Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scalarization via utility functions in multi-objective optimization

Lorenzo Lampariello, Simone Sagratella, Valerio Giuseppe Sasso, Vladimir Shikhman

Abstract

We study a general scalarization approach via utility functions in multi-objective optimization. It consists of maximizing utility which is obtained from the objectives' bargaining with regard to a disagreement reference point. The theoretical framework for a broad class of utility functions from microeconomics is developed. For that, we associate a utility-dependent single-objective optimization problem with the given multi-objective optimization problem. We show that Pareto optimal points of the latter can be recovered by solving the former. In particular, Cobb-Douglas, Leontief, and CES utility functions are considered. We prove that any Pareto optimal point can be obtained as a solution of scalarization via one of the mentioned utility functions. Further, we propose a numerical scheme to solve utility-dependent single-objective optimization problems. Here, the main difficulty comes from the necessity to address constraints which are associated with a disagreement reference point. Our crucial observation is that the explicit treatment of these additional constraints may be avoided. This is the case if the Slater condition is satisfied and the utility function under consideration has the so-called barrier property. Under these assumptions, we prove the convergence of our scheme to Pareto optimal points. Numerical experiments on real-world financial datasets in a portfolio selection context confirm the efficiency of our scalarization approach via utility functions.

Scalarization via utility functions in multi-objective optimization

Abstract

We study a general scalarization approach via utility functions in multi-objective optimization. It consists of maximizing utility which is obtained from the objectives' bargaining with regard to a disagreement reference point. The theoretical framework for a broad class of utility functions from microeconomics is developed. For that, we associate a utility-dependent single-objective optimization problem with the given multi-objective optimization problem. We show that Pareto optimal points of the latter can be recovered by solving the former. In particular, Cobb-Douglas, Leontief, and CES utility functions are considered. We prove that any Pareto optimal point can be obtained as a solution of scalarization via one of the mentioned utility functions. Further, we propose a numerical scheme to solve utility-dependent single-objective optimization problems. Here, the main difficulty comes from the necessity to address constraints which are associated with a disagreement reference point. Our crucial observation is that the explicit treatment of these additional constraints may be avoided. This is the case if the Slater condition is satisfied and the utility function under consideration has the so-called barrier property. Under these assumptions, we prove the convergence of our scheme to Pareto optimal points. Numerical experiments on real-world financial datasets in a portfolio selection context confirm the efficiency of our scalarization approach via utility functions.
Paper Structure (8 sections, 34 theorems, 133 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 8 sections, 34 theorems, 133 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Scalarization via utility functions
  3. Examples of utility functions
  4. Cobb-Douglas utility function
  5. Leontief utility function
  6. CES utility function
  7. Algorithmic developments
  8. Portfolio-selection and numerical results

Key Result

Proposition 2.4

If the Slater condition does not hold for (eq:ineq), then all its feasible points are weak Pareto optimal.

Figures (2)

  • Figure 1: Efficient frontier (in blue), reference level $a$ (cyan), starting point $x^0$ (green), optimal points for $u_{\text{CD}}$ (violet) and for $u_{\text{CES}}$ (red) in the objective space for the DIJA dataset
  • Figure 2: Efficient frontier (in blue), reference level $a$ (cyan), starting point $x^0$ (green), optimal points for $u_{\text{CD}}$ (violet) and for $u_{\text{CES}}$ (red) in the objective space for the NDX dataset

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 31 more