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**1 - 4**of**4**### Convexity Conditions of Kantorovich Function and Related Semi-infinite Linear Matrix Inequalities

"... Abstract. The Kantorovich function (x T Ax)(x T A −1 x), 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 Kantorovic ..."

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Abstract. The Kantorovich function (x T Ax)(x T A −1 x), 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 + 2 √ 2. Thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound ‘3 + 2 √ 2 ’ 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 + 2 √ 6, 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. Keywords. positive definite matrix. Matrix analysis, condition number, Kantorovich function, convex analysis,

### Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

, 2009

"... Abstract. While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is a ..."

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Abstract. While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called ‘scaled matrices ’ associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer’s fixed point of a mapping) with a special structure. Thus, a broader question than the open “Question 11 ” in [SIAM Review, 49 (2007), 225-273] is addressed in this paper.

### IT

, 2011

"... The Traffic Assignment Problem (TAP) is to find the traffic flow satisfying Wardrop’s user equilibrium principle, under which each driver selects his/her route with the minimum traffic cost. In order to solve the TAP, the data in the traffic network and each driver’s cost function must be evaluated ..."

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The Traffic Assignment Problem (TAP) is to find the traffic flow satisfying Wardrop’s user equilibrium principle, under which each driver selects his/her route with the minimum traffic cost. In order to solve the TAP, the data in the traffic network and each driver’s cost function must be evaluated exactly. However, in the real network, those data often involve uncertainties. For such an uncertain network, we consider the robust TAP based on the robust Wardrop equilibrium. Under such an equilibrium, each driver selects his/her route with taking the worst possible case into consideration. In this paper, we first study the existence condition for the robust Wardrop equilibria. To this end, we reformulate the robust TAP as a nonlinear complementarity problem (NCP), and apply the solvability theorem to such an NCP. Next we formulate the robust TAP with ellipsoidal uncertainty as the Second-Order Cone Complementarity Problem (SOCCP), which can be solved by using an existing algorithm based on the smoothing Newton method. Finally, by means of some numerical experiments, we observe the property of the robust Wardrop equilibria. 1 Contents