On approximate solutions for robust semi-infinite multi-objective convex symmetric cone optimization
A robust semi-infinite multi-objective symmetric programming problem with uncertain da ta (RSIMP) is investigated according to \(\min (f_1(x),\dots,f_K(x))\) s.t. \[ x\in \mathcal{F_P}=\left\{x\in \mathbb{R}^m \mid a_0^{(t)}+\sum_{i=1}^m x_i a_i^{(t)}\in \mathcal{K_J}\;(\text{def. }\succcurlyeq 0),\; a_i^{(t)}\in \mathcal{V}_i^{(t)},\; i=1,2,\dots,m,\; t\in T \right\} \] where \(a_i^{(t)}\) belong to an uncertain subset \(\mathcal{V}_i^{(t)}\) of an \(n\)-dimensional Jordan algebra \([\mathcal{J},\circ ]\), \(\mathcal{K_J}\) is the cone of \(a\circ a\) of all \(a\in \mathcal{J}\) and \(T\) is an arbitrary, may be, infinite index set. Initially, the unusual \(\epsilon\)-approximate solutions \(\bar x\in \mathcal{F_P}\) according to the coordinate functions of the objective \(f_k(x)\ge f_k(\bar x)-\epsilon, k=1,\dots,K\) for all \(x \in \mathcal{F_P}\) (known as \(\epsilon\) utopia point) are considered. However, from Prop. 2.4 the solution concept seems to be changed to \(E^6 \;\epsilon\)-efficiency of \textit{D. J. White} [J. Optim. Theory Appl. 49, 319--337 (1986; Zbl 0573.90085)], originally for max-problems. The authors use a sufficient condition of \textit{K. Yokoyama} [J. Math. Anal. Appl. 203, No. 1, 142--149 (1996; Zbl 0858.90114), Lem. 4.2], \(\epsilon\)-Efficiency \(E_1\) for such \(E^6\;\epsilon\)-efficiency as \(\epsilon\)-efficiency definition in the further context. Some main role in the paper plays the robust characteristic cone \(\mathcal D\) inconveniently defined in the paper. If \(\mathcal{V}_i^{(t)}\) is some linear map of a compact convex set of \(\mathbb{R}^l\rightarrow \mathcal{J}\) and some Slater condition holds then \(\mathcal{D}\) is a convex and closed cone in \(\mathbb{R}^{m+1}\). Using a Farkas like lemma and subgradients of the convex assumed functions \(f_k,k\in I\) the \(\epsilon\)-efficiency for (RSIMP) is equivalent to an \(\epsilon\)-subgradient relation (Theorem 3.1) which should tend to Pareto efficiency whenever \(\epsilon \rightarrow 0\). For arbitrary (RISMP) with convex \(f_k\) weak Wolfe duality is shown (Theorem 4.1.). For (RISMP) with closed and convex \(\mathcal{D}\) the Wolfe dual problem is \(2\epsilon\)-solvable whenever the primal problem has a \(\epsilon\)-solution (Theorem 4.2., Cor. 4.1.). One example demonstrates all results. The paper contains serious bugs. Unfortunately, Yokoyama cited for his \(E_1\) definition not correctly the \(E^6 \;\epsilon\)-efficiency of White. Doing transformation to min-problems and above notations one gets White: \[ E^6:=\{\bar x\in \mathcal{F_P}\mid \text{if }f(x)\leq f(\bar x) \text{ for some } x\in \mathcal{F_P} \text{ then } f(\bar x)\leq f(x) +\epsilon \}, \] Yokoyama: \[ E_1:=\{\bar x\in \mathcal{F_P}\mid \text{if there is } x\in \mathcal{F_P} \text{ such that }f(x) \leq f(\bar x) ,\; f(\bar x)\leq f(x) +\epsilon \}. \] \(E^6\) is much more restricted than \(E_1\) and describes an expected kind of \(\epsilon\)-efficiency. \(E_1\) contains a lot of points having nothing to do with \(\epsilon\)-efficiency. Therefore, the above reviewed results may be not true or not useful and need to be verified again, unless the mainly used Lemma 4.2 of Yokoyama is true for the correct cited \(E^6\) \(\epsilon\)-efficiency of White. Nowhere in the paper is said which, on \(t\) depending, variables in the sums are non zero for only finite \(t\in T\), otherwise the sums do not make any sense (see e.g. [\textit{J. H. Lee} and \textit{G. M. Lee}, Ann. Oper. Res. 269, No. 1--2, 419--438 (2018; Zbl 1446.90143)] for correct handling). Beyond that, the paper contains a lot of misprints and some double notations being a handicap for the reader to follow the development of statements.
scientific article; zbMATH DE number 7610120
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