# 1 convex sets and convex functions

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### Convex Sets

Convex combination Deﬁnition A convex combinationof the points x1 ⋅⋅⋅ xk is a point of the form 1x1 ⋅⋅⋅ kxk where 1 ⋅⋅⋅ k = 1 and i ≥ 0 for all i = 1 ⋅⋅⋅ k. A set is convex if and only if it contains every convex combinations

• ### mathlib docs analysisnvex.basic

Convex sets and functions on real vector spaces. In a real vector space we define the following objects and properties. segment x y is the closed segment joining x and y. A set s is convex if for any two points x y ∈ s it includes segment x y A function f E → β is convex on a set s if s is itself a convex set and for any two points x y ∈ s the segment joining x f x to y f y

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### Convex Functions and Sets

Convex Functions f Rn R is aconvex functionif domfis a convex set and for all xy2domfand 2 01 we have f x 1 y f x 1 f y fisstrictly convexif strict inequality above for all x6= yand 0 < <1 fisconcaveif fis convex A ne functions are convex and concave 12

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### Chapter 6

6.B. Convex Sets and Functions Each of the contour sets in Figure 6.1 bulges outward so that each bends away from the comment tangent at x∗ and cannot bend back to meet the other set once again. This property is called convexity. Formally Deﬁnition 6.B.1 deﬁnes convex sets. Deﬁnition 6.B.1 Convex Set .

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### Convex Functions

For certain functions de ned on convex sets it can be very easy to determine whether they have a global minimizer and if so to compute it. A class of functions that has this property is introduced through the following de nition. De nition Let C Rn be a convex set and let f C R. Then f x is convex on Cif

• ### Convex function

Like strictly convex functions strongly convex functions have unique minima on compact sets. Uniformly convex functions. A uniformly convex function with modulus is a function that for all x y in the domain and t ∈ 0 1 satisfies

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### 1 Convex Functions

This section focuses on convex functions while the next section focuses on convex sets. They are similar however in that convex functions and convex sets are extremely desirable. If the feasible region is a convex set and if the objective function is a convex function then it is much easier to nd the optimal solution.

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### 1.3 Convex Functions

Figure 1 Convex Function 1.3.2 Characterizations of Di erentiable Convex Functions We now give some characterizations of convexity for once or twice di erentiable Proposition Let Cbe a nonempty convex set ˆRn and f Rn R be twice continuously di erentiable over an open set that contains C. Then

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### Convex Optimization Theory

Sec. B.1.1 Convex Sets and Functions 471 Proposition 1.1.9 Projection Theorem Let C be a nonempty closed convex subset of ℜn and let z be a vector in ℜn.There exists a unique vector that minimizes kz−xk over x ∈ C called the projection

• ### II Convex Sets and Functions

Convex Sets and Functions function on R is a convex set relatively closed in 5. Q.E.D. PROOF.f Qf 5 is a convex function. 5.2. Convex Functions with One Variable In this section we consider the special properties of convex functions of one variable. It is obvious how the properties of convex functions investigated in the preceding

• ### Concepts about convex convex set/function/optimization

2 Convex function 2.1 Description. If the upper area of the image of the function is a convex set then the function is a convex function. Pay attention to the top because the concavity and convexity are the opposite for different viewing angles.For example a bowl placed on the square is convex and a bowl that is buckled upside down is concave.

• ### Relation between convex set and convex function

Relation between convex set and convex function. Ask Question. Asked 8 years 1 month ago. Active 8 years 1 month ago. Viewed 596 times.

