A Criterion Space Method for Biobjective Mixed Integer Programming: The Boxed Line Method

Perini, Tyler; Boland, Natashia; Pecin, Diego; Savelsbergh, Martin

doi:10.1287/ijoc.2019.0887

Cited by 19 publications

(8 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Earliest branchand-bound methods for multiobjective pure integer programs can be found in [KH82;KY83], but since then more sophisticated algorithms have been developed [BSW12; JLS12; Vin+13; SAD14; GNE19; PT19]. Objective space search methods have been the focus of study in the last few years separately for pure integer programs [KS14; BCS15a; DK15; BCS16a; BCS16b; TDM19] and mixed-integer programs [BCS15b;Per+20].…”

Section: Background On Existing Methodsmentioning

confidence: 99%

Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

Adelgren

Gupte

2022

INFORMS Journal on Computing

View full text Add to dashboard Cite

We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, checking node fathoming, presolve, and duality gap measurement. Our branch-and-bound is predominantly a decision space search method because the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and on a new set of instances that we generate using a benchmark library (MIPLIB2017) for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm. Summary of Contribution: Biobjective mixed-integer optimization problems have two linear objectives and a mixed-integer feasible region. Such problems have many applications in operations research, because many real-world optimization problems naturally comprise two conflicting objectives to optimize or can be approximated in such a manner and are even harder than single objective mixed-integer programs. Solving them exactly requires the computation of all the nondominated solutions in the objective space, whereas some applications may also require finding at least one solution in the decision space corresponding to each nondominated solution. This paper provides an exact algorithm for solving these problems using the branch-and-bound method, which works predominantly in the decision space. Of the many ingredients of this algorithm, some parts are direct extensions of the single-objective version, but the main parts are newly designed algorithms to handle the distinct challenges of optimizing over two objectives. The goal of this study is to improve solution quality and speed and show that decision-space algorithms perform comparably to, and sometimes better than, algorithms that work mainly in the objective-space.

show abstract

Section: Background On Existing Methodsmentioning

confidence: 99%

Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

Adelgren

Gupte

2022

INFORMS Journal on Computing

View full text Add to dashboard Cite

show abstract

“…For the setting in which we have continuous variables, only the case of two objectives has existing algorithms with theoretical performance analysis. These are two variants of the Boxed Line Method (BLM), which were proposed by [Perini et al, 2020]. It is important to note that the performance analysis of the BLM variants depends on the number of line segments in the EF, not on the number of stability regions.…”

Section: General Casementioning

confidence: 99%

On the Relationship Between the Value Function and the Efficient Frontier of a Mixed Integer Linear Optimization Problem

Fallah¹,

Ralphs²,

Boland³

2023

Preprint

View full text Add to dashboard Cite

In this paper, we investigate the connection between the efficient frontier (EF) of a general multiobjective mixed integer linear optimization problem (MILP) and the so-called restricted value function (RVF) of a closely related single-objective MILP. We demonstrate that the EF of the multiobjective MILP is comprised of points on the boundary of the epigraph of the RVF so that any description of the EF suffices to describe the RVF and vice versa. In the first part of the paper, we describe the mathematical structure of the RVF, including characterizing the set of points at which it is differentiable, the gradients at such points, and the subdifferential at all nondifferentiable points. Because of the close relationship of the RVF to the EF, we observe that methods for constructing so-called value functions and methods for constructing the EF of a multiobjective optimization problem, each of which have been developed in separate communities, are effectively interchangeable. By exploiting this relationship, we propose a generalized cutting plane algorithm for constructing the EF of a multiobjective MILP based on a generalization of an existing algorithm for constructing the classical value function. We prove that the algorithm is finite under a standard boundedness assumption and comes with a performance guarantee if terminated early.

show abstract

“…In industrial applications, the number of continuous variables typically clearly dominates the number of discrete variables. Existing algorithms for bicriteria optimization of mixed-integer programs are able to compute approximations of the Pareto frontier ( [2][3][4]). Although for many of these algorithms it is known that their results converge to the true Pareto frontier, state-of-the-art methods do not provide specific bounds on the convergence speed.…”

Section: Motivation and Conceptsmentioning

confidence: 99%

“…• Algorithms based on scalarization which generates special scalarized MIPs that are then solved to optimality to obtain a new point and corresponding information about the Pareto frontier. The papers [2,18] discuss the convex case, while [3,4,19] focus on the linear case.…”

Section: Related Workmentioning

confidence: 99%

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Diessel

2021

Optimization

View full text Add to dashboard Cite

Pareto frontiers of bicriteria continuous convex problems can be efficiently computed and optimal theoretical performance bounds have been established. In the case of bicriteria mixedinteger problems, the approximation of the Pareto frontier becomes, however, significantly harder. In this paper, we propose a new algorithm for approximating the Pareto frontier of bicriteria mixed-integer programs with convex constraints. Such Pareto frontiers are composed of patches of solutions with shared assignments for the discrete variables. By adaptively creating such a patchwork, our algorithm is able to create approximations that converge quickly to the true Pareto frontier. As a quality measure, we use the difference in hypervolume between the approximation and the true Pareto frontier. At least a certain number of patches is required to obtain an approximation with a given quality. This patch complexity gives a lower bound on the number of required computations. We show that our algorithm performs a number of optimization steps that are of a similar order as this lower bound. We provide an efficient MIP-based implementation of this algorithm. The efficiency of our algorithm is illustrated with numerical results showing that our algorithm has a strong theoretical performance guarantee while being competitive with other state-of-the-art approaches in practice.

show abstract

A Criterion Space Method for Biobjective Mixed Integer Programming: The Boxed Line Method

Cited by 19 publications

References 22 publications

Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

On the Relationship Between the Value Function and the Efficient Frontier of a Mixed Integer Linear Optimization Problem

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Contact Info

Product

Resources

About