5 - The primal and dual points of view

Summary

Consider the following polynomial optimization problem:

f* ≔ inf { f (x) : x ∈ K }

with feasible set K ⊂ ℝⁿ defined by

K = { x ∈ ℝⁿ : g_j (x) ≥ 0, j = 1, …, m },

and where f, g_j ∈ ℝ[x] are real-valued polynomials, j = 1, …, m.

In the real algebraic geometry terminology, such a set K defined by finitely many polynomial inequalities is called a basic closed semi-algebraic set.

Whenever K ≠ ℝⁿ and unless otherwise stated, we will assume that the set K is compact but we do not assume that K is convex or even connected. This is a rather rich modeling framework that includes linear, quadratic, 0/1, and mixed 0/1 optimization problems as special cases. In particular, constraints of the type x_i ∈ { 0, 1 } can be written as x²_i − x_i ≥ 0 and x_i − x²_i ≥ 0, or as the single equality constraint x²_i − x_i = 0.

When n = 1, we have seen in Chapter 2 that a univariate polynomial nonnegative on K = ℝ is a sum of squares and that a univariate polynomial nonnegative on an interval K = (−∞,b], K = [a, b] or K = [a, ∞), can be written in a specific form involving sums of squares whose degree is known. We will see that this naturally leads to reformulating problem (5.1) as a single semidefinite optimization problem for which efficient algorithms and software packages are available. Interestingly, this nonconvex problem can be reformulated as a tractable convex problem and underscores the importance of the representation theorems from Chapter 2.

On the other hand, the multivariate case differs radically from the univariate case because not every polynomial nonnegative on K = ℝⁿ can be written as a sum of squares of polynomials.