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### Convex Optimization

1. convex sets functions optimization problems 2. examples and applications 3. algorithms Introduction 1–13. Nonlinear optimization traditional techniques for general nonconvex problems involve compromises local optimization methods nonlinear programming

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### Convex Sets and Functions part II

Convex FunctionsBasics and Examples A extended value convex function f S R isproperly convexif f is not the constant function de ned by f x = 1. Thee ective domainof a convex function f S R is the set Dom f = fx 2S jf x <1g. 5/49

• ### Convex and Concave Function

Convex and Concave Function. Let f S → R where S is non empty convex set in R n then f x is said to be convex on S if f λ x 1 1 − λ x 2 ≤ λ f x 1 1 − λ f x 2 ∀ λ ∈ 0 1 . On the other hand Let f S → R where S is non empty convex set in R n then f x is said to be concave on S if f λ x 1

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### A Tutorial on Convex Optimization

convex sets functions and convex optimization problems so that the reader can more readily recognize and formulate engineering problems using modern convex optimization. This tutorial coincides with the publication of the new book on convex optimization by Boyd and Vandenberghe 7 who have made available a large amount of free course

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### Optimization Techniques in Finance

Convex sets and functions Convex optimization problems and duality Conic optimization Applications parameter estimation Convex sets A set C ˆRn is called convex if for any xy 2C the line segment joining x and y is contained in C. In other words if x 1 y 2Cfor any 0 1 1

• ### CiteSeerX

CiteSeerXDocument Details Isaac Councill Lee Giles Pradeep Teregowda Let C be a nonempty subset of ℜ n and let λ1 and λ2 be positive scalars. Show that if C is convex then λ1 λ2 C = λ1C λ2C cf. Prop. 1.1.1 c . Show by example that this need not be true when C is not convex. Solution We always have λ1 λ2 C ⊂ λ1C λ2C even if C is not convex.

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### 1 Convex Sets and Functions

approximately minimize convex functions and is applicable in many di erent settings. We also show how this gives an online algorithm with guarantees somewhat similar to the multiplicative weights algorithms.1 1 Convex Sets and Functions First recall the following standard de nitions De nition 1 A set K Rn is said to be convex if x 1 y

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### 1 Convexity Convex Relaxations and Global Opti

Convex Set Nonconvex Set x 1 N.B. Must hold for all pairs of points in the set. The intersection of two convex sets is convex. By induction the intersection of a nite number of convex sets is convex. 1.3 Convex functions De nition convex function . Let f S R where Sis a nonempty convex set in Rn. The function is said to be convex on Sif f

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### Lecture 3 Convex Functions

Lecture 3 Convex Functions Informally f is convex when for every segment x1 x2 as x α = αx1 1−α x2 varies over the line segment x1 x2 the points x α f x α lie below the segment connecting x1 f x1 and x2 f x2 Let f be a function from Rn to R f Rn → R The domain of f is a set in Rn deﬁned by dom f = x ∈ Rn f x is well deﬁned ﬁnite Def. A function f is

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### Chapter 2 Lecture 1 Convex sets

Math 484 Nonlinear Programming1 Mikhail Lavrov Chapter 2 Lecture 1 Convex sets February 4 2019 University of Illinois at Urbana Champaign 1 Convexity Earlier this semester we showed that if x is a critical point of f Rn R and Hf x 0 for all x 2Rn then x is a global minimizer. Our proof worked by being able to compare x to any other point x 2Rn along the line through x

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### Lecture 4 Convex Sets

2 Important a ne and convex sets 2.1 Lines The simplest example of a non trivial a ne set is probably a line in the space Rn. It is the set of all points yof the form y= x 1 1 x 2 Where x 1and x 2 are two points in the space and 2R is a scalar. This gives us the unique line passing

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### 2.1 Review 2.2 Convex Sets

1 x=2C is convex Support function for any set C convex or not its support function i C x = max y2C xT y is convex Max function f x = maxfx 1 x ngis convex 2.7 Key Properties of Convex Functions A function is convex if and only if its restriction to any line is convex.

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### 5.1 Convex Sets

5.1.4 Convex set representations Figure 5.1 Representation of a convex set as the convex hull of a set of points left and as the intersection of a possibly in nite number of halfspaces right . 5.1.4.1 Convex hull representation Let C Rnbe a closed convex set. Then Ccan be written as conv X the convex hull of possibly in nitely