# Questions tagged [sos]

Inequalities proofs by Sum of Squares method (SOS).

119 questions
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Given $a^2+b^2+c^2=1$ Prove $\sum_\text{cyc} \frac{1}{6ab+c^2}-\frac{1}{2+c^2}$ is nonnegative I have tried substituting 1 with $a^2+b^2+c^2$, but nothing is working. I’m trying to reduce it into a ...
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### Decomposing bivariate quadratic form into sum of two squares

I would like to decompose $$ax_1^2 + bx_2^2 + 2cx_1x_2$$ into two expressions, each involving only one variable. I'm trying to use a transform like $x_1 = x_+ + x_-$ and $x_2 = x_+ - x_-$ to ...
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### Duality gap in polynomial optimization problem Lasserre relaxation

Consider a polynomial optimization problem of the following type \begin{array}{cl} \text{maximize} & p(\mathbf{x}) \\ \text{subject to} & \mathbf{x} \in K \\ \end{array} \...
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### How can one prove that this polynomial is non-negative?

How one can prove the following inequality? $$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$ I plotted the graph on Wolfram Alpha and found that the inequality seems to ...
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### I have a inequality, I don't know where to start

Show that for $x,y,z > 0$ the inequality is true: $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}+x+y+z \geq \frac{(x+y)^2}{y+z}+\frac{(y+z)^2}{z+x}+\frac{(z+x)^2}{x+y}$ I have tried Holder, but i had ...
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### Prove the next cyclic inequality

Let $a,b,c>0$ such that $$a+b+c=1$$ prove that $$cyclic\sum \frac {ab}{\sqrt {c+ab}}\le\frac {1}{2}$$ By symetrie, i proved it by assuming $a=b=c$ but i cannot justify this hypothesis.
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### Inequality : $\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$

Let $a, b, c$ be positive real number such that $a+b+c = 3$. Prove that $$\displaystyle\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \displaystyle\sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$$ My attempt : ...
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### Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$

I am searching for an elementary proof of the AM-GM inequality in three variables: $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$ The inequality of the geometric mean vs the arithmetic mean of two variables ...
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### Convergence of sum of squares relaxations for global polynomial optimization

I have studied and understood the Moment-SOS hierarchy proposed by Lasserre where a sequence of semidefinite programs are required to be solved and a rank condition for the moment matrices is invoked ...
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### Sum-of-squares quartic polynomials in three variables

Suppose $f \in \mathbb R [x,y,z]$ is a homogeneous polynomial of degree $4$. Furthermore, suppose we can write $$f(x,y,z)=\sum_i p_i(x,y,z)^2$$ where each $p_i$ is a homogeneous polynomial of degree ...
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### Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$ are different positive real numbers. First, I tried to simplify the proof ...
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### Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Let $x,y,z$ be real numbers all greater than $1$, then prove that $$\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$$ My Attempt: I am trying to ...
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### Nessbit's and Triangle inequality

A question given in a book was as follows: Let $a$ $b$ and $c$ be the sides of a triangle. Prove that: $$\frac{3}{2}\leq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$$ My Attempt: The LHS is the ...
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### IMO 2006, A4 Problem, Idea behind the proof

IMO 2006, Problem A4, page 13: Prove the inequality: $$\sum_{i<j} \frac{a_{i}a_{j}}{a_{i}+ a_{j}} \le \frac{n}{2(a_{1} + a_{2} + \cdots + a_{n}) }\sum_{i<j}a_{i}a_{j}$$ for ...
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### Turkevicius inequality

Let $a$, $b$, $c$ and $d$ be positive real numbers. Prove that: $$a^4+b^4+c^4+d^4+2abcd \geq a^2b^2+ a^2c^2+ a^2d^2+ b^2c^2+ b^2d^2+ c^2d^2.$$ Source : Inequalities, Theorems, Techniques and ...
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### Write $(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$ as a sum of (three) squares of quadratic forms

The quartic form $$(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$$ is non-negative for all real $x$, $y$, $z$, as one can check (with some effort). A theorem of Hilbert implies that there exist ...
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### How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$

Let $x>0$, $y>0$ and $z>0$. Show that $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$ I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks?
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### If a, b, c>0 show that $\frac{a^2}{b^2+bc}+\frac{b^2}{c^2+ac}+\frac{c^2}{a^2+ab} \ge \frac{3}{2}$

For positive real numbers $a$, $b$, and $c$ prove that: $$\frac{a^2}{b^2+bc}+\frac{b^2}{c^2+ac}+\frac{c^2}{a^2+ab} \ge \frac{3}{2}.$$ I let $x=\frac{a}{b}$, $y=\frac{b}{c}$, and $z=\frac{c}{a}$. Then ...
Consider the AM-GM inequality in five variables $$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$ Can one write the LHS as a concrete (finite) sum $\,\sum_i h_i\,s_i\,$ with ...
### Inequality : $\sum_{cyc} \frac{a+b}{\sqrt{a+2c}} \geq 2 \sqrt{a+b+c}$
Let $a$, $b$ and $c$ be positive real numbers. Prove that: $$\displaystyle\sum_{cyc} \frac{a+b}{\sqrt{a+2c}} \geq 2 \sqrt{a+b+c}.$$ My attempt : By Holder inequality, \$ \left(\displaystyle\sum_{...