[Paper Review] A Benson-Type Algorithm for Bounded Convex Vector Optimization Problems with Vertex Selection

Daniel Dörfler, Andreas Löhne+2 more|arXiv (Cornell University)|Jun 28, 2020

Sparse and Compressive Sensing Techniques44 references22 citations

TL;DR

This paper proposes a vertex selection (VS) rule for a Benson-type algorithm solving bounded convex vector optimization problems, improving efficiency by strategically choosing which vertex to cut off in each iteration. By solving cheaper quadratic subproblems and verifying optimality via a single inequality, the method reduces the number of scalarizations needed—leading to faster convergence and coarser, more manageable solution sets without sacrificing approximation quality.

ABSTRACT

We present an algorithm for approximately solving bounded convex vector optimization problems. The algorithm provides both an outer and an inner polyhedral approximation of the upper image. It is a modification of the primal algorithm presented by L\"ohne, Rudloff, and Ulus in 2014. There, vertices of an already known outer approximation are successively cut off to improve the approximation error. We propose a new and efficient selection rule for deciding which vertex to cut off. Numerical examples are provided which illustrate that this method may solve fewer scalar problems overall and therefore may be faster while achieving the same approximation quality.

Motivation & Objective

• To address the inefficiency of existing Benson-type algorithms that require solving many scalar programs per iteration.
• To reduce the number of scalarizations in polyhedral approximation algorithms for bounded convex vector optimization problems.
• To provide a computationally efficient vertex selection rule that improves convergence speed while maintaining the same approximation accuracy.
• To enable real-time knowledge of the approximation error at every iteration, enhancing algorithm transparency and decision support.

Proposed method

• Introduces a heuristic vertex selection rule based on the Hausdorff distance between current inner and outer polyhedral approximations.
• Uses convex quadratic subproblems in the image space (not preimage space) to evaluate candidate vertices, which are cheaper to solve than standard scalarizations.
• Employs a single inequality check to verify optimality of solutions from prior iterations, avoiding redundant computations.
• Maintains both inner and outer polyhedral approximations of the upper image throughout the algorithm.
• Modifies the primal algorithm from L"ohne et al. (2014) by replacing arbitrary vertex selection with the new VS rule.
• Provides an efficient implementation strategy that tracks approximation error at every iteration at no additional cost.

Experimental results

Research questions

• RQ1 Can a smarter vertex selection rule reduce the number of scalarizations required in Benson-type algorithms for convex vector optimization?
• RQ2 Does the proposed vertex selection rule lead to faster convergence while preserving the same approximation quality?
• RQ3 Can the approximation error be monitored continuously during the algorithm’s execution without additional computation?
• RQ4 How does the new method compare to the original algorithm in terms of runtime and solution set size on real-world and academic problems?
• RQ5 Can the method be efficiently applied to problems with high-dimensional variable spaces, such as in machine learning?

Key findings

• The proposed vertex selection rule reduces the number of scalarizations required in all tested examples, leading to faster computation times.
• In the machine learning example (elastic net), the approximation error was smaller with vertex selection across all data sets, while test MSE remained comparable.
• For the truss design problem, the solution set size was significantly reduced—e.g., from 110 to 43 weak minimizers at ε = 0.2—without loss of accuracy.
• The algorithm provides knowledge of the current approximation error at every iteration, enabling better decision-making and monitoring.
• The method achieved the same approximation quality as the original algorithm but with fewer scalarizations, particularly benefiting high-dimensional problems where scalarizations are expensive.