min f(x) s.t. ax = b gi(x) 0i i - carnegie mellon school...
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![Page 1: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/1.jpg)
Convex program duality
• min f(x) s.t.Ax = bgi(x) ≤ 0 i ∈ I
![Page 2: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/2.jpg)
Properties
• Weak duality
• L(y, z) is closed, concave
![Page 3: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/3.jpg)
Duality example
• min 3x s.t. x2 ≤ 1• L(x, y) = 3x + y(x2 – 1)
![Page 4: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/4.jpg)
![Page 5: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/5.jpg)
Dual function
• L(y) = infx L(x,y) = infx 3x + y(x2 – 1)
![Page 6: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/6.jpg)
Dual function
![Page 7: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/7.jpg)
Duality w/ linear constraints
• min f(x) s.t. Ax = b, Cx ≤ d• L(x, y, z) =• L(y, z) = infx
• Dual problem
![Page 8: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/8.jpg)
CP duality w/ cone constraints
• min f(x) s.t.A0x + b0 = 0Aix + bi ∈ K i ∈ I
• Dual:
![Page 9: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/9.jpg)
Ex: 2nd order cone program
A1 =
b1 =
A2=
b2 =
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Strong duality
• v* = infx f(x) s.t. Ax=b, g(x) ≤ 0L(x, y, z) = L(y, z) =
• d* = supyz L(y, z) s.t. z ≥ 0• Strong duality:• Slater’s condition:
––
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Slater’s condition, part II
• If gi(x) affine, only need
• E.g., for infx f(x) s.t. Ax = b, Cx ≤ d
![Page 12: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/12.jpg)
Example: maxent
![Page 13: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/13.jpg)
Example: maxent
• Maxent problem:max H(p) s.t. T’p = b
H(p) =
H’(p) =
• Slater’s condition:
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Example of maxent solution
![Page 15: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/15.jpg)
Dual of maxent• max H(p) s.t. TTp = b• L(p, y) =• L(y) = infp L(p, y) =
![Page 16: min f(x) s.t. Ax = b gi(x) 0i I - Carnegie Mellon School ...ggordon/10725/10725-S08/slides/cp-duality-4... · Strong duality • v* = inf x f(x) s.t. Ax=b, g(x) ≤0 L(x, y, z) =](https://reader031.vdocuments.site/reader031/viewer/2022041518/5e2c7055c817e801e672ec89/html5/thumbnails/16.jpg)
Is Slater necessary?
• minx xTAx + 2bTx s.t. ||x||2 ≤ 1