Constrained Global Optimisation

Many real-world problems have a natural mathematical formulation where their solution is defined as the minimisation or maximisation of a particular function, known as theobjective function, usually subject to a number of constraints which can be specified by inequalities and equalities involving constraint functions. Physical problems include manufacturing

6 Problems Involving Polynomial Inequalities

optimisation, logistics optimisation (e.g. the travelling salesman problem), chemical engi-neering, the engineering verification of mechanical structures, and many more. Numerous abstract mathematical problems, such as curve-fitting or the Kepler conjecture [Hal05], can be formulated as optimisation problems. Some further examples are given in [Zum08].

A constrained global optimisation problem is canonically presented as

minx∈Ff(x), (6.6)

where the set offeasible solutionsF is defined by F := on S. The function f is referred to as the objective function, the g_i define inequality constraints, and the h_j equality constraints. We have to determine the global minimum over the entire feasible set, not just a local minimum, as is the case for a local optimisation problem.

A frequently used approach is the generation ofrelaxationsand their use in a branch and bound framework. Generally speaking, a relaxation of the given problem has the properties that

1. each feasible point of the given problem is feasible for the relaxation, 2. the relaxation is easier to solve than the original problem, and

3. the solutions of the relaxation converge to the solutions of the original problem, pro-vided the maximal width of the set of feasible points converges to zero.

We obtain a relaxation for the global optimisation problem (6.6) by proceeding as follows:

1. Replace the objective functionf and the functionsg_iby valid lower bounding functions f and g_i, respectively.

2. If the function hj defining the jth equality constraint is affine, then this equation is added to the constraints that define the relaxation. The remaining equations are rewritten as a pair of inequalities and are treated in the same way as above.

The following optimisation problem is then obtained:

x∈Fmin^relf(x) (6.7)

with the setF^rel of feasible solutions given by

g(x)≤0, i= 1, . . . , t⁰, x∈X,

6 Problems Involving Polynomial Inequalities wheret⁰ ≥tand s⁰ ≤s.

The optimisation problem (6.7) is a valid relaxation of (6.6) if its set of feasible solutions F^relsatisfiesF ⊆F^relandf(x)≤f(x)for allx∈F. The relaxed subproblem with its set of feasible solutions constitutes a simpler type of problem (for example, a linear programming problem) whose solution provides a lower bound for the solution of (6.6).

These relaxations are commonly used for solving constrained global optimisation prob-lems, e.g. [AMF95, AF96]. The computation of a good quality convex lower bounding function for a given function is thus of significant importance when a branch-and-bound approach is employed, such as in theCOCONUTsoftware package [BSV⁺01, Sch03]. Tight bounding functions and tight bounds for the ranges of the objective and constraint func-tions over sub-boxes are crucial for the efficient resolution of the associated subproblems.

Convex envelopes, the uniformly tightest underestimating convex functions, exhibit good performance, cf. [AF96, Flo00, TS02].

Because of their simplicity and ease of computation, constant and affine lower bounding functions are especially useful. Constant bound functions are often used when interval computation techniques are applied to global optimisation, cf. [RR88, Kea96b, Han03].

However, when using constant bound functions, all information about the shape of the given function is lost. A compromise between convex envelopes, which in general require significant computational effort, and constant lower bounding functions are affine lower bounding functions.

To generate an affine relaxation for problem (6.6), the functionsf, g_i (i= 1, . . . , t), and hj (j= 1, . . . , s) are replaced by affine lower bounding functionsf,g_i, andh_j, respectively.

Then the relaxed problem (6.7) with the respective set of feasible solutions yields a linear programming problem whose solution provides a lower bound for the solution of (6.6). Affine lower bounding functions are simpler to compute and work with than convex envelopes, preserving basic shape information and yielding linear programming subproblems that are relatively fast to resolve. A sequence of diverse methods for computing such affine bounding functions for polynomials based upon the Bernstein expansion is proposed in Chapter 10.

Apart from relaxations, many branch-and-bound based methods require bounds on the gradients and Hessians of constraint functions, which are preferably computed rigorously.

For example, theαBB algorithm, cf. [AMF95, Flo00], relies heavily on such bounds. Rigor-ous bounds for the partial derivatives of polynomials can also be provided by the Bernstein expansion, see Subsection 3.3.4.

Part II: Contributions

7 Computation of Topological Degree

This chapter consists of a detailed study of the recursive algorithm for the computation of topological degree, an outline of which was proposed by O. Aberth [Abe94]. Topological degree formed the subject of Chapter 4 and this algorithm was introduced in Subsection 4.4, with a simple example. The behavioural attributes in practice (especially the complexity) of Aberth’s method are unknown and remain to be investigated. Here we undertake an in-depth analysis of the algorithm and obtain an estimate of its complexity with the use of a geometric model coupled with a probabilistic analysis. The method is to be implemented and tested with a catalogue of examples, and the data is to be compared with the complexity study. Improvements to the algorithm, in particular the face subdivision strategy, will be considered and proposed. The implemented software is described in Appendix A.

To briefly recall from Section 4.4.2, we are given a boxXinRⁿand a continuous function F :X→Rⁿ given by component real-valued functions

f_i(x₁, . . . , x_n), i= 1, . . . n, (7.1) for which we assume efficient interval arithmetic implementations are available. Since inter-val arithmetic is a core low-level component of this method, we do not consider cases where the component functions f_i may have non-standard interval implementations, cf. Subsec-tion 2.1.7. It is desired to compute the Brouwer degree (see SecSubsec-tion 4.2) ofF over Xat0, deg(F,X,0), where the degree is calculated at 0without loss of generality.