Convex Optimization Course And Certification

What is Convex Optimization?

Convex Optimization is simply the process of optimizing Convex Functions and a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. 

Convex Function is a real-valued function defined on an n-dimensional interval in mathematics.

To fully understand what Convex Optimization is, we first need to understand what a Convex Function is. For Instance, if you draw a line segment between any two points on the graph of a function in a way that there is no point on this graph that is high above the line segment between this point, then the function drawn on the graph is called a Convex Function.

Convex Optimization, on the other hand, is a set of techniques with which we can reduce the time complexity and precision of the results of something.

From the definitions above, we can sum up that Convex Optimization is a subfield of Mathematical Optimization that deals with the problems of minimizing convex functions over convex sets. 

What are Convex Optimization Problems?

Convex Optimization Problems are problems 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. 

Convex Optimization Problems can be solved by some modern methods such as subgradient projection and interior-point methods or by some old methods such as cutting plane methods, ellipsoid methods, and subgradient methods.

Interior Point or Barrier methods are especially appropriate for convex problems because they treat linear, quadratic, conic, and smooth nonlinear functions in essentially the same way - they create and use a smooth convex nonlinear barrier function for the constraints, even for LP problems.

Classes of Convex Optimization Problems

The following are all classes of Convex Optimization Problems:

1. Least-squares

2. Linear programming

3. Convex quadratic minimization with linear constraints

4. Quadratic minimization with convex quadratic constraints

5. Conic optimization

6. Geometric programming

7. Second-order cone programming

8. Semidefinite programming

Entropy maximization with appropriate constraints

Applications of Convex Optimization: 

Convex Optimization has a lot of applications and used in a wide area of disciplines such as:

  1. Estimation and Signal Processing,
  2. Automatic Control Systems,
  3. Communications and Networks,
  4. Electronic Circuit Design, 
  5. Data Analysis and Modeling,
  6. Finance,
  7. Statistics, 
  8. Structural Optimization, etc.

Features of Convex Optimization: 

The following are some of the features and characteristics of Convex Optimization:

  1. Linear Programming: Linear Programming is a mathematical method that is used to determine the best feasible outcome or solution from a given set of inputs or a list of requirements, that are represented in the form of linear relationships. Linear programming, because of its nature is often referred to as Linear Optimization.
  2. Minima And Maxima: Minima And Maxima are defined sets for a mathematical function. Minima is the point of minimum value of the function while maxima's are the point of the maximum value of the function.
  3. Convex Set: In geometry, a convex set or convex region is a section of a Euclidean Space, or more generally, an Affine Space Over the reals that intersects every line into a line segment.
  4. Jensen's Inequality: Jensen's Inequality is an inequality involving the Convexity of a mathematical function. In its most basic form, the inequality states that the Convex transformation of a mean is less than or equal to the mean applied after the Convex transformation. it is a simple proposition that the opposite is true of concave transformations.

Benefits of Convex Optimization: 

Some of the Benefits of Convex Optimization include:

1. In a constrained problem, a Convex feasible area makes it possible that you do not generate solutions that are not easily attainable while searching for the best-optimized solution.

2. It allows local search algorithms to guarantee an optimized solution. Of which if it's not present, a local search algorithm would be stuck with a less optimized solution. 

Convex Optimization Study: 

The Convex Optimization Course is very useful for those who want to solve non-linear optimization problems that come up in numerous engineering and scientific applications.

This Course begins with the theory of linear programming and will introduce the concepts of convex sets and functions and related terminologies to teach and explain the various theorems that are needed to solve the programming problems.

This Course will introduce you to the various algorithms that are used to solve such problems. These type of problems comes up in various applications including machine learning, optimization problems in electrical engineering, etc. It requires the students to have previous knowledge of high school mathematical concepts and calculus.

In The Full Course, you will learn everything you need to know about Convex Optimization with Certification to showcase your knowledge and competence.  

Convex Optimization Course Outline:

Convex Optimization - Introduction

Convex Optimization - Linear Programming

Convex Optimization - Norm

Convex Optimization - Inner Product

Convex Optimization - Minima and Maxima

Convex Optimization - Convex Set

Convex Optimization - Affine Set

Convex Optimization - Convex Hull

Convex Optimization - Caratheodory Theorem

Convex Optimization - Weierstrass Theorem

Convex Optimization - Closest Point Theorem

Convex Optimization - Fundamental Separation Theorem

Convex Optimization - Convex Cones

Convex Optimization - Polar Cone

Convex Optimization - Conic Combination

Convex Optimization - Polyhedral Set

Convex Optimization - Extreme point of a convex set

Convex Optimization - Direction

Convex Optimization - Convex & Concave Function

Convex Optimization - Jensen's Inequality

Convex Optimization - Differentiable Convex Function

Convex Optimization - Sufficient & Necessary Conditions for Global Optima

Convex Optimization - Quasiconvex & Quasiconcave functions

Convex Optimization - Differentiable Quasiconvex Function

Convex Optimization - Strictly Quasiconvex Function

Convex Optimization - Strongly Quasiconvex Function

Convex Optimization - Pseudoconvex Function

Convex Optimization - Convex Programming Problem

Convex Optimization - Fritz-John Conditions

Convex Optimization - Karush-Kuhn-Tucker Optimality Necessary Conditions

Convex Optimization - Algorithms for Convex Problems

Convex Optimization - Exams and Certification

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