Glossary

2.1 Definitions and properties of convex sets

3 min read•august 9, 2024

Convex sets are fundamental in optimization, forming the basis for many algorithms. They're shapes without dents or holes, where any two points can be connected by a line entirely within the set. This concept is crucial for understanding feasible regions in optimization problems.

Convex combinations, interior points, and extreme points are key ideas within convex sets. These concepts help define the structure of convex sets and play a vital role in finding optimal solutions in various optimization scenarios.

Convex Sets and Combinations

Fundamental Concepts of Convex Sets

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  • encompasses all points on line segments connecting any two points within the set
    • Visualized as a shape without any indentations or concavities
    • Includes circles, ellipses, and rectangles
  • forms new points from weighted sum of existing points in set
    • Weights must sum to 1 and be non-negative
    • Expressed mathematically as x=∑i=1nλixix = \sum_{i=1}^n \lambda_i x_i where ∑i=1nλi=1\sum_{i=1}^n \lambda_i = 1 and λi≥0\lambda_i \geq 0
  • resides within set, surrounded by other points in all directions
    • Can move slightly in any direction while remaining in set
    • Mathematically defined using open balls: ∃ϵ>0\exists \epsilon > 0 such that B(x,ϵ)⊆SB(x, \epsilon) \subseteq S

Boundary and Extreme Points

  • lies on edge of set, not surrounded by other points in all directions
    • Can be approached arbitrarily closely by points both inside and outside set
    • Formally defined: ∀ϵ>0,B(x,ϵ)∩S≠∅\forall \epsilon > 0, B(x, \epsilon) \cap S \neq \emptyset and B(x,ϵ)∩Sc≠∅B(x, \epsilon) \cap S^c \neq \emptyset
  • cannot be expressed as convex combination of other points in set
    • Represents "corners" or vertices of convex set
    • Crucial in optimization problems, often optimal solutions found at extreme points
    • Mathematically: xx is extreme if x=λy+(1−λ)zx = \lambda y + (1-\lambda)z with 0<λ<10 < \lambda < 1 implies x=y=zx = y = z

Hyperplanes and Half-spaces

Hyperplanes: Definition and Properties

  • divides space into two half-spaces
    • Generalizes concept of line in 2D and plane in 3D to higher dimensions
    • Defined by linear equation aTx=ba^Tx = b where a≠0a \neq 0 is normal vector
  • Hyperplane properties include flatness and extension to infinity
    • Dimensionality one less than ambient space (2D line in 3D space)
    • Separates space into two distinct regions

Half-spaces and Separation Theorem

  • represents all points on one side of hyperplane, including hyperplane itself
    • Closed half-space defined by aTx≤ba^Tx \leq b or aTx≥ba^Tx \geq b
    • Open half-space defined by aTx<ba^Tx < b or aTx>ba^Tx > b
  • Separating hyperplane theorem proves existence of hyperplane separating disjoint convex sets
    • Crucial in support vector machines and other machine learning algorithms
    • States: For disjoint convex sets AA and BB, ∃a≠0,b\exists a \neq 0, b such that aTx≤ba^Tx \leq b for all x∈Ax \in A and aTx≥ba^Tx \geq b for all x∈Bx \in B

Polyhedra and Simplices

Polyhedra: Characteristics and Representations

  • defined as intersection of finite number of half-spaces
    • Includes familiar shapes (cubes, pyramids, prisms)
    • Can be bounded or unbounded
  • Polyhedron representations include:
    • (H-representation): {x∣Ax≤b}\{x | Ax \leq b\}
    • (V-representation): {x∣x=∑i=1kλivi,∑i=1kλi=1,λi≥0}\{x | x = \sum_{i=1}^k \lambda_i v_i, \sum_{i=1}^k \lambda_i = 1, \lambda_i \geq 0\}
  • Polyhedra play crucial role in linear programming and optimization

Simplices: Special Polyhedra

  • represents simplest possible polyhedron in given dimension
    • 0-simplex (point), 1-simplex (line segment), 2-simplex (triangle), 3-simplex (tetrahedron)
  • Simplex properties include:
    • Having n+1n+1 vertices in nn-dimensional space
    • Every face of simplex also simplex
    • of n+1n+1 affinely independent points
  • Simplices used in triangulation methods and simplicial approximation in topology

Key Terms to Review (15)

