Category Archives: Optimization

Least squares in a non-ordinary sense

Posted on 13/01/2018 by HL

Simple ordinary least squares regression (SOLSR) means the following. Given data , , find a line in represented by that fits the data in the following sense. The loss of each data point to the line is       … Continue reading →

Posted in Applied mathematics, Calculus, Optimization, Statistics | Leave a comment

Polynomial Optimization 3: Why do we need generalize?

Posted on 11/10/2014 by HL

It has been a while since the last post. Let us recall what we have done. We study the unconstrained polynomial optimization problem ()           where is a real polynomial. This problem is equivalent to () … Continue reading →

Posted in Algebra, Applied mathematics, Optimization | Leave a comment

AM-GM-HM Inequality: A Statistical Point of View

Posted on 21/03/2014 by HL

In this post we shall give another proof of the famous AM-GM-HM inequality: If are positive real numbers, then AM GM HM, precisely .

Posted in Applied mathematics, Calculus, Optimization, Statistics | 1 Comment

Polynomial Optimization 2: SOS and SDP

Posted on 21/03/2014 by HL

In this article we shall describe something called Gram-matrix method which can decompose a polynomial into sum of squares. The notation means is a square symmetric positive semidefinite matrix.  Proposition 1.          Let , be a polynomial … Continue reading →

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S-Lemma

Posted on 23/12/2013 by HL

This post aims at proving a standard result in quadratic programming called S-Lemma. The use of this result will be obvious in a future post. We begin with a lemma. Lemma 1.           Let be two … Continue reading →

Posted in Applied mathematics, Linear Algebra, Optimization | 4 Comments

Polynomial Optimization 1: Motivation

Posted on 29/11/2013 by HL

(Nov 28, 2013) A (constrained) polynomial minimization problem is in the form  ()          subject to where are polynomials in with real coefficients. If and , then the above problem becomes , an unconstrained polynomial minimization problem. … Continue reading →

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The World of Complexity

Posted on 15/11/2013 by HL

(Sept 17, 2013) Our little plan is to write a series of posts discussing a sequence of optimization problems. Before we dive into them, it is cool to learn a rigorous treatment of standard complexity notions P, NP, NP-hard and … Continue reading →

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Notes on Optimization

Posted on 05/07/2011 by HL

This is my first time trying to upload my notes here. If the Acrobat Reader does not show all words, please close the file and open the file again.  Foundations of Optimization (5 July 2011)

Posted in Analysis, Applied mathematics, Functional analysis, Optimization | Leave a comment