Convex optimization theory

  • What is a convex function in optimization theory?

    A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing.
    Linear functions are convex, so linear programming problems are convex problems..

  • What is convexity theory?

    An axiomatic setting for the theory of convexity is provided by taking an arbitrary set X and constructing a family ^ of subsets of X which is closed under inter- sections.
    The pair consisting of any ordered vector space and its family of convex subsets thus become the prototype for all such pairs (X, ^)..

  • Why convexity is the key to optimization?

    Convex functions are helpful in optimization and other fields of mathematics due to a variety of key features.
    For example, they are always continuous and have a unique global minimum, implying that convex function optimization issues are often simple to solve..

  • A convex set is a collection of points in which the line AB connecting any two points A, B in the set lies completely within the set.
    In other words, A subset S of En is considered to be convex if any linear combination θx1 + (1 − θ)x2, (0 ≤ θ ≤ 1) is also included in S for all pairs of x1, x2 ∈ S.
  • Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
  • Convex optimization is a powerful tool used to solve optimization problems in various fields such as finance, engineering, and machine learning.
    In a convex optimization problem, the goal is to find a point that maximizes or minimizes the objective function.
(a) Convex analysis, particularly as it relates to optimization. (b) Duality theory for optimization and minimax problems, mainly within a convexity framework.
An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory. Convexity theory is first developed in a Google BooksOriginally published: 2009Author: Dimitri Bertsekas

Definition

A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set is a convex set. A function m…

Properties

The following are useful properties of convex optimization problems:

Applications

The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations:

Lagrange multipliers

Consider a convex minimization problem given in standard form by a cost function and inequality constraints for . Then the domain is:

Algorithms

Unconstrained convex optimization can be easily solved with gradient descent (a special case of steepest descent) or Newton's method, combine…


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