Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Biobjective optimization with M-convex functions

Ellen H. Fukuda, Satoru Iwata, Itsuki Nakagawa

TL;DR

It is shown that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M$^\natural$-convex function and a linear function with binary coefficients.

Abstract

In this paper, we deal with two ingredients that, as far as we know, have not been combined until now: multiobjective optimization and discrete convex analysis. First, we show that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M$^\natural$-convex function and a linear function with binary coefficients. We also observe that a more efficient algorithm can be obtained in the special case where the M$^\natural$-convex function is M-convex. Additionally, we present a polynomial-time method for biobjective optimization problems that combine M$^\natural$-convex function minimization with lexicographic optimization.

Biobjective optimization with M-convex functions

TL;DR

It is shown that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M-convex function and a linear function with binary coefficients.

Abstract

In this paper, we deal with two ingredients that, as far as we know, have not been combined until now: multiobjective optimization and discrete convex analysis. First, we show that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M-convex function and a linear function with binary coefficients. We also observe that a more efficient algorithm can be obtained in the special case where the M-convex function is M-convex. Additionally, we present a polynomial-time method for biobjective optimization problems that combine M-convex function minimization with lexicographic optimization.

Paper Structure

This paper contains 10 sections, 8 theorems, 54 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized matroids
  4. Discrete convex functions
  5. Multiobjective optimization
  6. A specific biobjective optimization on matroids
  7. Biobjective optimization with M-natural-convex and binary linear functions
  8. Biobjective optimization with M-convex and binary linear functions
  9. Biobjective optimization with M-natural-convex and lexicographic functions
  10. Conclusions

Key Result

Lemma 3.3

For all $k$ with $\underline k \leq k \leq \bar{k}$ we have $\mathcal{T}_k \neq \varnothing$.

Figures (4)

  • Figure 1: The path through points in $\mathcal{T}_k$. Each black dot represents $(g(x_k),\langle b,x_k\rangle)$ that corresponds to $x_k \in \mathcal{T}_k$, while the white dots correspond to other feasible points. The red thick line is the path through the Pareto optimal points.
  • Figure 2: Example of a branching problem with arc and root costs. We consider $V = \{1,2,3,4,5,6\}$, and the arc costs are indicated on the corresponding arcs of the graph. The subset of vertices $S=\{2,4,6\}$ is indicated in red. We illustrate the minimal-cost branchings for each $R \subseteq S$, shown in blue. In this case, we have two Pareto optimal solutions.
  • Figure 3: Example of a bipartite matching problem with edge and vertex costs. We consider $S = \{1,2,3\}$, $T=\{4,5\}$, and the edge costs are indicated next to the corresponding edges. In this case, we have four possible $T$-perfect matchings, shown as thick red lines, and two of them are Pareto optimal. Considering the improved algorithm, note that given a perfect matching that minimizes $c(M)$ -- in this case, the leftmost matching above -- the shortest path (visiting vertices 1, 4, and 2 in this order) and the corresponding update of the matching yield the rightmost matching, which indeed has $b(R)$ decreased by one.
  • Figure 4: Lexicographic cone in $\mathbb{R}^2$

Theorems & Definitions (17)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3: Tak23
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Theorem 3.7
  • ...and 7 more