Matrix analysis
Linear multiplicative programming
Mathematical Programming: Series A and B
A global Newton method II: analytic centers
Mathematical Programming: Series A and B - Special issue: Festschrift in Honor of Philip Wolfe part II: studies in nonlinear programming
The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions
Automatica (Journal of IFAC)
Multiplicative programming problems: analysis and efficient point search heuristic
Journal of Optimization Theory and Applications
Robust Solutions to Least-Squares Problems with Uncertain Data
SIAM Journal on Matrix Analysis and Applications
Mathematics of Operations Research
Robust Solutions to Uncertain Semidefinite Programs
SIAM Journal on Optimization
Operations Research
Convex Optimization
Tractable Approximations to Robust Conic Optimization Problems
Mathematical Programming: Series A and B
Explicit Reformulations for Robust Optimization Problems with General Uncertainty Sets
SIAM Journal on Optimization
The Legendre-Fenchel Conjugate of the Product of Two Positive Definite Quadratic Forms
SIAM Journal on Matrix Analysis and Applications
| Hi-index | 7.29 |
The Kantorovich function (x^TAx)(x^TA^-^1x), where A is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: when is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only if the condition number of its matrix is less than or equal to 3+22. Thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound '3+22' is turned out to be a necessary condition for the convexity of the Kantorovich function in any finite-dimensional spaces. We also point out that when the condition number of the matrix (which can be any dimensional) is less than or equal to 5+26, the Kantorovich function is convex. Furthermore, we prove that this general sufficient convexity condition can be improved to 2+3 in 3-dimensional space. Our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semi-infinite linear matrix inequality or 'robust positive semi-definiteness' of symmetric matrices. In fact, our main result for 3-dimensional cases has been proved by finding an explicit solution range to some semi-infinite linear matrix inequalities.