Boundary Point: A boundary point is a point in a set that lies on the edge or limit of that set. It can either be part of the set itself or can be an accumulation point of the set, meaning every neighborhood around it contains points from the set. Understanding boundary points is crucial for analyzing convex sets, as they help define the boundaries within which all points in the set lie.
Convex Combination: A convex combination is a linear combination of points where all the coefficients are non-negative and sum to one. This concept is key when discussing convex sets, as it helps to define the properties that characterize these sets, such as closure under linear combinations. By using convex combinations, one can explore how various points within a set relate to each other and how they can form new points that remain within the same set.
Convex Hull: The convex hull of a set of points in a Euclidean space is the smallest convex set that contains all the points. Visually, you can think of it as the shape formed by stretching a rubber band around the outermost points, which creates a minimal enclosing polygon or polyhedron. This concept plays a crucial role in understanding convex sets and their properties, and it helps to determine whether a given set is convex.
Convex set: A convex set is a subset of a vector space such that, for any two points within the set, the line segment connecting them lies entirely within the set. This property is crucial in optimization, as convex sets often lead to simpler and more tractable problems, especially when it comes to finding optimal solutions and verifying optimality conditions.
Extreme Point: An extreme point refers to a point in a convex set that cannot be expressed as a combination of other points within the set. This concept is critical when analyzing convex sets because extreme points serve as the vertices or corner points where the maximum or minimum values of a function are often found. Understanding extreme points helps in recognizing the shape and boundaries of convex sets, which is fundamental in optimization problems.
Feasible Region: The feasible region is the set of all possible solutions that satisfy a given set of constraints in an optimization problem. This region is crucial because it defines the boundaries within which optimal solutions can be found, and it relates directly to concepts such as convexity, constraint types, and optimization methods.
Half-space: A half-space is a concept in geometry and optimization that refers to one side of a hyperplane in a multidimensional space. It is defined as the set of points that satisfy a linear inequality, and it divides the space into two distinct regions: one containing the points that satisfy the inequality and the other containing those that do not. Understanding half-spaces is crucial for grasping the properties of convex sets, as they help form the basic building blocks for defining convexity and understanding feasible regions in optimization problems.
Half-Space Representation: Half-space representation refers to a method of defining convex sets using linear inequalities. In this representation, a convex set can be expressed as the intersection of half-spaces, each defined by a linear inequality of the form $$a^T x \leq b$$, where $$a$$ is a vector of coefficients, $$x$$ is a point in the space, and $$b$$ is a scalar. This approach emphasizes the geometric interpretation of convex sets as regions formed by intersecting these half-spaces, which is crucial in understanding their properties and applications in optimization.
Hyperplane: A hyperplane is a flat affine subspace of one dimension less than its ambient space, commonly used in geometry and optimization. It serves as a boundary that separates different regions of the space and can be defined mathematically as the set of points satisfying a linear equation. Understanding hyperplanes is crucial for analyzing the properties of convex sets and for applications such as classification in machine learning algorithms.
Interior Point: An interior point of a set is a point that lies within the set and is not on its boundary. It implies that there exists a neighborhood around this point that also lies entirely within the set. This concept is crucial in understanding the properties of convex sets, as it helps to characterize feasible regions in optimization problems and provides insights into convergence behavior of algorithms.
Optimal Solution Set: The optimal solution set refers to the collection of feasible solutions in an optimization problem that yield the best possible outcome, typically the maximum or minimum value of the objective function. This concept is crucial in understanding how various feasible solutions relate to each other, especially in contexts involving convex sets, where any local minimum is also a global minimum.
Polyhedron: A polyhedron is a three-dimensional solid figure composed of flat polygonal faces, straight edges, and vertices. Each face of a polyhedron is a polygon, and the edges are where two faces meet, while vertices are the points where edges converge. Understanding polyhedra is essential because they represent a fundamental class of shapes in geometry that can help explain more complex structures and concepts in convex sets.
Separation Theorem: The Separation Theorem is a fundamental concept in convex analysis that states if two convex sets do not intersect, there exists a hyperplane that separates them. This theorem is crucial for understanding how convex sets can be distinguished from one another and lays the groundwork for characterizing convex functions and optimization problems involving these sets. It also emphasizes the geometric properties of convexity, showcasing the relationship between separation and the existence of optimal solutions in nonlinear optimization.
Simplex: The simplex is a fundamental geometric structure used in optimization, particularly in linear programming. It represents the simplest form of a convex set with the minimum number of vertices, which are crucial for finding optimal solutions in multidimensional spaces. The properties of a simplex are closely tied to the definitions of convex sets, as each vertex corresponds to a potential solution in the feasible region defined by the constraints of an optimization problem.
Vertex Representation: Vertex representation refers to the way of expressing a convex set as the convex hull of its vertices, which are the extreme points or corner points of that set. This concept is essential in understanding the structure of convex sets and highlights how every point in a convex set can be formed as a combination of these vertices. The vertex representation emphasizes the importance of these extreme points in defining the shape and boundaries of the convex set.
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