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Multi-Objective Evolutionary Optimization of Chance-Constrained Multiple-Choice Knapsack Problems with Implicit Probability Distributions

Xuanfeng Li, Shengcai Liu, Wenjie Chen, Yew-Soon Ong, Ke Tang

TL;DR

NHILS, a hybrid evolutionary algorithm that integrates specialized initialization and local search into NSGA-II to navigate sparse feasible regions, is developed and demonstrated that NHILS consistently outperforms several state-of-the-art multi-objective optimizers in convergence, diversity, and feasibility.

Abstract

The multiple-choice knapsack problem (MCKP) is a classic combinatorial optimization with wide practical applications. This paper investigates a significant yet underexplored extension of MCKP: the multi-objective chance-constrained MCKP (MO-CCMCKP) under implicit probability distributions. The goal of the problem is to simultaneously minimize the total cost and maximize the confidence level of satisfying the capacity constraint, capturing essential trade-offs in domains like 5G network configuration. To address the computational challenge of evaluating chance constraints under implicit distributions, we first propose an order-preserving efficient resource allocation Monte Carlo (OPERA-MC) method. This approach adaptively allocates sampling resources to preserve dominance relationships while reducing evaluation time significantly. Further, we develop NHILS, a hybrid evolutionary algorithm that integrates specialized initialization and local search into NSGA-II to navigate sparse feasible regions. Experiments on synthetic benchmarks and real-world 5G network configuration benchmarks demonstrate that NHILS consistently outperforms several state-of-the-art multi-objective optimizers in convergence, diversity, and feasibility. The benchmark instances and source code will be made publicly available to facilitate research in this area.

Multi-Objective Evolutionary Optimization of Chance-Constrained Multiple-Choice Knapsack Problems with Implicit Probability Distributions

TL;DR

NHILS, a hybrid evolutionary algorithm that integrates specialized initialization and local search into NSGA-II to navigate sparse feasible regions, is developed and demonstrated that NHILS consistently outperforms several state-of-the-art multi-objective optimizers in convergence, diversity, and feasibility.

Abstract

The multiple-choice knapsack problem (MCKP) is a classic combinatorial optimization with wide practical applications. This paper investigates a significant yet underexplored extension of MCKP: the multi-objective chance-constrained MCKP (MO-CCMCKP) under implicit probability distributions. The goal of the problem is to simultaneously minimize the total cost and maximize the confidence level of satisfying the capacity constraint, capturing essential trade-offs in domains like 5G network configuration. To address the computational challenge of evaluating chance constraints under implicit distributions, we first propose an order-preserving efficient resource allocation Monte Carlo (OPERA-MC) method. This approach adaptively allocates sampling resources to preserve dominance relationships while reducing evaluation time significantly. Further, we develop NHILS, a hybrid evolutionary algorithm that integrates specialized initialization and local search into NSGA-II to navigate sparse feasible regions. Experiments on synthetic benchmarks and real-world 5G network configuration benchmarks demonstrate that NHILS consistently outperforms several state-of-the-art multi-objective optimizers in convergence, diversity, and feasibility. The benchmark instances and source code will be made publicly available to facilitate research in this area.
Paper Structure (39 sections, 1 theorem, 17 equations, 7 figures, 8 tables, 2 algorithms)

This paper contains 39 sections, 1 theorem, 17 equations, 7 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. The Deterministic MCKP
  4. Chance-Constrained Optimization Paradigms
  5. Multi-Objective Chance-Constrained Knapsack Problems
  6. MOEAs and the Sparsity-of-Feasibility Challenge
  7. Problem Formulation
  8. Mathematical Model
  9. Solution Representation and Fitness Evaluation
  10. Solution Encoding
  11. Fitness Evaluation
  12. OPERA-MC: Efficient Evaluation of Implicit Chance Constraints
  13. Motivation and Problem Setting
  14. The OPERA-MC Algorithm
  15. Theoretical Analysis
  16. ...and 24 more sections

Key Result

Theorem 4.1

Let $\mathbf{x}_A$ and $\mathbf{x}_B$ be two solutions with true confidence levels $p_A > p_B$. If $\mathbf{x}_A$ receives $N_A$ samples and $\mathbf{x}_B$ receives $N_B \leq N_A$ samples under OPERA-MC, the probability of erroneously concluding that $\hat{p}_B \geq \hat{p}_A$ satisfies:

Figures (7)

  • Figure 1: Technical architectures and requirement spectrum of heterogeneous 5G network service scenarios: Remote Surgery (mission-critical); Cloud Gaming (balanced); and Video Streaming (cost-sensitive).
  • Figure 2: Trade-off and solution distribution heatmap between cost and CL.
  • Figure 3: The block diagram of NHILS
  • Figure 4: Probability density function. The horizontal axis represents the time delay and the vertical axis represents the probability density.
  • Figure 5: Pareto front comparisons of algorithms on LAB instances. From left to right and top to bottom: LAB-ls1 to LAB-ls6.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 4.1: Pair-wise Order-Preservation Error Bound