4.51 Monotone transformation of objective in vector optimization. Consider the vector optimization problem (4.56). Suppose we form a new vector optimization problem by replacing the objective $f_0$ with $\phi \circ f_0$, where $\phi: \mathbf{R}^q \rightarrow \mathbf{R}^q$ satisfies
$$
u \preceq K v, u \neq v \Longrightarrow \phi(u) \preceq_K \phi(v), \phi(u) \neq \phi(v) \text {. }
$$
Show that a point $x$ is Pareto optimal (or optimal) for one problem if and only if it is Pareto optimal (optimal) for the other, so the two problems are equivalent. In particular, composing each objective in a multicriterion problem with an increasing function does not affect the Pareto optimal points.

4.52 Pareto optimal points and the boundary of the set of achievable values. Consider a vector optimization problem with cone $K$. Let $\mathcal{P}$ denote the set of Pareto optimal values, and let $\mathcal{O}$ denote the set of achievable objective values. Show that $\mathcal{P} \subseteq \mathcal{O} \cap \mathbf{b d} \mathcal{O}$, i.e., every Pareto optimal value is an achievable objective value that lies in the boundary of the set of achievable objective values.

4.53 Suppose the vector optimization problem (4.56) is convex. Show that the set
$$
\mathcal{A}=\mathcal{O}+K=\left{t \in \mathbf{R}^q \mid f_0(x) \preceq_K t \text { for some feasible } x\right},
$$
is convex. Also show that the minimal elements of $\mathcal{A}$ are the same as the minimal points of $\mathcal{O}$.

4.54 Scalarization and optimal points. Suppose a (not necessarily convex) vector optimization problem has an optimal point $x^$. Show that $x^$ is a solution of the associated scalarized problem for any choice of $\lambda \succ K * 0$. Also show the converse: If a point $x$ is a solution of the scalarized problem for any choice of $\lambda \succ \kappa * 0$, then it is an optimal point for the (not necessarily convex) vector optimization problem.