Convex Optimization · Problem 4.51

Duality — Boyd & Vandenberghe

Mathematics — Optimization · monotone-transformation-vector-optimization

Problem

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 \to \mathbf{R}^q

satisfies

u \preceq_K v, \; u \neq v \Longrightarrow \phi(u) \preceq_K \phi(v), \; \phi(u) \neq \phi(v).

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.

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