This tutorial provides a comprehensive introduction to (possibly infinite) linear systems containing strict inequalities and evenly convex sets, with applications to convex optimization and economics. It discusses evenly convex polyhedra, finite linear systems, and conjugacy and duality perspectives.

Format: Paperback / softback
Length: 181 pages
Publication date: 28 October 2021
Publisher: Springer Nature Switzerland AG

This tutorial serves as an extensive and comprehensive introduction to (potentially infinite) linear systems that incorporate strict inequalities and evenly convex sets. The book delves into their application in convex optimization, with a particular focus on evenly convex polyhedra and finite linear systems with strict inequalities. It also explores evenly convex and quasiconvex functions from a conjugacy and duality perspective, discussing their applications in economics. Written in an expository style, the main concepts and fundamental results are illustrated with appropriate examples and figures, making it accessible to a wide range of readers.

Introduction

Linear systems, which are mathematical models describing the relationships between variables, play a crucial role in various fields of study. In this tutorial, we will focus on a particular type of linear system known as linear systems with strict inequalities and evenly convex sets. These systems arise in various applications, including optimization, control theory, and economics.

Definition and Notation

A linear system with strict inequalities and evenly convex sets is defined as follows:

Let A be a matrix of coefficients and x be a vector of variables. The system is said to have strict inequalities if there exist constants c1, c2,..., cn such that Ax - b ≥ 0 for all b ∈ Rn, where Rn is the real vector space of n dimensions.

Additionally, the set of all solutions to the system is assumed to be evenly convex, meaning that any line segment connecting two solutions lies within the set.

Examples

Consider the following linear system:

Ax - b = 0

y = cx

where A is a matrix with coefficients a1, a2, and a3, b is a vector of constants, and c is a scalar constant. This system is a linear system with strict inequalities because the inequality Ax - b ≥ 0 holds for all b ∈ Rn.

On the other hand, the system

x = Ay

is a linear system with strict inequalities because the inequality Ax - b ≥ 0 holds for all b ∈ Rn.

However, the system

x = Ay

is not a linear system with strict inequalities because the set of solutions is not evenly convex.

Properties of Linear Systems with Strict Inequalities and Evenly Convex Sets

Linear systems with strict inequalities and evenly convex sets have several important properties. These properties include:

Stability: The system is stable if the matrix A is positive definite.

Convergence: The system converges to a unique solution if it is initialized close to a solution.

Optimal Control: The system can be controlled to achieve a desired output by designing an appropriate feedback controller.

Application to Convex Optimization

Linear systems with strict inequalities and evenly convex sets find applications in convex optimization. Convex optimization is a branch of mathematics that deals with the optimization of non-linear functions subject to constraints. It is widely used in engineering, finance, and other fields to solve complex problems.

In convex optimization, linear systems with strict inequalities and evenly convex sets are used to model optimization problems. The objective is to find the optimal solution to the problem, which is the solution that minimizes or maximizes a given objective function while satisfying the constraints.

One example of convex optimization is the quadratic programming problem. In quadratic programming, we are given a set of linear inequalities and a quadratic objective function. We want to find the optimal solution to the problem, which is the solution that minimizes the objective function while satisfying the constraints.

Linear systems with strict inequalities and evenly convex sets are also used in linear programming. In linear programming, we are given a set of linear inequalities and a linear objective function. We want to find the optimal solution to the problem, which is the solution that minimizes or maximizes the objective function while satisfying the constraints.

Evenly Convex Polyhedra and Finite Linear Systems

Evenly convex polyhedra are a special type of convex set that is defined as the intersection of finitely many convex polyhedra. They are important in linear systems with strict inequalities and evenly convex sets because they provide a way to represent and analyze the system.

Finite linear systems with strict inequalities and evenly convex sets are a special type of linear system that arises in many applications. They are characterized by the presence of a finite number of variables and a finite number of constraints.

In finite linear systems with strict inequalities and evenly convex sets, we can use the theory of convex optimization to find the optimal solution to the problem. We can also use the theory of linear programming to find the optimal solution to the problem.

Evenly Convex and Quasiconvex Functions

Evenly convex and quasiconvex functions are important in linear systems with strict inequalities and evenly convex sets because they provide a way to represent and analyze the system.

Evenly convex functions are functions that are convex on a closed interval. They have a single global maximum and a single global minimum. Quasiconvex functions are functions that are convex on a half-open interval. They have a single global maximum and a single global minimum, but they may have local minima and maxima.

In linear systems with strict inequalities and evenly convex sets, we can use the theory of convex optimization to find the optimal solution to the problem. We can also use the theory of linear programming to find the optimal solution to the problem.

Applications in Economics

Linear systems with strict inequalities and evenly convex sets have applications in economics. They are used to model economic problems such as production, consumption, and investment.

In economics, we are often interested in finding the optimal solution to a problem that involves maximizing a given objective function while satisfying a set of constraints. Linear systems with strict inequalities and evenly convex sets provide a way to model these problems and find the optimal solution.

One example of an economic problem that can be modeled using linear systems with strict inequalities and evenly convex sets is the production problem. In the production problem, we are given a set of inputs and a set of outputs. We want to find the optimal production plan that minimizes the cost of producing the outputs while satisfying the given constraints.

Linear systems with strict inequalities and evenly convex sets can also be used to model investment problems. In investment problems, we are given a set of assets and a set of liabilities. We want to find the optimal investment plan that maximizes the expected return while satisfying the given constraints.

In conclusion, this tutorial serves as an extensive and comprehensive introduction to (potentially infinite) linear systems that incorporate strict inequalities and evenly convex sets. The book delves into their application in convex optimization, with a particular focus on evenly convex polyhedra and finite linear systems with strict inequalities. It also explores evenly convex and quasiconvex functions from a conjugacy and duality perspective, discussing their applications in economics. Written in an expository style, the main concepts and fundamental results are illustrated with appropriate examples and figures, making it accessible to a wide range of readers.

Weight: 314g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030534585
Edition number: 1st ed. 2020