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Cardinality-Constrained Multi-Objective Optimization: Novel Optimality Conditions and Algorithms

Matteo Lapucci, Pierluigi Mansueto

TL;DR

The concept of L-stationarity is defined and its relationships with other existing conditions and Pareto optimality concepts are analyzed and two novel algorithmic approaches are proposed for solving multi-objective optimization problems with a sparsity constraint on the vector of variables.

Abstract

In this paper, we consider multi-objective optimization problems with a sparsity constraint on the vector of variables. For this class of problems, inspired by the homonymous necessary optimality condition for sparse single-objective optimization, we define the concept of L-stationarity and we analyze its relationships with other existing conditions and Pareto optimality concepts. We then propose two novel algorithmic approaches: the first one is an Iterative Hard Thresholding method aiming to find a single L-stationary solution, while the second one is a two-stage algorithm designed to construct an approximation of the whole Pareto front. Both methods are characterized by theoretical properties of convergence to points satisfying necessary conditions for Pareto optimality. Moreover, we report numerical results establishing the practical effectiveness of the proposed methodologies.

Cardinality-Constrained Multi-Objective Optimization: Novel Optimality Conditions and Algorithms

TL;DR

The concept of L-stationarity is defined and its relationships with other existing conditions and Pareto optimality concepts are analyzed and two novel algorithmic approaches are proposed for solving multi-objective optimization problems with a sparsity constraint on the vector of variables.

Abstract

In this paper, we consider multi-objective optimization problems with a sparsity constraint on the vector of variables. For this class of problems, inspired by the homonymous necessary optimality condition for sparse single-objective optimization, we define the concept of L-stationarity and we analyze its relationships with other existing conditions and Pareto optimality concepts. We then propose two novel algorithmic approaches: the first one is an Iterative Hard Thresholding method aiming to find a single L-stationary solution, while the second one is a two-stage algorithm designed to construct an approximation of the whole Pareto front. Both methods are characterized by theoretical properties of convergence to points satisfying necessary conditions for Pareto optimality. Moreover, we report numerical results establishing the practical effectiveness of the proposed methodologies.
Paper Structure (19 sections, 13 theorems, 29 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 19 sections, 13 theorems, 29 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Proximal Operator in Multi-Objective Optimization
  4. Optimality Conditions
  5. New Algorithmic Approaches for Sparse MOO Problems
  6. Multi-Objective Iterative Hard Thresholding
  7. Convergence Analysis
  8. Sparse Front Steepest Descent
  9. Phase One: Initialization
  10. Phase Two: Front steepest descent
  11. Algorithm Theoretical Analysis
  12. Computational Experiments
  13. Experiments Setup
  14. Quadratic Problems
  15. Preliminary Assessment of MOIHT, MOSPD and MOHyb
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\bar{x} \in \Omega$ be locally weakly Pareto optimal for problem eq::mo-prob. Then, $\bar{x}$ is Pareto-stationary for eq::mo-prob.

Figures (7)

  • Figure 1: Pareto optimal solutions and Pareto front of problem of Example \ref{['ex::L-stat']}.
  • Figure 2: $L$-stationary points in the Pareto front of problem of Example \ref{['ex::L-stat']} ($L(F) = [1, 1]^\top$) for different values of $L$.
  • Figure 3: Results achieved by MOIHT, MOSPD and MOHyb with $\tau_0 \in \{1, 100\}$, starting at 25 random initial solutions, on a selection of quadratic problems. The filled markers denote $L$-stationary solutions ($L = 1.1\kappa$). The small black dots form the reference front.
  • Figure 4: Results of SFSD phase two compared to simple MOSD refinement of solutions retrieved in phase one. We show one example instance for each considered multi-start/phase one strategy.
  • Figure 5: Performance profiles for SFSD with different initialization strategies, i.e., MOIHT, MOSPD, MOHyb (best executions w.r.t. purity) and MIQP on the quadratic problems.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3: lapucci22
  • Lemma 1: lapucci22
  • Lemma 2
  • proof
  • Definition 4: lapucci22
  • Lemma 3: lapucci22
  • Lemma 4: lapucci22
  • Lemma 5
  • ...and 24 more