An interactive algorithm for nonlinear vector optimization
The following sets and definitions are given: \(\emptyset \neq M\subseteq R^{\ell}\), \(\emptyset \neq K\subseteq R^{\ell}\), with K a closed convex pointed cone. For \(x,y\in R^{\ell}:\) \[ x\leq_ ky\leftrightarrow^{Def}y-x\in K. \] \[ EFF(M,K)=^{Def}\{y\in M:\;(\{y\}-K)\cap M=\{x\}\}. \] For \((p,q)\in R^{\ell}\times K^ 0\), where \(K^ 0\) is the relative interior of K, the following problem MP(p,q) is defined: \[ MP(p,q):\;\min [t]=E(p,q)\text{ subject to } t\in R,\quad x\in M,\quad x-tq\leq_ KP. \] The following sets are defined: \[ \beta =\{(p,q)\in R^{\ell}\times K^ 0:\;MP(p,q)\quad has\quad a\quad feasible\quad solution\}, \] \[ {\mathcal L}=\{(p,q)\in R^{\ell}\times K^ 0:\;M(p,q)\quad has\quad an\quad optimal\quad solution\}, \] \[ Q(p,q)=\{x\in M:\;(x,E(p,q))\quad solves\quad MP(p,q)\}. \] The following theorems are proved. Theorem 2.2. If \(M+K\) is convex and closed, and EFF(M,K)\(\neq \emptyset\), then \(\beta ={\mathcal L}\). If int \(K\neq \emptyset\), then \(\beta =R^{\ell}\times K^ 0.\) Theorem 2.3. If \(M+K\) is convex and closed, then the following statements are equivalent: (a) EFF(M,K)\(\neq \emptyset\); (b) Q(p,q) is bounded for each (p,q)\(\in {\mathcal L}\); (c) Q(p,q) is bounded for at least one (p,q)\(\in {\mathcal L}.\) Theorem 2.4. If \(M+K\) is convex and closed, and (p,q)\(\in {\mathcal L}\), then: (a) EFF(M,K)\(\neq \emptyset\) implies that MP(r,E(p,q)) is solvable for each \(r\in int(dual\) cone of K); (b) If x solves MP(r,E(p,q) then \(x\in EFF(M,K).\) Two illustrations and a sketch of an algorithm are given.
scientific article; zbMATH DE number 4172752