Sum-of-squares optimization


A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficients in certain polynomials, those polynomials should have the polynomial SOS property. When fixing the maximum degree of the polynomials involved, sum-of-squares optimization is also known as the Lasserre hierarchy of relaxations in semidefinite programming.
Sum-of-squares optimization techniques have been successfully applied by researchers in the control engineering field, in particular in the search for polynomial Lyapunov functions for dynamical systems described by polynomial vector fields.

Optimization problem

The problem can be expressed as
subject to
Here "SOS" represents the class of sum-of-squares polynomials.
The vector and polynomials are given as part of the data for the optimization problem. The quantities are the decision variables. SOS programs can be converted to semidefinite programs using the
duality of the SOS polynomial program and a relaxation for constrained polynomial optimization using positive-semidefinite matrices, see the following section.

Dual problem: constrained polynomial optimization

Suppose we have an -variate polynomial , and suppose that we would like to minimize this polynomial over a subset.
Suppose furthermore that the constraints on the subset can be encoded using polynomial equalities of degree at most, each of the form where is a polynomial of degree at most.
A natural, though generally non-convex program for this optimization problem is the following:
subject to:
where is the -dimensional vector with one entry for every monomial in of degree at most, so that for each multiset , is a matrix of coefficients of the polynomial that we want to minimize, and is a matrix of coefficients of the polynomial encoding the th constraint on the subset. The additional, fixed constant index in our search space,, is added for the convenience of writing the polynomials and in a matrix representation.
This program is generally non-convex, because the constraints are not convex. One possible convex relaxation for this minimization problem uses semidefinite programming to replace the rank-one matrix of variables with a positive-semidefinite matrix : we index each monomial of size at most by a multiset of at most indices, . For each such monomial, we create a variable in the program, and we arrange the variables to form the matrix, where is the set of real matrices whose rows and columns are identified with multisets of elements from of size at most. We then write the following semidefinite program in the variables :
subject to:
where again is the matrix of coefficients of the polynomial that we want to minimize, and is the matrix of coefficients of the polynomial encoding the th constraint on the subset.
The third constraint ensures that the value of a monomial that appears several times within the matrix is equal throughout the matrix, and is added to make respect the symmetries present in the quadratic form.

Duality

One can take the dual of the above semidefinite program and obtain the following program:
subject to:
We have a variable corresponding to the constraint , a real variable for each polynomial constraint and for each group of multisets, we have a dual variable for the symmetry constraint. The positive-semidefiniteness constraint ensures that is a sum-of-squares of polynomials over : by a characterization of positive-semidefinite matrices, for any positive-semidefinite matrix, we can write for vectors. Thus for any,
where we have identified the vectors with the coefficients of a polynomial of degree at most. This gives a sum-of-squares proof that the value over.
The above can also be extended to regions defined by polynomial inequalities.

Sum-of-squares hierarchy

The sum-of-squares hierarchy, also known as the Lasserre hierarchy, is a hierarchy of convex relaxations of increasing power and increasing computational cost. For each natural number the corresponding convex relaxation is known as the th level or th round of the SOS hierarchy. The st round, when, corresponds to a basic semidefinite program, or to sum-of-squares optimization over polynomials of degree at most. To augment the basic convex program at the st level of the hierarchy to th level, additional variables and constraints are added to the program to have the program consider polynomials of degree at most.
The SOS hierarchy derives its name from the fact that the value of the objective function at the th level is bounded with a sum-of-squares proof using polynomials of degree at most via the dual. Consequently, any sum-of-squares proof that uses polynomials of degree at most can be used to bound the objective value, allowing one to prove guarantees on the tightness of the relaxation.
In conjunction with a theorem of Berg, this further implies that given sufficiently many rounds, the relaxation becomes arbitrarily tight on any fixed interval. Berg's result states that every non-negative real polynomial within a bounded interval can be approximated within accuracy on that interval with a sum-of-squares of real polynomials of sufficiently high degree, and thus if is the polynomial objective value as a function of the point, if the inequality holds for all in the region of interest, then there must be a sum-of-squares proof of this fact. Choosing to be the minimum of the objective function over the feasible region, we have the result.

Computational cost

When optimizing over a function in variables, the th level of the hierarchy can be written as a semidefinite program over variables, and can be solved in time using the ellipsoid method.

Sum-of-squares background

A polynomial is a sum of squares if there exist polynomials
such that . For example,
is a sum of squares since
where
Note that if is a sum of squares
then for all. Detailed descriptions of polynomial SOS are available.
Quadratic forms can be expressed as where is a symmetric matrix. Similarly, polynomials of degree ≤ 2d can be expressed as
where the vector contains all monomials of degree. This is known as the Gram matrix form. An important fact is that
is SOS if and only if there exists a symmetric and positive-semidefinite matrix such that.
This provides a connection between SOS polynomials and positive-semidefinite matrices.

Software tools