2 where. x 1 θ = 1 θ = 0.6 x 2 θ = 0 θ = 0.2

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1 Convex et I a ne and convex et I ome important example I operation that preerve convexity I eparating and upporting hyperplane I generalized inequalitie I dual cone and generalized inequalitie IOE 6: Nonlinear Programming, Fall Convex et Page 2 A ne et a ne combination of x and x 2 : any point x of the form x = x +( )x 2 where 2 R θ =.2 x θ = θ =.6 x 2 θ = θ =.2 line through x, x 2 :alla ne combination of x and x 2 a ne et: contain the line through any two ditinct point in the et (i.e., a et that cloed under a ne combination) example: olution et of linear equation {x Ax = b} (converely, every a ne et can be expreed a olution et of ytem of linear equation) IOE 6: Nonlinear Programming, Fall Convex et Page 2 2

2 Convex et convex combination of x and x 2 : any point x of the form x = x +( )x 2 where apple apple line egment between x and x 2 : all point x = x +( )x 2 with apple apple convex et: contain line egment between any two point in the et x, x 2 2 C, apple apple =) x +( )x 2 2 C example (one convex, two nonconvex et) IOE 6: Nonlinear Programming, Fall Convex et Page 2 3 Convex combination and convex hull convex combination of x,..., x k : any point x of the form x = x + 2 x k x k with + + k =, i convex hull conv S: et of all convex combination of point in S IOE 6: Nonlinear Programming, Fall Convex et Page 2 4

3 Convex cone cone: aetc uch that for any x 2 C and, x 2 C conic (nonnegative) combination of x and x 2 : any point of the form x = x + 2 x 2 where, 2 t x x 2 convex cone: a cone that i convex, i.e., et that contain all conic combination of point in the et { x + 2 x k x k i } IOE 6: Nonlinear Programming, Fall Convex et Page 2 5 Example of a ne/convex et and convex cone Before we get to intereting example, ome trivial one: I An empty et ;, a ingleton {x } R n, the whole pace R n are a ne and convex et in R n I Any line i a cone ne. If it pae through, it i alo a convex I A line egment i convex, but not a ne (unle a ingleton) I A ray (et {x + v }) i convex but not a ne; if x =, it i a convex cone I A ubpace (a ubet of R n cloed under um and calar multiplication) i a ne, convex, and i a convex cone IOE 6: Nonlinear Programming, Fall Convex et Page 2 6

4 Hyperplane and halfpace hyperplane: et of the form {x a T x = b} (a 6= ) { } a x x a T x = b halfpace: et of the form {x a T x apple b} (a 6= ) { } a x a T x b a T x b I a i the normal vector of the hyperplane/halfpace I hyperplane are a ne and convex; halfpace are convex IOE 6: Nonlinear Programming, Fall Convex et Page 2 7 Polyhedra olution et of finitely many linear inequalitie and equalitie Ax b, Cx = d (A 2 R m n, C 2 R p n, i componentwie inequality) nt a a2 a 5 P a 3 a 4 polyhedron i interection of finite number of halfpace and hyperplane IOE 6: Nonlinear Programming, Fall Convex et Page 2 8

5 Poitive emidefinite cone notation: I S n i et of ymmetric n n matrice I S n + = {X 2 S n X }: poitive emidefinite n n matrice X 2 S n + () z T Xz forallz I S n ++ = {X 2 S n X S n + i a convex cone }: poitive definite n n matrice ent example: apple x y y z 2 S 2 + z.5 y x.5 IOE 6: Nonlinear Programming, Fall Convex et Page 2 9 Euclidean ball and ellipoid (Euclidean) ball with center x c and radiu r: B(x c, r) ={x kx x c k 2 apple r} = {x c + ru kuk 2 apple } ellipoid: et of the form {x (x x c ) T P (x x c ) apple } with P 2 S n ++ (i.e., P ymmetric poitive definite) x c other repreentation: {x c + Au kuk 2 apple } with A quare and noningular IOE 6: Nonlinear Programming, Fall Convex et Page 2

6 Norm ball and norm cone norm: a function k k that atifie I kxk ; kxk = if and only if x = I ktxk = t kxk for t 2 R I kx + yk applekxk + kyk notation: k k i a general (unpecified) norm; k k ymb i a particular norm norm ball with center x c and radiu r: {x kx x c kappler} } norm cone: {(x, t) kxk applet} R n+ Euclidean norm cone i called econd-order cone norm ball and cone are convex t.5 x 2 x IOE 6: Nonlinear Programming, Fall Convex et Page 2 Operation that preerve convexity practical method for etablihing convexity of a et C. apply definition x, x 2 2 C, apple apple =) x +( )x 2 2 C 2. how that C i obtained from imple convex et (hyperplane, halfpace, norm ball,... ) by operation that preerve convexity I Carteian product (S S 2 = {(x, x 2 ) x 2 S, x 2 2 S 2 };if S and S 2 are convex, o i S S 2 ) I interection I a ne function I perpective function I linear-fractional function IOE 6: Nonlinear Programming, Fall Convex et Page 2 2

