MATRIZ DE ANÁLISIS (PRIMER VACIADO DE INFORMACIÓN)

In the recent years, the need to deal with uncertain data has become crucial, especially when real life applications are involved. In these circumstances, robust optimisation aims to recover an optimal solution whose feasibility must be guaranteed for any realisation of the uncertain data [20]. Robust optimisation, for which the data are not specified exactly, explicitly incorporates uncertainty to protect the decision-maker against parameter ambiguity and stochastic uncertainties [65].

We have seen that convex optimisation offers the possibility of getting around the problems that linear and non-linear approaches have in terms of convergence. We have also seen that the cost function in convex optimisation is geometrically meaningful, and has a single global minimum [25, 94]. In this thesis, most problems are formulated in terms of parameter estimation from image-based measurements. Since these measurements are subject to deterministic perturbations, more studies have started to focus on how this parameter estimation might be improved if uncertainties in these data are integrated [26].

Therefore, in this thesis more interest is given to robust optimisation and robust convex optimisation in particular. Throughout this thesis we will be dealing with uncertain data resulting from inexact measurements. Inaccuracy of the devices could be considered as an important source of uncertainties as well. Furthermore, data uncertainty results in uncertain constraints and an uncertain objective function as well.

Robust optimisation is a recent approach to optimisation under uncertain data, where the uncertainty model is not stochastic, but rather deterministic. Even though it is still considered as a new approach to optimisation problems under uncertainty, an increasing number of real applications have already proved its efficiency. Robust optimisation is mainly designed to allow uncertainty-affected optimisation problems to provide guarantees about the performance of the solution [20, 65]. In other words, in this optimisation, instead of recovering the solution in some probabilistic sense under stochastic uncertainty, the optimiser builds a solution that is optimal for any realisation of the uncertainty in a given set [18]. In cases where the optimality of a solution is affected by the uncertainty, the robust optimisation main goal will be then to seek a solution that performs well enough for any value taken by the unknown coefficients. Many studies are conducted toward different robustness methods, however, the most known one is to optimise the worst-case objective function [65].

Robust optimisation, in general form, deals with two sets of entities, decision and uncertain variables. The first aim of robust optimisation is to recover the optimal

3.8. Robust convex optimisation 73

solution on the decision variables such that the worst-case is minimised and the constraints are robustly feasible, while the uncertainty is allowed to take arbitrary values in a defined uncertainty set [116]. The optimal solution is evaluated using the realisation of the uncertainty that is most unfavourable [65].

The general form of this robust optimisation is given by: min

x maxω f0(x, ω)

subject to fi(x, ωi) ≤ 0 ; ∀ωi ∈ W, i = 1, . . . , m

(3.44)

where x ∈ Rn _{are the decision variable, f}

0 and fi are objective function and the

inequality constraint functions, ωi ∈ Rk, are the uncertain variables and W ⊆ Rk

are the uncertainty sets. If the objective function and the inequality constraints are convex, the problem (3.44) is considered as a robust convex optimisation problem. The aim of problem (3.44) is to recover the optimal solution, x∗, of the cost function among all feasible solutions, allowing the uncertain variables ωi to take any realisation

within Wi.

3.8.1

Robust solution using SOCP

In this thesis we focus on the SOCP, as most problems are solved using this approach. In our work, uncertainties, and their propagation through all multiple-view geometry algorithms, are estimated. Therefore, uncertainties are included in our optimisation problems. Let us consider the following SOCP problem:

min

x f

⊤

x

subject to ∥Aix + bi∥ ≤ c⊤i x + di, for i = 1, . . . , m

g_i⊤x = hi, for i = 1, . . . , p

(3.45)

Due to these uncertainties, the problem parameters Ai, bi, ci and di are allowed

to accept any values within the set. In these situations, we refer to checking the feasibility of the SOCP for all realisations of the uncertain parameters as a robust feasibility problem. The problem of finding the set of robust feasible solutions is called the robust counterpart. Thus, problem (3.45) will be written as:

min

x f

⊤

x

subject to ∥Aix + bi∥ ≤ c⊤i x + di; ∀(Ai, bi, ci, di) ∈ Wi

g⊤_i x = hi, for i = 1, . . . , p

(3.46)

74 Chapter

3.

3.8.2

Robust least-squares via SOCP

First let us remind the reader on least squares (LS) minimisation. Consider the LS problem given in (3.3): min x fo  x=∥ Ax − b ∥2₂=X i  a⊤_i x − bi 2 (3.47)

The objective function here is the sum of squares (a⊤_i x − bi). Note that the LS

problem has no constraints. The objective function, in fact, is a (quadratic) convex function since f0(x) may be written as x⊤A⊤Ax − 2b⊤Ax + b⊤b. Thus, a set of linear

equations can be used to solve the least-squares problem (3.47):

(A⊤A)x = A⊤b (3.48) Thus,

Ax = b (3.49)

where A = (A⊤A) and b = A⊤b. This is similar to finding a solution to an

overdetermined set of equations Ax ∼= b. Figure 3.9a (page 63) shows a plot of the objective function. Obviously, the problem has a single minimum. Similarly to other problems, this optimal solution xopt is obtained by putting its gradient ▽f0(x) =

2x⊤A⊤A − 2b⊤A = 0. Therefore, the analytical solution xopt = (A⊤A)−1A⊤b = A‡b,

where A‡= (A⊤A)−1A⊤ is the pseudo-inverse of A.

Now, in the scenario where the parameters of the equation (3.49) are subject to unknown but bounded uncertainties ∆A and ∆b, where ∥∆A∥ ≤ ρ and ∥∆b∥ ≤ ξ, then our new Least-Squares problem has the form:

min

x ∥∆A∥≤ρ,∥∆b∥≤ξmax ∥(A + ∆A)x − (b + ∆b)∥ (3.50)

This problem is refereed as the Robust Least-Squares and introduced by El Ghaoui and Lebret in [52], Miguel Soma Lobo in [135], Chandrasekaran in [32, 33] and Sayad et al. in [5]. The problem (3.50) may be formulated in closed form as:

max

∥∆A∥≤ρ,∥∆b∥≤ξ∥(A + ∆A)x − (b + ∆b)∥

= max

∥∆A∥≤ρ,∥∆b∥≤ξ ∥y∥≤1max y ⊤ (Ax − b) + y⊤∆Ax − y⊤∆b = max ∥z∥≤1∥y∥≤1maxy ⊤ (Ax − b) + z⊤x + ξ = ∥Ax − b∥ + ρ∥x∥ + ξ (3.51)