π≤π≤πβΉ{
π−π≤ π
π−π ≤ π
πΏ (π, π1 , π2 ) = π0 (π) + ππ1 (π − π) + ππ2 (π − π)
πΎπΎπ (1): ∇π± πΏ (π∗ , π∗1 , π∗2 ) = ∇ π± π0 (π∗ ) + (π∗1 + π∗2 ) = π
πΎπΎπ (2): π1∗ ≥ π , π∗2 ≥ π
ππ π∗1 = π ⇒ π ≠ π
ππ (π − π) = π
ππ π = π ⇒ π∗1 ≠ π
πΎπΎπ (3): { 1π
βΉ{
ππ π∗2 = π ⇒ π ≠ π
π2 (π − π) = π
ππ π = π ⇒ π∗2 ≠ π
πΎπΎπ (4): π ≤ π , π ≥ π
πΏ (π, π 1 , π 2 ) =
1 2 1 2
π₯ + π₯ − π₯ 1 π₯2 − 2π₯ 2 + π 1 (π₯1 + π₯ 2 − 1) + π 2 (−π₯1 + π₯ 2 − 1)
2 1 2 2
πΎπΎπ (1): (
π₯ 1∗ − π₯ 2∗ + π∗1 − π∗2
) = (0)
π₯ 2∗ − π₯ 1∗ + π∗1 + π∗2
0
πΎπΎπ (2): π∗1 ≥ 0 , π ∗2 ≥ 0
π ∗ ( π₯ ∗ + π₯ ∗ − 1) = 0
πΎπΎπ (3): { ∗ 1 1 ∗ 2 ∗
π 2 (−π₯1 + π₯ 2 − 1) = 0
π₯∗ + π₯∗ − 1 ≤ 0
πΎπΎπ (4): { 1 ∗ 2∗
−π₯ 1 + π₯ 2 − 1 ≤ 0
β«βͺβ
β¬β¬
β«βͺβ
‘β¬β¬
β«βͺβ
’β¬β¬
β«βͺβ
£β¬β¬
β«βͺπ1 = 0β¬β¬
β«βͺπ1 ≠ 0β¬β¬
β«βͺπ1 = 0β¬β¬
β«βͺπ1 ≠ 0β¬β¬
β«βͺπ2 = 0β¬β¬
β«βͺπ2 = 0β¬β¬
β«βͺπ2 ≠ 0β¬β¬
β«βͺπ2 ≠ 0β¬β¬
β«βͺKKT (3):β¬β¬
β«βͺπ∗1 = π∗2 = 0:β¬β¬
β«βͺπ₯ 1∗ − π₯ 2∗ = 0β¬β¬
β«∗βͺπ₯ 1∗ = π₯ 2β¬β¬
β«βͺπ₯ 2∗ − π₯ 1∗ = 0β¬β¬
β«∗ {β¬
β«βͺ⇒{ ∗ 1β¬β¬
β«βͺπ₯1 + π₯ 2∗ ≤ 1β¬β¬
β«≤ βͺπ₯1β¬β¬
β«∗ {β¬
β«βͺ2β¬β¬
β«∗β¬
β«βͺ−π₯1 + π₯2 ≤ 1β¬β¬
β«{β¬
β«βͺπ∗1 ≠ 0 , π∗2 = 0:β¬β¬
β«βͺπ₯ 1∗ − π₯ 2∗ + π∗1 = 0β¬β¬
β«βͺ1β¬β¬
β«∗βͺβΉ 2π₯1∗ = 2π₯ 2∗ ⇒ π₯ 1∗ = π₯ 2β¬β¬
