Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Outer-space branch-and-bound algorithm for generalized linear multiplicative programs

Bo Zhang

TL;DR

This work tackles global optimization of the generalized linear multiplicative program ($GLMP$), a nonconvex problem arising in diverse applications. It transforms GLMP to an equivalent problem ($EP$) via a logarithmic lift and develops an outer-space simplicial branch-and-bound framework (OSSBBA) that relies on a simplex-based convex relaxation to obtain tight lower bounds while performing branching on simplices. The algorithm provides a finite termination guarantee with a derived iteration bound and demonstrates convergence and complexity analyses, complemented by numerical experiments. Across small-scale problems, OSSBBA is competitive with existing solvers, and on large-scale instances it often outperforms BARON and related BB methods, highlighting its potential for scalable global optimization of GLMP. These results suggest a practical, theoretically grounded approach for GLMPs, with room for enhancement through improved relaxations and branching strategies.

Abstract

This paper introduces a new global optimization algorithm for solving the generalized linear multiplicative problem (GLMP). The algorithm starts by introducing $\bar{p}$ new variables and applying a logarithmic transformation to convert the problem into an equivalent problem (EP). By using the strong duality of linear program, a new convex relaxation subproblem is formulated to obtain the lower bounds for the optimal value of EP. This relaxation subproblem, combined with a simplicial branching process, forms the foundation of a simplicial branch-and-bound algorithm that can globally solve the problem. The paper also includes an analysis of the theoretical convergence and computational complexity of the algorithm. Additionally, numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm in various test instances.

Outer-space branch-and-bound algorithm for generalized linear multiplicative programs

TL;DR

This work tackles global optimization of the generalized linear multiplicative program (), a nonconvex problem arising in diverse applications. It transforms GLMP to an equivalent problem () via a logarithmic lift and develops an outer-space simplicial branch-and-bound framework (OSSBBA) that relies on a simplex-based convex relaxation to obtain tight lower bounds while performing branching on simplices. The algorithm provides a finite termination guarantee with a derived iteration bound and demonstrates convergence and complexity analyses, complemented by numerical experiments. Across small-scale problems, OSSBBA is competitive with existing solvers, and on large-scale instances it often outperforms BARON and related BB methods, highlighting its potential for scalable global optimization of GLMP. These results suggest a practical, theoretically grounded approach for GLMPs, with room for enhancement through improved relaxations and branching strategies.

Abstract

This paper introduces a new global optimization algorithm for solving the generalized linear multiplicative problem (GLMP). The algorithm starts by introducing new variables and applying a logarithmic transformation to convert the problem into an equivalent problem (EP). By using the strong duality of linear program, a new convex relaxation subproblem is formulated to obtain the lower bounds for the optimal value of EP. This relaxation subproblem, combined with a simplicial branching process, forms the foundation of a simplicial branch-and-bound algorithm that can globally solve the problem. The paper also includes an analysis of the theoretical convergence and computational complexity of the algorithm. Additionally, numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm in various test instances.
Paper Structure (10 sections, 6 theorems, 46 equations, 8 tables, 1 algorithm)

This paper contains 10 sections, 6 theorems, 46 equations, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem reformulation
  3. Novel convex relaxation approach
  4. A new branch and bound algorithm
  5. Simplicial bisection method
  6. Simplicial branch-and-bound algorithm
  7. Numerical Experiments
  8. Feasibility tests
  9. Computational efficiency test
  10. Conclusions

Key Result

lemma thmcounterlemma

For any positive real number $a\in (0,+\infty)$, the following inequality holds:

Theorems & Definitions (15)

  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • theorem 4
  • proof
  • ...and 5 more