On the convergence of reference point methods in multiobjective programming
This paper is concerned with the multiobjective program: \[ (P_ 1)\quad Max[f_ 1(x),...,f_ q(x)],\quad s.t.\quad g_ i(x)\leq 0\quad (i=1,...,m),\quad x\in {\mathbb{R}}^ n \] where the \(f_ k's\) are concave and the \(g_ i's\) convex. Choosing a reference (eventually ideal) point \(\bar y\in {\mathbb{R}}^ n\), the program \((P_ 1)\) is closely related to the convex program: \[ (P_ 2)\quad Min d(y,\bar y),\quad s.t.\quad f_ k(x)\geq y_ k\quad and\quad g_ i(x)\leq 0 \] where d is a quasi-convex distance. On the one hand, a reminiscence of Philip's theorem says that if the Lagrange vector associated with \((P_ 2)\) is strictly positive for the constraints \(f_ k(x)\geq y_ k\), then the solution of \((P_ 2)\) is efficient. On the other hand, one has the idea of using these Lagrange multipliers in order to direct the decision maker's tradeoffs. Starting from these components, the authors give a cutting-plane procedure in which the reference point, at each step, is obtained by \[ Maximize\quad U(y)\quad s.t.\quad \lambda^*y\leq \lambda^*y^* \] where U is the quasiconcave utility function of the decision maker, \(y^*\) an optimal solution of \((P_ 2)\) at the preceeding step and \(\lambda^*\) the associated Lagrange multipliers. In this case it is shown that the procedure converges to the optimal point of \[ Max U(y),\quad y\in Y=\{y_ k=f_ k(x)| \quad x\quad feasible\}. \] The authors close their paper by giving two other procedures in which the decision maker uses an improvement direction v, where v is any vector spanning the subspace parallel to the supporting hyperplane to the set Y at a solution point \(y^*\) of \((P_ 2)\). These two algorithms exploit the vein of the curve-search procedure as popularized by \textit{P. J. Korhonen} and \textit{J. Laasko} [Eur. J. Oper. Res. 24, 277-287 (1986; Zbl 0581.90088)]. But in the latter cases, one cannot ensure the convergence.
scientific article; zbMATH DE number 4076986
scientific article