Convex Optimization Tutorial
Convex Optimization Resources
Selected Reading
- Algorithms for Convex Problems
- Karush-Kuhn-Tucker Optimality Necessary Conditions
- Fritz-John Conditions
- Convex Programming Problem
- Pseudoconvex Function
- Strongly Quasiconvex Function
- Strictly Quasiconvex Function
- Differentiable Quasiconvex Function
- Quasiconvex & Quasiconcave functions
- Sufficient & Necessary Conditions for Global Optima
- Differentiable Convex Function
- Convex & Concave Function
- Direction
- Extreme point of a convex set
- Polyhedral Set
- Conic Combination
- Polar Cone
- Convex Cones
- Fundamental Separation Theorem
- Closest Point Theorem
- Weierstrass Theorem
- Caratheodory Theorem
- Convex Hull
- Affine Set
- Convex Set
- Minima and Maxima
- Inner Product
- Norm
- Linear Programming
- Introduction
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Introduction
Convex Optimization - Introduction
This course is useful for the students who want to solve non-pnear optimization problems that arise in various engineering and scientific apppcations. This course starts with basic theory of pnear programming and will introduce the concepts of convex sets and functions and related terminologies to explain various theorems that are required to solve the non pnear programming problems. This course will introduce various algorithms that are used to solve such problems. These type of problems arise in various apppcations including machine learning, optimization problems in electrical engineering, etc. It requires the students to have prior knowledge of high school maths concepts and calculus.
In this course, the students will learn to solve the optimization problems pke $min fleft ( x ight )$ subject to some constraints.
These problems are easily solvable if the function $fleft ( x ight )$ is a pnear function and if the constraints are pnear. Then it is called a pnear programming problem (LPP). But if the constraints are non-pnear, then it is difficult to solve the above problem. Unless we can plot the functions in a graph, then try to analyse the optimization can be one way, but we can t plot a function if it s beyond three dimensions. Hence there comes the techniques of non-pnear programming or convex programming to solve such problems. In these tutorial, we will focus on learning such techniques and in the end, a few algorithms to solve such problems. first we will bring the notion of convex sets which is the base of the convex programming problems. Then with the introduction of convex functions, we will some important theorems to solve these problems and some algorithms based on these theorems.
Terminologies
The space $mathbb{R}^n$ − It is an n-dimensional vector with real numbers, defined as follows − $mathbb{R}^n=left { left ( x_1,x_2,...,x_n ight )^{ au }:x_1,x_2,....,x_n in mathbb{R} ight }$
The space $mathbb{R}^{mXn}$ − It is a set of all real values matrices of order $mXn$.