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Convex sets
! (^) affine and convex sets ! (^) some important examples ! (^) operations that preserve convexity ! (^) separating and supporting hyperplanes ! (^) generalized inequalities ! (^) dual cones and generalized inequalities IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Affine set
line through x 1 , x 2 : all points x = θx 1 + (1 − θ)x 2 (θ ∈ R) x 1 x 2 θ = 1. 2 θ = 1 θ = 0. 6 θ = 0 θ = − 0. 2 affine set: contains the line through any two distinct points in the set example: solution set of linear equations {x | Ax = b} (conversely, every affine set can be expressed as solution set of system of linear equations)
Convex set
line segment between x 1 and x 2 : all points x = θx 1 + (1 − θ)x 2 with 0 ≤ θ ≤ 1 convex set: contains line segment between any two points in the set x 1 , x 2 ∈ C , 0 ≤ θ ≤ 1 =⇒ θx 1 + (1 − θ)x 2 ∈ C examples (one convex, two nonconvex sets) IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Convex combination and convex hull
convex combination of x 1 ,... , xk : any point x of the form x = θ 1 x 1 + θ 2 x 2 + · · · + θk xk with θ 1 + · · · + θk = 1, θi ≥ 0 convex hull conv S: set of all convex combinations of points in S
Euclidean balls and ellipsoids
(Euclidean) ball with center xc and radius r : B(xc , r ) = {x | ‖x − xc ‖ 2 ≤ r } = {xc + ru | ‖u‖ 2 ≤ 1 } ellipsoid: set of the form {x | (x − xc )T^ P−^1 (x − xc ) ≤ 1 } with P ∈ Sn ++ (i.e., P symmetric positive definite) xc other representation: {xc + Au | ‖u‖ 2 ≤ 1 } with A square and nonsingular IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Norm balls and norm cones
norm: a function ‖ ·‖ that satisfies ! (^) ‖x‖ ≥ 0; ‖x‖ = 0 if and only if x = 0 ! (^) ‖tx‖ = |t| ‖x‖ for t ∈ R ! (^) ‖x + y ‖ ≤ ‖x‖ + ‖y ‖ notation: ‖ ·‖ is general (unspecified) norm; ‖ ·‖ (^) symb is particular norm norm ball with center xc and radius r : {x | ‖x − xc ‖ ≤ r } norm cone: {(x, t) | ‖x‖ ≤ t} Euclidean norm cone is called second-order cone x (^2) x 1 t − 1 0 1 − 1 0 1 0
- 5 1 norm balls and cones are convex
Polyhedra
solution set of finitely many linear inequalities and equalities Ax ( b, Cx = d (A ∈ Rm×n, C ∈ Rp×n, ( is componentwise inequality) a 1 a 2 a 3 a 4 a 5 P polyhedron is intersection of finite number of halfspaces and hyperplanes IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Positive semidefinite cone
notation: ! (^) Sn^ is set of symmetric n × n matrices ! (^) Sn + = {X ∈ Sn^ | X * 0 }: positive semidefinite n × n matrices X ∈ S n
- ⇐⇒^ z T Xz ≥ 0 for all z S n
- is a convex cone ! (^) Sn ++ = {X ∈ Sn^ | X , 0 }: positive definite n × n matrices example:
[
x y y z
]
∈ S^2 +
y x z 0
- 5 1 − 1 0 1 0
- 5 1
Affine function
suppose f : Rn^ → Rm^ is affine (f (x) = Ax + b with A ∈ Rm×n, b ∈ R m ) ! (^) the image of a convex set under f is convex S ⊆ Rn^ convex =⇒ f (S) = {f (x) | x ∈ S} convex ! (^) the inverse image f −^1 (C ) of a convex set under f is convex C ⊆ R m convex =⇒ f − 1 (C ) = {x ∈ R n | f (x) ∈ C } convex examples ! (^) scaling, translation, projection ! (^) solution set of linear matrix inequality {x | x 1 A 1 + · · · + xmAm ( B} (with Ai , B ∈ Sp^ ) ! (^) hyperbolic cone {x | xT^ Px ≤ (cT^ x)^2 , cT^ x ≥ 0 } (with P ∈ S n +) IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Perspective and linear-fractional function
perspective function P : Rn+1^ → Rn: P(x, t) = x/t, dom P = {(x, t) | t > 0 } images and inverse images of convex sets under perspective are convex linear-fractional function f : R n → R m : f (x) = Ax + b cT^ x + d , dom f = {x | c T x + d > 0 } images and inverse images of convex sets under linear-fractional functions are convex
example of a linear-fractional function f (x) =
x 1 + x 2 + 1 x example of a linear-fractional function f (x) =
x 1 + x 2 + 1 x PSfrag replacements x 1 x 2 C − 1 0 1 − 1 0 1 PSfrag replacements x 1 x 2 f (C) − 1 0 1 − 1 0 1 Convex sets 2 – 15 IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Separating hyperplane theorem