7 Interection the interection of (any number of) convex et i convex example: S = {x 2 R m p x (t) apple for t apple /3} where p x (t) =x co t + x 2 co 2t + + x m co mt for m = 2: ment 2 p(t) x2 S π/3 2π/3 π t x S t = {x 2 R m apple (co t,...,co mt) T x apple }; S = \ t apple /3 S t. IOE 6: Nonlinear Programming, Fall Convex et Page 2 3 A ne function uppoe f : R n! R m i a ne (f (x) =Ax + b with A 2 R m n, b 2 R m ) I the image of a convex et under f i convex S R n convex =) f (S) ={f (x) x 2 S} R m convex I example: caling, tranlation, projection, um of et I the invere image f (C): f (C) ={x 2 R n f (x) 2 C} the invere image of a convex et under f i convex C R m convex =) f (C) convex I example: olution et of linear matrix inequality {x x A + + x m A m B} (with A i, B 2 S p ) I example: hyperbolic cone {x x T Px apple (c T x) 2, c T x } (with P 2 S n +) IOE 6: Nonlinear Programming, Fall Convex et Page 2 4

8 Perpective and linear-fractional function perpective function P : R n+! R n : P(x, t) =x/t, dom P = {(x, t) t > } image and invere image of convex et under perpective are convex linear-fractional function f : R n! R m : f (x) = Ax + b c T x + d, dom f = {x ct x + d > } image and invere image of convex et under linear-fractional function are convex IOE 6: Nonlinear Programming, Fall Convex et Page 2 5 example of a linear-fractional function f (x) = x + x 2 + x : R2! R 2 nt PSfrag replacement x2 C x2 f(c) x x IOE 6: Nonlinear Programming, Fall Convex et Page 2 6

9 Separating hyperplane theorem if C and D are dijoint convex et, then there exit a 6=, b uch that a T x apple b for x 2 C, a T x b for x 2 D a T x b a T x b D C a the hyperplane {x a T x = b} eparate C and D trict eparation (a T x < b for x 2 C, a T x > b for x 2 D) require additional aumption (e.g., C i cloed, D i a ingleton; ee example 2.2) IOE 6: Nonlinear Programming, Fall Convex et Page 2 7 Supporting hyperplane theorem upporting hyperplane to et C at boundary point x : {x a T x = a T x } where a 6= anda T x apple a T x for all x 2 C a C x upporting hyperplane theorem: if C i convex, then there exit a upporting hyperplane at every boundary point of C IOE 6: Nonlinear Programming, Fall Convex et Page 2 8

10 A theorem of alternative for linear ytem Theorem Given A 2 R m n and b 2 R m, exactly one of (i) Ax < b and (ii) A T =, b T apple,, 6= ha a olution. Proof I Eay to ee (i) and (ii) can t both have olution. I If (i) doe not have a olution, C = {b Ax x 2 R n } and D = {y 2 R m ++} are convex dijoint et and can be eparated. IOE 6: Nonlinear Programming, Fall Convex et Page 2 9 Generalized inequalitie a convex cone K R n i a proper cone if I K i cloed (contain it boundary) I K i olid (ha nonempty interior) I K i pointed (contain no line) example I nonnegative orthant K = R n + = {x 2 R n x i, i =,...,n} I poitive emidefinite cone K = S n + I nonnegative polynomial of degree n on [, ]: K = {x 2 R n x +x 2 t +x 3 t 2 + +x n t n fort 2 [, ]} IOE 6: Nonlinear Programming, Fall Convex et Page 2 2

11 Generalized inequalitie a proper cone K can be ued to define a generalized inequality: x K y () y x 2 K, x K y () y x 2 int K example I componentwie inequality (K = R n +) x R n + y () x i apple y i, i =,...,n I matrix inequality (K = S n +) X S n + Y () Y X poitive emidefinite thee two type are o common that we drop the ubcript in propertie: many propertie of K are imilar to apple on R, e.g., x K y, u K v =) x + u K y + v K IOE 6: Nonlinear Programming, Fall Convex et Page 2 2 Minimum and minimal element K i not in general a linear ordering: wecanhavex 6 y 6 K x x 2 S i the minimum element of S with repect to K y and K if y 2 S =) x K y (equivalently, S x + K). x 2 S i a minimal element of S with repect to K if y 2 S, y K x =) y = x (equivalently, (x K)\S = {x}). example (K = R 2 +) x i the minimum element of S x 2 i a minimal element of S 2 nt x S 2 S x 2 IOE 6: Nonlinear Programming, Fall Convex et Page 2 22

12 optimal production frontier I di erent production method ue di erent amount of reource x 2 R n I production et P: reource vector x for all poible production method I e cient (Pareto optimal) method correpond to reource vector x that are minimal w.r.t. R n + fuel example (n = 2) x, x 2, x 3 are e cient; x 4, x 5 are not x x x5 2 x 4 λ P x 3 labor IOE 6: Nonlinear Programming, Fall Convex et Page 2 23 Dual cone and generalized inequalitie dual cone of a cone K: example K = {y hx, yi forallx 2 K} I K = R n +: K = R n + (for x, y 2 R n, hx, yi = y T x) I K = S n +: K = S n + (for X, Y 2 S n, hx, Y i = P i,j X ijy ij = tr(xy )) I K = {(x, t) kxk 2 apple t}: K = {(y, ) kyk 2 apple } I K = {(x, t) kxk apple t}: K = {(y, ) kyk apple } firt three example are elf-dual cone dual cone of proper cone are proper, hence define generalized inequalitie: y K () y T x forallx K IOE 6: Nonlinear Programming, Fall Convex et Page 2 24

13 Minimum and minimal element via dual inequalitie minimum element w.r.t. K x i minimum element of S i for all K, x i the unique minimizer of T z over S x S minimal element w.r.t. K I if x minimize T z over S for ome K, then x i minimal r S for ome λ K λ, the x S x 2 λ 2 I if x i a minimal element of a convex et S, thenthereexit a nonzero K uch that x minimize T z over S IOE 6: Nonlinear Programming, Fall Convex et Page 2 25

Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004

Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004 18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem