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 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.
Optimization problem.
The problem can be expressed as
subject to
Here "SOS" represents the class of sum-of-squares (SOS) polynomials.
The vector formula_3 and polynomials formula_4 are given as part of the data for the optimization problem. The quantities formula_5 are the decision variables. SOS programs can be converted to semidefinite programs (SDPs) 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 formula_6-variate polynomial formula_7 , and suppose that we would like to minimize this polynomial over a subset formula_8.
Suppose furthermore that the constraints on the subset formula_9 can be encoded using formula_10 polynomial equalities of degree at most formula_11, each of the form formula_12 where formula_13 is a polynomial of degree at most formula_11. 
A natural, though generally non-convex program for this optimization problem is the following:
subject to:
where formula_18 is the formula_19-dimensional vector with one entry for every monomial in formula_20 of degree at most formula_21, so that for each multiset formula_22 formula_23, formula_24 is a matrix of coefficients of the polynomial formula_25 that we want to minimize, and formula_26 is a matrix of coefficients of the polynomial formula_27 encoding the formula_28th constraint on the subset formula_29. The additional, fixed constant index in our search space, formula_17, is added for the convenience of writing the polynomials formula_25 and formula_27 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 formula_33 with a positive-semidefinite matrix formula_34: we index each monomial of size at most formula_11 by a multiset formula_36 of at most formula_11 indices, formula_38. For each such monomial, we create a variable formula_39 in the program, and we arrange the variables formula_39 to form the matrix formula_41, where formula_42is the set of real matrices whose rows and columns are identified with multisets of elements from formula_43 of size at most formula_21. We then write the following semidefinite program in the variables formula_39:
subject to:
where again formula_24 is the matrix of coefficients of the polynomial formula_25 that we want to minimize, and formula_26 is the matrix of coefficients of the polynomial formula_27 encoding the formula_28th constraint on the subset formula_29.
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 formula_34 respect the symmetries present in the quadratic form formula_33.
Duality.
One can take the dual of the above semidefinite program and obtain the following program:
subject to:
We have a variable formula_62corresponding to the constraint formula_63(where formula_64is the matrix with all entries zero save for the entry indexed by formula_65), a real variable formula_66for each polynomial constraint formula_67and for each group of multisets formula_68, we have a dual variable formula_69for the symmetry constraint formula_70. The positive-semidefiniteness constraint ensures that formula_71 is a sum-of-squares of polynomials over formula_29: by a characterization of positive-semidefinite matrices, for any positive-semidefinite matrix formula_73, we can write formula_74 for vectors formula_75. Thus for any formula_76,
where we have identified the vectors formula_78 with the coefficients of a polynomial of degree at most formula_79. This gives a sum-of-squares proof that the value formula_80 over formula_29.
The above can also be extended to regions formula_82 defined by polynomial inequalities.
Sum-of-squares hierarchy.
The sum-of-squares hierarchy (SOS hierarchy), also known as the Lasserre hierarchy, is a hierarchy of convex relaxations of increasing power and increasing computational cost. For each natural number formula_83 the corresponding convex relaxation is known as the "formula_84th level" or "formula_84th round of the SOS hierarchy." The formula_86st round, when formula_87, corresponds to a basic semidefinite program, or to sum-of-squares optimization over polynomials of degree at most formula_88. To augment the basic convex program at the formula_86st level of the hierarchy to formula_84th level, additional variables and constraints are added to the program to have the program consider polynomials of degree at most formula_91.
The SOS hierarchy derives its name from the fact that the value of the objective function at the formula_84th level is bounded with a sum-of-squares proof using polynomials of degree at most formula_93 via the dual (see "Duality" above). Consequently, any sum-of-squares proof that uses polynomials of degree at most formula_93 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 formula_95 on that interval with a sum-of-squares of real polynomials of sufficiently high degree, and thus if formula_96 is the polynomial objective value as a function of the point formula_97, if the inequality formula_98 holds for all formula_97 in the region of interest, then there must be a sum-of-squares proof of this fact. Choosing formula_100 to be the minimum of the objective function over the feasible region, we have the result.
Computational cost.
When optimizing over a function in formula_101 variables, the formula_84th level of the hierarchy can be written as a semidefinite program over formula_103 variables, and can be solved in time formula_103 using the ellipsoid method.
Sum-of-squares background.
A polynomial formula_105 is a "sum of squares" ("SOS") if there exist polynomials formula_106
such that formula_107. For example,
is a sum of squares since
where 
Note that if formula_105 is a sum of squares
then formula_112 for all formula_113. Detailed descriptions of polynomial SOS are available.
Quadratic forms can be expressed as formula_114 where formula_115 is a symmetric matrix. Similarly, polynomials of degree ≤ 2"d" can be expressed as 
where the vector formula_117 contains all monomials of degree formula_118. This is known as the Gram matrix form. An important fact is that
formula_105 is SOS if and only if there exists a symmetric and positive-semidefinite matrix formula_115 such that formula_121.
This provides a connection between SOS polynomials and positive-semidefinite matrices.