An Extension of the Carathéodory Differentiability to Set-Valued Maps

This paper uses the generalization of the Hukuhara difference for compact convex set to extend the classical notions of Carathéodory differentiability to multifunctions (set-valued maps). Using the Hukuhara difference and affine multifunctions as a local approximation, we introduce the notion of CH-differentiability for multifunctions. Finally, we tackle the study of the relation among the Fréchet differentiability, Hukuhara differentiability, and CH-differentiability.

Müller–Zhang truncation for general linear constraints with first or second order potential

Let $$\mathcal {B}$$ B be a homogeneous differential operator of order $$l=1$$ l = 1 or $$l=2$$ l = 2 . We show that a sequence of functions of the form $$(\mathcal {B}u_j)_j$$ ( B u j ) j converging in the $$L^1$$ L 1 -sense to a compact, convex set K can be modified into a sequence converging uniformly to this set provided that the derivatives of order l are uniformly bounded. We prove versions of our result on the whole space, an open domain, and for K varying uniformly continuously on an open, bounded domain. This is a conditional generalization of a theorem proved by S. Müller for sequences of gradients. Moreover, a potential of order two for the linearized isentropic Euler system is constructed.

Constructing a subgradient from directional derivatives for functions of two variables

For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally differentiable, it is shown that a subgradient may always be constructed from the function's directional derivatives in the four compass directions, arranged in a so-called "compass difference". When the original function is nonconvex, the obtained subgradient is an element of Clarke's generalized gradient, but the result appears to be novel even for convex functions. The function is not required to be represented in any particular form, and no further assumptions are required, though the result is strengthened when the function is additionally L-smooth in the sense of Nesterov. For certain optimal-value functions and certain parametric solutions of differential equation systems, these new results appear to provide the only known way to compute a subgradient. These results also imply that centered finite differences will converge to a subgradient for bivariate nonsmooth functions. As a dual result, we find that any compact convex set in two dimensions contains the midpoint of its interval hull. Examples are included for illustration, and it is demonstrated that these results do not extend directly to functions of more than two variables or sets in higher dimensions.

Asymptotic properties of the (convex) hyperspaces

It is known that the hyperspaces of compact sets and compact convex set of the Euclidean space $\mathbb R^n$, $n\ge2$, both are homeomorphic to the puctured Hilbert cube. The main result of this note states that these hyperspaces are not coarsely equivalent.

On duality for a second-order multiobjective fractional programming problem involving type-I functions

The purpose of this paper is to study a nondifferentiable multiobjective fractional programming problem (MFP) in which each component of objective functions contains the support function of a compact convex set. For a differentiable function, we introduce the class of second-order {(C,\alpha,\rho,d)-V} -type-I convex functions. Further, Mond–Weir- and Wolfe-type duals are formulated for this problem and appropriate duality results are proved under the aforesaid assumptions.

Higher-order symmetric duality in nondifferentiable multiobjective optimization over cones

In this paper, a new pair of higher-order nondifferentiable multiobjective symmetric dual programs over arbitrary cones is formulated, where each of the objective functions contains a support function of a compact convex set. We identify a function lying exclusively in the class of higher-order K-?-convex and not in the class of K-?-bonvex function already existing in literature. Weak, strong and converse duality theorems are then established under higher-order K-?-convexity assumptions. Self duality is obtained by assuming the functions involved to be skew-symmetric. Several known results are also discussed as special cases.

Global Optimization for the Sum of Concave-Convex Ratios Problem

This paper presents a branch and bound algorithm for globally solving the sum of concave-convex ratios problem (P) over a compact convex set. Firstly, the problem (P) is converted to an equivalent problem (P1). Then, the initial nonconvex programming problem is reduced to a sequence of convex programming problems by utilizing linearization technique. The proposed algorithm is convergent to a global optimal solution by means of the subsequent solutions of a series of convex programming problems. Some examples are given to illustrate the feasibility of the proposed algorithm.

Ranks and eigenvalues of states with prescribed reduced states

For a quantum state represented as an $n\times n$ density matrix $\sigma \in M_n$, let $\cS(\sigma)$ be the compact convex set of quantum states $\rho = (\rho_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $\sigma$, i.e., $\tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma$. It is known that if $m\ge n$ then there is a rank one matrix $\rho \in \cS(\sigma)$ satisfying $\tr_1(\rho) = \sigma$. If $m < n$, there may not be any rank one matrix in $\cS(\sigma)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set $\cS$. We also determine $\rho^* \in \cS(\sigma)$ with rank bounded by $k$ such that $\|\tr_1(\rho^*) - \sigma\|$ is minimum for a given unitary similarity invariant norm $\|\cdot\|$. Furthermore, the relation between the eigenvalues of $\sigma$ and those of $\rho \in \cS(\sigma)$ is analyzed. Extension of the results and open problems will be mentioned.