Introduction

Again let P be the polynomial optimization problem f* = inf{f(x) : x ∈ K}, whose feasible set is the basic semi-algebraic set:

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

for some polynomials f, (gj) ⊂ ℝ[x]. As already mentioned, P is in general NP-hard and one goal of this book is precisely to describe methods to obtain (or at least approximate) f* and whenever possible a global minimizer x* ∈ K.

However, in many cases one is often satisfied with a local minimum only (for instance because the methods described in Chapter 6 are computationally too expensive and cannot be implemented for the problem on hand). On the other hand a local minimum can be obtained by running some local minimization algorithm choosen among those available in the literature. Typically in suchalgorithms, at a current iterate (i.e., some feasible solution y ∈ K) one checks whether some optimality conditions (e.g. the Karush–Kuhn–Tucker (KKT) optimality conditions of Chapter 7) are satisfied within some ε-tolerance. However, those conditions are valid for any local minimum, and in fact, even for any stationary point of the Lagrangian. Moreover, in some practical situations the criterion f to minimize is subject to modeling errors or is questionable. In such a situation the practical meaning of a local (or global) minimum f* (and local (or global) minimizer) also becomes questionable. It could well be that the current solution y is in fact a global minimizer of an optimization problem P′ with the same feasible set as P but with a different criterion f. Therefore if f is “close enough” to f one might not be willing to spend an enormous amount of computing time and effort to find the global (or even local) minimum f* because one might be already satisfied with the current iterate y as a global minimizer of P′.

The moment-SOS approach described in Chapter 6 as well as its extensions in Chapter 11 aim at solving difficult nonconvex (and in general NP-hard) optimization problems. On the other hand, a large class of convex optimization problems can be solved efficiently and sometimes in time polynomial in the input size of the problem. Therefore the moment-SOS approach should have the highly desirable feature of somehow recognizing “easy” problems, for example convex problems. That is, when applied to such problems it should show some significant improvement or a particular nice behavior not necessarily valid in the general case. Otherwise

would one trust an optimization method aimed at solving difficult problems which behaves poorly on relatively “easy” problems?

Probably not! The impact of convexity on the moment-SOS approach is one of the issues investigated in this chapter. We will show that indeed the moment-SOS approach recognizes a large class of convex problems. In particular, the first semidefinite relaxation of the hierarchy (6.3) is exact for SOS-convex problems (among which are linear and convex quadratic programs), i.e., finite convergence takes place at the first step in the hierarchy. But we also provide algebraic characterization of convexity for basic semi-algebraic sets and polynomials as well as some of their specific properties.

Convexity and polynomials

We first introduce some material which shows that in the presence of convexity some of the results presented in Chapter 2 have a nice specialization.

13.1.1 Algebraic certificates of convexity for sets

We first consider the problem of detecting whether some given basic semialgebraic set K as in (5.2) is convex. By detecting, we mean that if K is convex then one can obtain a certificate (or a proof) of convexity by some algorithm.

The geometric characterization of convexity

λx + (1 − λ) y ∈ K, ∀x, y ∈ K, λ ∈ (0, 1)

is not a certificate because it cannot be checked by an algorithm. In fact, detection of convexity is a difficult problem.

Introduction

Recall from Chapter 2 that the convex cone Cd ⊂ ℝ[x]2d of nonnegative polynomials of degree at most 2d (a nonnegative polynomial has necessarily even degree) is much harder to characterize than its subcone ∑[x]d of sums of squares (SOS). Indeed, while we have seen that the latter has a simple semidefinite representation in a higher dimensional space, so far there is no such simple representation for the former. In addition, when d is fixed Blekherman (2006) has shown that after proper normalization, the “gap” between Cd and Σ[x]d increases unboundedly with the number of variables.

Similarly, for a subset K ⊂ ℝn, the (finite-dimensional) convex cone Cd (K) of polynomials of degree at most d and nonnegative on K does not have a simple and tractable representation. This is why when f ∈ ℝ[x]d the optimization problem f* = inf { f (x) : x ∈ K }, which is equivalent to f * = sup { λ : f (x) − λ ≥ 0, ∀x ∈ K }, is very difficult to solve in general even though it is also the finite-dimensional convex optimization problem sup { λ : f − λ ∈ Cd (K) }. Finite-dimensional convex optimization problems are not always tractable!

However, we next show that the results of Chapter 2 and Chapter 3 provide us with tractable inner and outer approximations of Cd (K), respectively. Those approximations will be very useful to approximate as closely as desired the global optimum f* of the optimization problem P in (1.1).

Introduction

The power of the moment-SOS approach for polynomial optimization rests upon the fact that the powerful Putinar positivity certificate (2.10)

1. • is stated in terms of SOS polynomial weights, and

2. • is linear in the polynomials that describe the feasible set K,

which allows a practical and efficient computational procedure to be defined, as described in Chapter 6. For instance, the stronger and more general Stengle positivity certificates of Theorem 2.12 do not have this property.

From an optimization (and practical) viewpoint there should be a trade off between the generality (or strength) of a positivity certificate and whether it can be implemented in some practical efficient numerical procedure. In this respect Putinar's Positivstellensatz in Theorem 2.15 achieves a good compromise since by being restricted to compact basic semi-algebraic sets K and polynomial objective functions, it still applies to a large class of important optimization problems while being implementable in an efficient practical procedure.

However, even though such polynomial optimization problems encompass many important applications of practical interest, one may wonder whether the moment-SOS approach described in Chapter 6 can be extended to a more general class of optimization problems that involve some algebra of functions larger than the polynomials.

In Chapter 10 we have already seen how to minimize a sum of rational functions on a compact basic semi-algebraic set. However, in several applications one has to deal with functions like absolute values, square roots, fractions with rational powers, etc., which appear in both the objective function to be minimized an the description of the feasible set K. Again, to extend the moment-SOS approach to this new framework one needs an analogue of Putinar's positivity certificate for such functions which also translates into a practical procedure for optimization purposes.

In this chapter we consider the algebra A which consists of functions on a basic semi-algebraic set K ⊂ ℝn, generated by basic monadic and dyadic relations on polynomials.