A KKT-Based Approach to Kantorovich-Type Inequalities
Abstract
We develop a unified optimization-based framework for Kantorovich-type inequalities using Karush–Kuhn–Tucker (KKT) conditions. For two commuting positive definite matrices A and B, we derive a sharp eigenvalue-based upper bound for the product (Ax, x)(Bx, x), which recovers the classical Kantorovich inequality as a special case when B = A−1 .
A central feature of our approach is a structural characterization of extremal configurations: We show that every global maximizer is supported on at most two indices. This reveals that the reduction to pairs of eigenvalues arises naturally from first-order optimality conditions.
In addition, we establish a weighted inequality involving explicit optimal constants, extending known results related to the generalized Kantorovich constant K(h, p). Our results provide a systematic variational framework for deriving and understanding Kantorovichtype inequalities, offering both alternative proofs and a basis for further generalizations in matrix analysis.
Keywords
References
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Details
Primary Language
English
Subjects
Mathematical Optimisation
Journal Section
Research Article
Publication Date
July 31, 2026
Submission Date
January 12, 2026
Acceptance Date
May 23, 2026
Published in Issue
Year 2026 Volume: 7 Number: 2
APA
Gümüş, İ. H., & Gezen, A. (2026). A KKT-Based Approach to Kantorovich-Type Inequalities. Fundamentals of Contemporary Mathematical Sciences, 7(2), 156-168. https://doi.org/10.54974/fcmathsci.1861947
AMA
1.Gümüş İH, Gezen A. A KKT-Based Approach to Kantorovich-Type Inequalities. FCMS. 2026;7(2):156-168. doi:10.54974/fcmathsci.1861947
Chicago
Gümüş, İbrahim Halil, and Asiye Gezen. 2026. “A KKT-Based Approach to Kantorovich-Type Inequalities”. Fundamentals of Contemporary Mathematical Sciences 7 (2): 156-68. https://doi.org/10.54974/fcmathsci.1861947.
EndNote
Gümüş İH, Gezen A (July 1, 2026) A KKT-Based Approach to Kantorovich-Type Inequalities. Fundamentals of Contemporary Mathematical Sciences 7 2 156–168.
IEEE
[1]İ. H. Gümüş and A. Gezen, “A KKT-Based Approach to Kantorovich-Type Inequalities”, FCMS, vol. 7, no. 2, pp. 156–168, July 2026, doi: 10.54974/fcmathsci.1861947.
ISNAD
Gümüş, İbrahim Halil - Gezen, Asiye. “A KKT-Based Approach to Kantorovich-Type Inequalities”. Fundamentals of Contemporary Mathematical Sciences 7/2 (July 1, 2026): 156-168. https://doi.org/10.54974/fcmathsci.1861947.
JAMA
1.Gümüş İH, Gezen A. A KKT-Based Approach to Kantorovich-Type Inequalities. FCMS. 2026;7:156–168.
MLA
Gümüş, İbrahim Halil, and Asiye Gezen. “A KKT-Based Approach to Kantorovich-Type Inequalities”. Fundamentals of Contemporary Mathematical Sciences, vol. 7, no. 2, July 2026, pp. 156-68, doi:10.54974/fcmathsci.1861947.
Vancouver
1.İbrahim Halil Gümüş, Asiye Gezen. A KKT-Based Approach to Kantorovich-Type Inequalities. FCMS. 2026 Jul. 1;7(2):156-68. doi:10.54974/fcmathsci.1861947