β«βͺ{ π₯ 2∗ − π₯ 1∗ + π∗1 = 0β¬β¬
β«βͺ⇒ π₯ 1∗ = π₯ 2∗ = ⇒π∗ = −1β¬β¬
β«βͺ2β¬β¬
β«βͺπ∗1 ≠ 0 βΉ (π₯ 1∗ + π₯ 2∗ − 1) = 0 ⇒ π₯ 1∗ + π₯ 2∗ = 1β¬β¬
β«{β¬
β«Χ Χ¨ΧΧ Χ©Χ©ΧΧ¨ ΧΧͺΧ ΧΧΧ ΧΧͺΧ§ΧΧΧΧΧ Χ’ΧΧΧ¨ ΧΧ€ΧͺΧ¨ΧΧ ΧΧ "Χβͺ:β¬β¬
β«βͺ(−π₯ 1∗ + π₯ 2∗ − 1) = − 1 + 1 − 1 = −1 ≤ 0, π∗1 = π₯ 2∗ − π₯ 1∗ = 0 ≥ 0 ⇒ π∗1 = 0β¬β¬
β«βͺ2β¬β¬
β«βͺ2β¬β¬
β«ΧΧΧΧΧ¨ Χ§ΧΧΧΧ Χ ΧͺΧΧ¦ΧΧ Χ’ΧΧΧ¨ ΧΧΧ¦Χ ΧΧ¨ΧΧ©ΧΧ Χ©ΧΧ βͺ ,π∗1 = π∗2 = 0β¬ΧΧΧ ΧΧΧ ΧΧ¦ΧΧΧ ΧΧΧΧβͺ.β¬β¬
β«βͺπ∗1 = 0 , π∗2 ≠ 0:β¬β¬
β«βͺπ₯ ∗ − π₯ 2∗ − π∗2 = 0β¬β¬
β«βͺ{ ∗1β¬β¬
β«βͺβΉ π₯ 2∗ − π₯ 1∗ + π∗2 = 0β¬β¬
β«βͺπ₯ 2 − π₯ 1∗ + π∗2 = 0β¬β¬
β«{β¬
β«βͺ⇒ π∗2 = −1 ≤ 0β¬β¬
β«∗β¬
β«∗β¬
β«∗β¬
β«∗β¬
β«∗β¬
β«βͺπ 2 ≠ 0 βΉ (−π₯ 1 + π₯ 2 − 1) = 0 ⇒ π₯ 1 = π₯ 2 − 1β¬β¬
β«Χ€ΧͺΧ¨ΧΧ ΧΧ Χ€Χ‘ΧΧ ΧΧΧΧΧΧ Χ©ΧͺΧ ΧΧ )βͺ KKT(2β¬ΧΧΧ¨Χ© βͺπ∗2 ≥ 0β¬β¬
β«βͺπ∗1 ≠ 0, π∗2 ≠ 0:β¬β¬
β«βͺπ₯ 1∗ + π₯ 2∗ − 1 = 0β¬β¬
β«∗βͺπ₯ 1∗ = 1 − π₯2β¬β¬
β«∗ {β¬
β«∗ {⇒β¬
β«βͺ⇒ π₯ 2∗ = 1 ⇒ π₯ 1∗ = 0β¬β¬
β«βͺ−π₯ 1 + π₯ 2∗ − 1 = 0β¬β¬
β«βͺπ₯ 1 = π₯ 2∗ − 1β¬β¬
β«ΧΧ Χ€Χ ΧͺΧ¨ΧΧ ΧΧ ΧΧ ΧΧ§ΧΧΧ ΧΧͺ ΧΧΧ¨ΧΧ©Χ Χ© βͺπ∗2 ≥ 0β¬β¬
β«βͺ−1 + π∗1 − π∗2 = 0β¬β¬
β«{β¬
β«βͺ⇒ π∗1 = 0 ≤ 0, π∗2 = −1 ≤ 0β¬β¬
β«βͺ1 + π∗1 + π∗2 = 0β¬β¬
β«ΧΧΧ ΧΧ€ΧΧͺΧ¨ΧΧ ΧΧ‘ΧΧ€Χ Χ© Χ§ΧΧΧΧ Χ ΧΧΧβ¬
β«βͺ1β¬β¬
β«βͺ2β¬β¬
β«= ∗βͺπ ∗ = −1 , π₯ 1∗ = π₯ 2β¬β¬
