Coercive functions from a topological viewpoint and properties of minimizing sets of convex functions appearing in image restoration

Many tasks in image processing can be tackled by modeling an appropriate data fidelity term $Φ: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$ and then solve one of the regularized minimization problems \begin{align*} &{}(P_{1,τ}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ Φ(x) \;{\rm s.t.}\; Ψ(x) \leq τ\big\} \\ &{}(P_{2,λ}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ Φ(x) + λΨ(x) \}, \; λ> 0 \end{align*} with some function $Ψ: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$ and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets ${\rm SOL}(P_{1,τ})$ and ${\rm SOL}(P_{2,λ})$ of the minimization problems.} \\ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals $(0,c)$ and $(0,d)$ such that the setvalued curves \begin{align*} τ\mapsto {}& {\rm SOL}(P_{1,τ}), \; τ\in (0,c) \\ {} λ\mapsto {}& {\rm SOL}(P_{2,λ}), \; λ\in (0,d) \end{align*} are the same, besides an order reversing parameter change $g: (0,c) \rightarrow (0,d)$. Moreover we show that the solver sets are changing all the time while $τ$ runs from $0$ to $c$ and $λ$ runs from $d$ to $0$. In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity. Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.