Geometric Duality Results and Approximation Algorithms for Convex Vector Optimization Problems
研究了凸向量优化问题的几何对偶，提出了一个不依赖方向参数的对偶问题，并基于此设计了同时求解原问题和对偶问题的几何对偶算法，通过随机实例测试了性能。
.We study geometric duality for convex vector optimization problems. For a primal problem with a \(q\) -dimensional objective space, we formulate a dual problem with a \((q+1)\) -dimensional objective space. Consequently, different from an existing approach, the geometric dual problem does not depend on a fixed direction parameter, and the resulting dual image is a convex cone. We prove a one-to-one correspondence between certain faces of the primal and dual images. In addition, we show that a polyhedral approximation for one image gives rise to a polyhedral approximation for the other. Based on this, we propose a geometric dual algorithm which solves the primal and dual problems simultaneously and is free of direction-biasedness. We also modify an existing direction-free primal algorithm in such a way that it solves the dual problem as well. We test the performance of the algorithms for randomly generated problem instances by using the so-called primal error and hypervolume indicator as performance measures.Keywordsconvex vector optimizationmultiobjective optimizationapproximation algorithmscalarizationgeometric dualityhypervolume indicatorMSC codes90B5090C2590C29