78 lines
2.6 KiB
Python
78 lines
2.6 KiB
Python
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#!/usr/bin/env python3
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# Dualitaet_Nachweis.py
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"""
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Kapitel LP: Primales und duales Problem unabhaengig loesen und den starken
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Dualitaetssatz sowie den komplementaeren Schlupf numerisch nachweisen.
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Modell aus der Simplex-Handrechnung (Bot-Allokation, LP-Relaxation):
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max 150*x1 + 250*x2 u.d.N. 2*x1+5*x2<=40, 4*x1+6*x2<=60, x1<=8, x>=0
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"""
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import numpy as np
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from scipy.optimize import linprog
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# --- Primales Problem --------------------------------------------------------
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C_PRIMAL = np.array([150.0, 250.0])
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A_PRIMAL = np.array([[2.0, 5.0], [4.0, 6.0], [1.0, 0.0]])
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B_PRIMAL = np.array([40.0, 60.0, 8.0])
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# --- Duales Problem: min b^T y u.d.N. A^T y >= c, y >= 0 ------------------
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# linprog kennt nur <=, also A^T y >= c <=> -A^T y <= -c
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C_DUAL = B_PRIMAL
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A_DUAL = -A_PRIMAL.T
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B_DUAL = -C_PRIMAL
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def loese():
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primal = linprog(c=-C_PRIMAL, A_ub=A_PRIMAL, b_ub=B_PRIMAL,
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bounds=[(0, None)] * 2, method="highs")
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dual = linprog(c=C_DUAL, A_ub=A_DUAL, b_ub=B_DUAL,
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bounds=[(0, None)] * 3, method="highs")
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if not primal.success or not dual.success:
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raise SystemExit("Primal oder Dual nicht loesbar.")
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return primal, dual
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if __name__ == "__main__":
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primal, dual = loese()
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x = primal.x
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y = dual.x
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z_primal = -primal.fun
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z_dual = dual.fun
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print("=" * 78)
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print(" PRIMALES PROBLEM")
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print("=" * 78)
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print(f"x* = ({x[0]:.4f}, {x[1]:.4f})")
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print(f"Z* = {z_primal:.4f}")
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print("\n" + "=" * 78)
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print(" DUALES PROBLEM")
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print("=" * 78)
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print(f"y* = ({y[0]:.4f}, {y[1]:.4f}, {y[2]:.4f})")
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print(f"W* = {z_dual:.4f}")
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print("\n" + "=" * 78)
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print(" STARKER DUALITAETSSATZ: c^T x* == b^T y* ?")
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print("=" * 78)
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print(f" Primal Z* = {z_primal:.6f}")
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print(f" Dual W* = {z_dual:.6f}")
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differenz = abs(z_primal - z_dual)
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print(f" Differenz = {differenz:.2e} -> "
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f"{'BESTAETIGT' if differenz < 1e-6 else 'VERLETZT!'}")
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print("\n" + "=" * 78)
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print(" KOMPLEMENTAERER SCHLUPF: s_i * y_i == 0 fuer alle i ?")
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print("=" * 78)
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schlupf = B_PRIMAL - A_PRIMAL @ x
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ressourcen = ["vCPU (s1)", "RAM (s2)", "Marktlimit (s3)"]
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for name, s, yi in zip(ressourcen, schlupf, y):
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produkt = s * yi
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print(f" {name:<16} Schlupf s={s:6.4f} Schattenpreis y={yi:6.4f} "
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f"s*y={produkt:.2e} {'OK' if abs(produkt) < 1e-6 else 'VERLETZT!'}")
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print("\nFazit: Das dual geloeste y* stimmt exakt mit den Schattenpreisen")
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print("überein, die die Simplex-Rechnung von Hand in der Z-Zeile")
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print("ablas - unabhaengig voneinander berechnet, identisches Ergebnis.")
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print("=" * 78)
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