if C and D are disjoint convex sets, then there exists a &= 0, b such that a T x ≤ b for x ∈ C , a T x ≥ b for x ∈ D Separating hyperplane theorem if C and D are disjoint convex sets, then there exists a != 0 , b such that aT^ x ≤ b for x ∈ C, aT^ x ≥ b for x ∈ D PSfrag replacements D C a aT^ x ≥ b aT^ x ≤ b the hyperplane {x | aT^ x = b} separates C and D strict separation requires additional assumptions (e.g., C is closed, D is a singleton) Convex sets 2 – 19 the hyperplane {x | aT^ x = b} separates C and D strict separation (aT^ x < b for x ∈ C , aT^ x > b for x ∈ D) requires additional assumptions (e.g., C is closed, D is a singleton)
Generalized inequalities
generalized inequality defined by a proper cone K : x (K y ⇐⇒ y −x ∈ K , x ≺K y ⇐⇒ y −x ∈ int K examples ! (^) componentwise inequality (K = Rn +) x (Rn
y ⇐⇒ xi ≤ yi , i = 1,... , n ! (^) matrix inequality (K = Sn +) X (Sn
Y ⇐⇒ Y − X positive semidefinite these two types are so common that we drop the subscript in (K properties: many properties of (K are similar to ≤ on R, e.g., x (K y , u (K v =⇒ x + u (K y + v IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Minimum and minimal elements
(K is not in general a linear ordering: we can have x &(K y and y &(K x x ∈ S is the minimum element of S with respect to (K if y ∈ S =⇒ x (K y (equivalently, S ⊆ x + K ). x ∈ S is a minimal element of S with respect to (K if y ∈ S, y (K x =⇒ y = x (equivalently, (x−K )∩S = {x}). example (K = R^2 +) x 1 is the minimum element of S 1 x 2 is a minimal element of S 2 Minimum and minimal elements !K is not in general a linear ordering : we can have x "!K y and y "!K x x ∈ S is the minimum element of S with respect to !K if y ∈ S =⇒ x !K y x ∈ S is a minimal element of S with respect to !K if y ∈ S, y !K x =⇒ y = x example (K = R^2 +) x 1 is the minimum element of S 1 x 2 is a minimal element of S 2 PSfrag replacements x 1 S 1 x 2 S 2 IOE 611: Nonlinear Programming, Winter 2008 Convex sets 2. Convex sets Page 2–20 2 – 18
Dual cones and generalized inequalities
dual cone of a cone K : K ∗ = {y | y T x ≥ 0 for all x ∈ K } examples ! (^) K = Rn +: K ∗^ = Rn + ! (^) K = Sn +: K ∗^ = Sn + ! (^) K = {(x, t) | ‖x‖ 2 ≤ t}: K ∗^ = {(x, t) | ‖x‖ 2 ≤ t} ! (^) K = {(x, t) | ‖x‖ 1 ≤ t}: K ∗^ = {(x, t) | ‖x‖∞ ≤ t} first three examples are self-dual cones dual cones of proper cones are proper, hence define generalized inequalities: y *K ∗^0 ⇐⇒ y T x ≥ 0 for all x *K 0 IOE 611: Nonlinear Programming, Winter 2008 2. Convex sets Page 2–
Minimum and minimal elements via dual inequalities
minimum element w.r.t. (K x is minimum element of S iff for all λ ,K ∗ 0, x is the unique minimizer of λT^ z over S Minimum and minimal elements via dual inequalities minimum element w.r.t. !K x is minimum element of S iff for all λ "K∗ 0 , x is the unique minimizer of λT^ z over S PSfrag replacements (^) x S minimal element w.r.t. #K
- if x minimizes λT^ z over S for some λ "K∗^0 , then x is minimal PSfrag replacements x (^1) S x 2 λ 1 λ 2
- if x is a minimal element of a convex set S, then there exists a nonzero λ #K∗^0 such that x minimizes λT^ z over S Convex sets 2 – 22 minimal element w.r.t. (K ! (^) if x minimizes λT^ z over S for some λ ,K ∗ (^) 0, then x is minimal Minimum and minimal elements via dual inequalities minimum element w.r.t. !K x is minimum element of S iff for all λ "K∗^0 , x is the unique minimizer of λT^ z over S PSfrag replacements (^) x S minimal element w.r.t. #K
- if x minimizes λT^ z over S for some λ "K∗^0 , then x is minimal PSfrag replacements x (^1) S x 2 λ 1 λ 2
- if x is a minimal element of a convex set S, then there exists a nonzero λ #K∗ 0 such that x minimizes λT^ z over S Convex sets 2 – 22 ! (^) if x is a minimal element of a convex set S, then there exists a nonzero λ *K ∗ 0 such that x minimizes λT^ z over S
