PROCEDIMIENTO PARA LOS INGRESOS Y EGRESOS DE EFECTIVO EN EL ÁREA DE TESORERÍA.

Chapter 2: Mathematical preliminaries

In Chapter 2, we define notation and basic mathematical tools used throughout the thesis. Specifically, we provide background material for linear algebra, convex and nonconvex optimality analysis, tools for first-order optimisation methods, and finally discrete gradient methods and geometric numerical integration.

Chapter 3: The foundations of discrete gradient methods for smooth

optimisation

After the introduction, we present a new existence result for the discrete gradient equation, based on the Brouwer fixed point theorem in Section 3.3. In Section 3.4 we study fixed point iterative methods for solving the discrete gradient equation, including a relaxed fixed point method with improved efficiency, while in Section 3.5, we study the dependence of the update xk+1←_{[ x}k _{on the time step τ}

k> 0 for the mean value discrete gradient and the

Itoh–Abe methods. In Sections 3.6 and 3.7, we prove convergence rates for the discrete gradient methods, and convergence guarantees for functions that satisfy the strong Kurdyka– Łojasiewicz inequality, respectively. Before a brief discussion of preconditioned methods in Section 3.8, we present numerical results in Section 3.9.

Chapter 4: Discrete gradient methods for nonsmooth, nonconvex opti-

misation

In Section 4.1, we discuss the background for nonsmooth, nonconvex optimisation, and the purpose of the chapter. In Section 4.2 we provide the main theoretical results of the chapter, namely that the generalised Itoh–Abe methods converge to a connected set of Clarke stationary points. In Section 4.4 we propose an algorithm for solving black-box optimisation problems, and in Section 4.5 we provide numerical examples.

Chapter 5: Discrete gradient methods for nonsmooth, nonconvex, con-

strained optimisation

In Section 5.1 we propose a modification of the Itoh–Abe methods for constrained problems and discuss related works, while in Section 5.2 we provide preliminary results on epi- Lipschitzian sets and Clarke subdifferential analysis in the constrained setting. In Section 5.3

1.3 Outline of chapters 15

we study the proposed optimisation algorithm and prove that the methods converge to a connected set of Clarke stationary points in the constrained setting as well. In Section 5.4 we present numerical results.

Chapter 6: Bregman discrete gradient methods for sparse optimisation

After introducing the inverse scale space flow in Section 6.1, we propose to solve it using a Bregman discrete gradient method based on the ISS flow in Section 6.2. In Section 6.3 we prove well-posedness and convergence results for this method in a nonconvex, nons- mooth framework. Furthermore, in Sections 6.4 and 6.5, we discuss particular examples of Bregman discrete gradient methods and prove equivalencies between methods derived from different numerical integration schemes, respectively, before providing numerical results in Section 6.6.

Chapter 7: Differentiation for nonsmooth bilevel optimisation

In Section 7.1 we discuss bilevel optimisation of nonsmooth variational problems and motivations for studying the differential properties of the solution mapping. In Section 7.2, we review examples of nonsmooth bilevel problems and existing approaches for solving them in literature. In Section 7.3 we provide preliminary concepts for the subdifferential analysis, and prove for a sufficiently general class of variational problems that they are subdifferentially regular. In Section 7.4 we define partly smooth functions, and show under reasonable assumptions that the solution mapping is piecewise differentiable. In Section 7.5 we study algorithmic differentiation of various first-order methods for solving nonsmooth variational methods, and prove convergence guarantees to the implicit derivative. In Section 7.6 we present some numerical results, and in Section 7.7 we conclude.

Chapter 8: Conclusion & outlook

In Sections 8.1 and 8.2 we summarise and discuss the results of this thesis. In Section 8.3 we discuss future directions of research building on the work in this thesis. In particular, in Section 8.3.1, we consider solving the Wasserstein gradient flow with discrete gradients. In Section 8.3.2, we propose the use of mean value discrete gradient methods for nonsmooth objective functions, under assumptions of partial smoothness. In Section 8.3.3, we discuss future work for gradient-based approaches to bilevel problems, considering algorithmic differentiation of primal-dual methods, and studying stability of algorithmic differentiation methods when the number of iterations is determined by a stopping rule.

Chapter 2

Mathematical preliminaries

In this section, we provide mathematical preliminaries which will be used throughout the thesis. We first consider basic properties of differentiable functions, followed by theory of the class of convex, proper, lower semicontinuous functions. Next we provide an overview of nonconvex generalised differential theory, which in comparison to its convex counterpart is rather less unified. We conclude with an overview of geometric numerical integration and in particular discrete gradients.

2.1

Basic notation and conventions

In a Euclidean space setting, we denote by ∥ · ∥ and ⟨·, ·⟩ the norm and associated inner product. For x ∈ Rnand p > 0, the ℓp-norm ∥ · ∥pis defined as

∥x∥p:= p

q

x₁p+ x₂p+ . . . + xnp,

while ∥x∥∞:= maxi=1,...,n|xi|, and ∥x∥0:= | supp(x)|.

We denote by (ei)n_{i=1the standard coordinate vectors in R}n. We denote by [x, y] the line segment between two points x, y ∈ Rni.e.

[x, y] :=λ x + (1 − λ )y : λ ∈ [0, 1] .

For ε > 0, x ∈ Rn, we denote by Bε(x) the open ball of radius ε at x, {y ∈ Rn : ∥x − y∥ < ε},

and by Bε(x) the closed ball of radius ε at x, {y ∈ Rn : ∥x − y∥ ≤ ε}. We denote by Sn−1

We summarise big and small o-notation. For two functions f : Rn→ Rm_{and g : R}n_{→ R}l_,

if ∥ f (x)∥/∥g(x)∥ → 0 as ∥x∥ → 0, then f (x) = o(g(x)). If there is ε > 0 and C > 0 such that ∥x∥ < ε implies ∥ f (x)∥ ≤ C∥g(x)∥, then f (x) = O(g(x)).

Similarly, suppose (xn)_{k∈N⊂ R}nand (yn)_{k∈N⊂ R}mare two sequences. If ∥xk∥/∥yk_{∥ → 0}

as k → ∞, then xk= o(yk_{), while if there is C > 0 and K ∈ N such that ∥x}k∥ ≤ C∥yk_{∥ for all}

k≥ K, then xk_{= O(y}k_).