mathematics

Sum-of-squares optimization

Sum-of-squares optimization is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sum-of-squares optimization rather than just read about it. In short: A sum-of-squares optimization program is an optimization problem with a linear cost function and constraints that certain polynomials constructed from the decision variables should be sums of squares. When the maximum degree of the polynomials involved is fixed, sum-of-squares optimization is also known as the Lasserre hierarchy of semidefinite programming relaxations.

Key takeaways

Sum-of-squares optimization belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sum-of-squares optimization to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sum-of-squares optimization from memory before moving on to harder problems.

Reference excerpt

A sum-of-squares optimization program is an optimization problem with a linear cost function and constraints that certain polynomials constructed from the decision variables should be sums of squares. When the maximum degree of the polynomials involved is fixed, sum-of-squares optimization is also known as the Lasserre hierarchy of semidefinite programming relaxations. Sum-of-squares optimization techniques have been applied across a variety of areas, including control theory (in particular, for searching for polynomial Lyapunov functions for dynamical systems described by polynomial vector fields), statistics, finance and machine learning.

Background

A polynomial p {\displaystyle p} is a sum of squares (SOS) if there exist polynomials { f i } i = 1 m {\displaystyle \{f_{i}\}_{i=1}^{m}} such that p = ∑ i = 1 m f i 2 {\textstyle p=\sum _{i=1}^{m}f_{i}^{2}} . For example,

p = x 2 − 4 x y + 7 y 2 {\displaystyle p=x^{2}-4xy+7y^{2}}

is a sum of squares since

p = f 1 2 + f 2 2 {\displaystyle p=f_{1}^{2}+f_{2}^{2}}

where

f 1 = ( x − 2 y ) and f 2 = 3 y . {\displaystyle f_{1}=(x-2y){\text{ and }}f_{2}={\sqrt {3}}y.}

Note that if p {\displaystyle p} is a sum of squares then p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} . Detailed descriptions of polynomial SOS are available. Quadratic forms can be expressed as p ( x ) = x T Q x {\displaystyle p(x)=x^{T}Qx} where Q {\displaystyle Q} is a symmetric matrix. Similarly, polynomials of degree ≤ 2d can be expressed as

p ( x ) = z ( x ) T Q z ( x ) , {\displaystyle p(x)=z(x)^{\mathsf {T}}Qz(x),}

where the vector z {\displaystyle z} contains all monomials of degree ≤ d {\displaystyle \leq d} . This is known as the Gram matrix form. An important fact is that p {\displaystyle p} is SOS if and only if there exists a symmetric and positive-semidefinite matrix Q {\displaystyle Q} such that p ( x ) = z ( x ) T Q z ( x ) {\displaystyle p(x)=z(x)^{\mathsf {T}}Qz(x)} . This provides a connection between SOS polynomials and positive-semidefinite matrices.

Optimization problem A sum-of-squares optimization problem is a conic optimization problem with respect to the cone of sum-of-squares polynomials. Concretely, given a vector c ∈ R n {\displaystyle c\in \mathbb {R} ^{n}} and polynomials a k , j {\displaystyle a_{k,j}} for k = 1 , … N s {\displaystyle k=1,\dots N_{s}} , j = 0 , 1 , … , n {\displaystyle j=0,1,\dots ,n} , a sum-of-squares optimization problem is written as

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Sum-of-squares optimization

Start with the simplest possible case. Write down what Sum-of-squares optimization claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sum-of-squares optimization before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sum-of-squares optimization ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sum-of-squares optimization

In research: Sum-of-squares optimization appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sum-of-squares optimization in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sum-of-squares optimization is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical optimization, Real algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Sum-of-squares optimization outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Sum-of-squares optimization in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sum-of-squares optimization means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sum-of-squares optimization out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sum-of-squares optimization in simple terms?

Why does Sum-of-squares optimization matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sum-of-squares optimization?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sum-of-squares optimization.