Convex Optimization · Problem 4.55

Duality — Boyd & Vandenberghe

Mathematics — Optimization · generalization-weighted-sum-scalarization

Problem

Generalization of weighted-sum scalarization. In \S 4.7.4 we showed how to obtain Pareto optimal solutions of a vector optimization problem by replacing the vector objective f0:Rn→Rqf_0 : \mathbf{R}^n \to \mathbf{R}^q with the scalar objective λTf0\lambda^T f_0, where λ≻K∗0\lambda \succ_{K^*} 0. Let ψ:Rq→R\psi : \mathbf{R}^q \to \mathbf{R} be a KK-increasing function, i.e., satisfying u⪯Kv,  u≠v⟹ψ(u)<ψ(v).u \preceq_K v, \; u \neq v \Longrightarrow \psi(u) < \psi(v). Show that any solution of the problem minimizeψ(f0(x))subject tofi(x)≤0,i=1,…,mhi(x)=0,i=1,…,p\begin{aligned} \text{minimize} & \quad \psi(f_0(x)) \\ \text{subject to} & \quad f_i(x) \leq 0, \quad i = 1, \ldots, m \\ & \quad h_i(x) = 0, \quad i = 1, \ldots, p \end{aligned} is Pareto optimal for the vector optimization problem minimize (w.r.t. K)f0(x)subject tofi(x)≤0,i=1,…,mhi(x)=0,i=1,…,p.\begin{aligned} \text{minimize (w.r.t. } K) & \quad f_0(x) \\ \text{subject to} & \quad f_i(x) \leq 0, \quad i = 1, \ldots, m \\ & \quad h_i(x) = 0, \quad i = 1, \ldots, p. \end{aligned} Note that ψ(u)=λTu\psi(u) = \lambda^T u, where λ≻K∗0\lambda \succ_{K^*} 0, is a special case. As a related example, show that in a multicriterion optimization problem (i.e., a vector optimization problem with f0=F:Rn→Rqf_0 = F : \mathbf{R}^n \to \mathbf{R}^q, and K=R+qK = \mathbf{R}^q_+), a unique solution of the scalar optimization problem minimizemax⁡i=1,…,qFi(x)subject tofi(x)≤0,i=1,…,mhi(x)=0,i=1,…,p,\begin{aligned} \text{minimize} & \quad \max_{i=1,\ldots,q} F_i(x) \\ \text{subject to} & \quad f_i(x) \leq 0, \quad i = 1, \ldots, m \\ & \quad h_i(x) = 0, \quad i = 1, \ldots, p, \end{aligned} is Pareto optimal.

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