A Three-Term Conjugate Gradient-Type Method with Sufficient Descent Property for Vector Optimization

The field of vector optimization represents a critical domain within the broader spectrum of optimization problems. Extensive research efforts are currently dedicated to developing solution methods for vector optimization problems. A range of classical approaches, originally designed for scalar optimization, have been adapted to address issues in vector optimization. These include techniques such as the steepest descent method, Newton’s method, quasi-Newton method and conjugate gradient method, among others. However, limited attention has been given to the three-term conjugate gradient method in the context of vector optimization. In this paper, based on the modified self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) method proposed by Kou and Dai (J. Optim. Theory Appl., 165(1): 209-224, 2015), we propose a novel three-term conjugate gradient-type method specifically designed for vector optimization problems. This method ensures the sufficient descent property independent of any line search strategy. Furthermore, the improved Wolfe line search is extended to vector optimization. The global convergence of the proposed method under the improved Wolfe line search is analyzed, demonstrating that at least one accumulation point of the sequence generated by the proposed algorithm is a K-critical point of vector optimization problem. Numerical experiments conducted on a set of benchmark test problems highlight the effectiveness of the proposed method compared to some existing gradient-based approaches.