Review on Line Search Techniques for Nonlinear Multi-objective Optimization Problems
摘要
A general line search technique for an optimization problem (P) : \(\underset{x\in X\subseteq \mathbb {R}^n}{\min }\) f(x), \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) generates a sequence \(\{x^k\}\) with update formula \(x^{k+1}=x^k+t_k v^{k}\) starting with an initial point \(x^0\) . The sequence converges to a local minimum point under certain assumptions. We say \(v^{k}\in \mathbb {R}^n\) as a descent direction to f at \(x^k\) and \(t_k\) is the step length along the direction \(v^{k}\) . Recently several researchers have developed new line search techniques for nonlinear multi-objective optimization problems. In this chapter, some recent line search techniques for multi-objective programming problems are summarized. In every method, a sequence of points \(\{x^k\}\) is generated, which converges to a critical point of the multi-objective programming problem under some mild assumptions. Improvement in the objective function is measured with respect to a partial ordering in the cone \(\mathbb {R}^m_{+}\) .