Existence and Optimality for Matrix Optimization Problems
Fixed Point Methods and Optimization, Volume 3, Issue 3, December 2026, Pages 166–173
Gwi Soo Kim
Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea
Moon Hee Kim
College of General Education, Tongmyong University, Busan 48520, Korea
Gue Myung Lee
Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea
Jen-Chih Yao
Center for General Education, China Medical University, Taichung, Taiwan
Abstract
To calculate the closed cone induced by epigraph of conjugate functions of sum of linear mappings and matrix norms, we give the formula of the subdifferential of matrix norm \(\Vert \cdot \Vert_1\) at \(0\). We consider a sublinear matrix optimization problem (P) involving a matrix norm \(\Vert \cdot \Vert_1\). and then we show that the existence of optimal solutions for (P) is closely related to its zero solution. Moreover we consider a convex matrix optimization problem (CP) involving matrix norm \(\Vert \cdot \Vert_1\), and establish an optimality theorem (CP) which holds without any constraint qualification and expressed with the subdifferential. We give an example illustrating the optimality theorem.
Cite this Article as
Gwi Soo Kim, Moon Hee Kim, Gue Myung Lee, and Jen-Chih Yao, Existence and Optimality for Matrix Optimization Problems, Fixed Point Methods and Optimization, 3(3), 166–173, 2026