(P, q, name):
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n = len(q)
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w = cp.Variable(n)
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ziel = cp.Minimize(0.5 * cp.quad_form(w, P) + q @ w)
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bedingungen = [cp.sum(w) == 1, w >= 0]
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problem = cp.Problem(ziel, bedingungen)
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print(f"\n--- {name} ---")
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eigenwerte = np.linalg.eigvalsh(P)
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print(f"Eigenwerte von P: {np.round(eigenwerte, 4)}")
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print(f"DCP-konvex (CVXPY-Pruefung)? {problem.is_dcp()}")
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if not problem.is_dcp():
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print("-> CVXPY lehnt das Problem ab, BEVOR ueberhaupt ein Solver laeuft.")
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return
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problem.solve()
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print(f"Status: {problem.status}")
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print(f"w* = {np.round(w.value, 4)}")
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print(f"Zielwert = {problem.value:.6f}")
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if __name__ == "__main__":
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q = np.zeros(2)
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# Fall 1: P positiv definit -> eindeutiges Minimum
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P_definit = np.array([[2.0, 0.5], [0.5, 1.0]])
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loese_qp(P_definit, q, "P positiv definit")
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# Fall 2: P singulaer/semidefinit (zwei "identische" Assets) -> unendlich viele Minima
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P_semidefinit = np.array([[1.0, 1.0], [1.0, 1.0]])
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loese_qp(P_semidefinit, q, "P positiv semidefinit (singulaer)")
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# Fall 3: P mit negativem Eigenwert -> nicht konvex
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P_indefinit = np.array([[1.0, 2.0], [2.0, 1.0]])
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loese_qp(P_indefinit, q, "P indefinit (negativer Eigenwert)")
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print("\n--- Nachweis: 'unendlich viele Minima' im semidefiniten Fall ---")
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for punkt in [np.array([1.0, 0.0]), np.array([0.0, 1.0]), np.array([0.3, 0.7])]:
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wert = 0.5 * punkt @ P_semidefinit @ punkt
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print(f" w = {punkt} -> Zielwert = {wert:.4f} (identisch, obwohl w verschieden)")
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