On Parallel and Batch-Cutting Strategies for Norm-Minimization-Based Convex Vector Optimization

Abstract

We develop parallel and batch-cutting variants of the norm-minimization-based outer approximation algorithm for convex vector optimization. The standard algorithm solves $N_k$ independent subproblems at each iteration~$k$ to evaluate all vertices of the current polyhedral approximation, but processes only the single best cut. We propose two improvements. First, we parallelize the \revise{subproblem evaluations} across $\nw$ workers, reducing per-iteration wall-clock time. Second, we introduce a batch-cutting strategy that adds up to $K$ supporting halfspaces per iteration, using information from all solved subproblems rather than discarding it. We prove that the batch-cutting variant inherits the convergence rate $O(k^{2/(1-q)})$ of the standard algorithm, where $k$ is the number of outer iterations and $q$ is the number of objectives. Computational experiments on eight test problems with $q\in \{2,3,4,5\}$ show that parallelism on 8 cores \revise{increases the speed by a factor of 1.1 to 4.2}, and batch cutting consistently reduces the iteration count by 62--80\%. However, the wall-clock benefit of batch cutting is problem-dependent: the additional cuts per iteration accelerate vertex count growth, so batch cutting is most effective when per-vertex subproblem cost dominates.

18 pages, 2 figures