Questions tagged [sum-of-squares]

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2
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0answers
33 views

Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related? [closed]

This is a crosspost of mathoverflow/345282 I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy: $$\inf_{x\in\mathbb{R}^n}\quad p(x)$$ ...
10
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3answers
394 views

When is the duality gap of semidefinite programming (SDP) zero?

I haven't been able to find in the literature a precise characterization of the vanishing of the SDP duality gap. Or, when does "strong duality" hold? For example, when one goes back and forth ...
4
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0answers
186 views

Testing emptiness property complexity in Sum of Squares Proof systems

Take the set $$\mathcal T=\{f_1(x_1,\dots,x_n)=\dots=f_m(x_1,\dots,x_n)=0, h_1(x_1,\dots,x_n)\geq a_1,\dots,h_t(x_1,\dots,x_n)\geq a_t\}$$ where $$h_1(x_1,\dots,x_n),\dots,h_t(x_1,\dots,x_n)\in\mathbb ...
1
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0answers
98 views

Complexity of planted root of a system of quadratic homogeneous polynomials?

Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
12
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0answers
441 views

Is satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ NP Complete?

Question I would like to show that satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ is NP-Complete. Consider $L= \{(\bar{y},r):\exists \bar{x} \text{ such that } \sum_{i=1}^{n}{x_i^{y_i}}=r\}$. Where $\...
3
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0answers
77 views

Primal/Dual of the Lasserre/ SOS SDP hierarchy

Sum of Squares proofs and the Lasserre hierarchy can both be stated as SDPs. It is often claimed without proof that these SDPs are dual to each other, although I do not see that this is obvious. I was ...
3
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0answers
90 views

Some questions about the Ryan O'Donnel and Yuan Zhou's paper “Approximability and proof complexity”

My question is particularly about the set-up in section $8$ (``Analysis of the KV Max-Cut instances") of the paper, https://arxiv.org/pdf/1211.1958.pdf. What they call the Khot-Vishnoi UG instance ...
3
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0answers
108 views

Can the Lasserre relaxation be defined over the reals?

If one wants to say minimize a function $f : \{-1,1\}^n \rightarrow \mathbb{R}$ on its domain then a degree$-d$ Lasserre relaxation of it would be to solve the problem of $\min \mathbb{E}_\mu [f(x)]$ ...
3
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0answers
113 views

SOS and the small set expansion property

For what graphs do we know that their small set expansion property has a low degree SOS proof? Is this known to be true for say the complete graphs? A terminology issue about what is ``low degree" :...
2
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0answers
65 views

Is there a relationship between the probabilistic interepretation of Sherali-Adams SDP hierarchy and the Lasserre SDP hierarchy?

Firstly note this paper http://ttic.uchicago.edu/~madhurt/Papers/reductions.pdf where a Lasserre SDP is being setup for the independent set probblem at the bottom of page 4 where the author says says, ...
3
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1answer
171 views

SOS hardness of $Max-2-Lin(\mathbb{Z}_2)$?

Do we know of instances of $Max-2-Lin(\mathbb{Z}_2)$ which have a integrality gaps w.r.t to high degree (> 4) SOS relaxations? Or if we specialize to Max-CUT do we know of graphs whose Max-CUT ...
1
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0answers
27 views

How does one know what is not in a certain class of pseudo-distributions?

We consider working in the function space $\mathbb{R}^{\{ -1,1\}^n}$ where the expectation inner-product makes the juntas form a $2^n$ dimensional orthonormal basis. Now say one has found a degree $...
7
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0answers
294 views

Why is it so difficult to study Sum of Squares (SoS) algorithms with degree $d>4$?

In many publications on the computational complexity of Sum of Squares (SoS) algorithms, it is typical to study the degree-$4$ relaxation; e.g. Rounding Sum-of-Squares Relaxations Sum-of-Squares ...
4
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1answer
100 views

From Lasserre maps to pseudo-distributions

Let me define a ``Lasserre map of degree $d$" as a linear map $L : \mathbb{R}_n[x] \rightarrow \mathbb{R}$ i.e a real valued linear map on polynomials over $n$ variables with real coeffients. This is ...
13
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2answers
564 views

Numerical precision in sum-of-squares method?

I have been reading a bit about the sum-of-squares method (SOS) from the survey of Barak & Steurer and the lecture notes of Barak. In both cases they sweep issues of numerical accuracy under the ...
1
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0answers
53 views

About increasing the objective values of certificates for Max-Clique SDP

Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and ...
7
votes
1answer
749 views

Positivstellensatz and sum of squares method

This question comes from many online resources that introduce Sum-of-Squares method, such as the survey of Barak and Steurer (http://arxiv.org/abs/1404.5236). Let me focus on Theorem 2.1 of this ...
2
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1answer
162 views

How to prove $\tilde{\mathbb{E}}PQ=0$ when $\tilde{\mathbb{E}}P^2 = 0$?

Let $\tilde{\mathbb{E}}$ be a degree-$r$ pseudo-distribution. If $\tilde{\mathbb{E}}P^2 = 0$, how to prove that $\tilde{\mathbb{E}}PQ=0$ for all polynomials $Q$ such that $\mathrm{deg}(PQ)\le r$? We ...
23
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2answers
2k views

Sum-of-squares proof system

Recently I have seen several articles on arxiv that refer to a proof system called sum-of-squares. Can someone explain what is a sum-of-squares proof and why such proofs are important/interesting? ...
4
votes
1answer
241 views

Complexity of iterative least squares regression

Given a set of points $P = \{(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n) \}$ one can use least squares method to fit a polynomial to $P$. In particular I am interested in linear and quadratic regression. I ...