Category Archives: Optimization

Polynomial Optimization 3: Why do we need generalize?

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

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

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

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

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

(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

(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

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