Category Archives: Optimization

Least squares in a non-ordinary sense

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

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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 .

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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

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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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