On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ?? n, given by F ₁(z)?=?(0, I _n;m (z)) with the isometric constant equal to (m?+?1)/(n?+?1) and F ₂(z)?=?(I _n;m (z), I _n;m (z)) with the isometric constant equal to 2(m?+?1)/(n?+?1). Here ${I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}$ is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.     

A robust optimization model for multi-site production planning problem in an uncertain environment

This paper addresses the multi-site production planning problem for a multinational lingerie company in Hong Kong subject to production import/export quotas imposed by regulatory requirements of different nations, the use of manufacturing factories/locations with regard to customers’ preferences, as well as production capacity, workforce level, storage space and resource conditions at the factories. In this paper, a robust optimization model is developed to solve multi-site production planning problem with uncertainty data, in which the total costs consisting of production cost, labor cost, inventory cost, and workforce changing cost are minimized. By adjusting penalty parameters, production management can determine an optimal medium-term production strategy including the production loading plan and workforce level while considering different economic growth scenarios. The robustness and effectiveness of the developed model are demonstrated by numerical results. The trade-off between solution robustness and model robustness is also analyzed.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.     

Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions (and of some modified ones) for the variational inequality problem (VIP) defined by a locally Lipschitz but not necessarily differentiable function on a closed convex set in an Euclidean space. As applications we show that, under the strong monotonicity assumption, the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of Armijo type converge to the solution of the (VIP). The research of this author was supported by an Earmarked Grant from the Research Council of Hong Kong.     

A functional equation arising from ranked additive and separable utility

All strictly monotonic solutions of a general functional equation are determined. In a particular case, which plays an essential role in the axiomatization of rank-dependent expected utility, all nonnegative solutions are obtained without any regularity conditions. An unexpected possibility of reduction to convexity makes the present proof possible.

     

Preemptive multiprocessor order scheduling to minimize total weighted flowtime

Consider m identical machines in parallel, each of which can produce k different product types. There is no setup cost when the machines switch from producing one product type to another. There are n orders each of which requests various quantities of the different product types. All orders are available for processing at time t = 0, and preemption is allowed. Order i has a weight w_i and its completion time is the time when its last requested product type finishes. Our goal is to find a preemptive schedule such that the total weighted completion time ∑_wiCi$\sum w_i{C}_i$ is minimized. We show that this problem is NP-hard even when all jobs have identical weights and there are only two machines. Motivated by the computational complexity of the problem, we propose a simple heuristic and show that it obeys a worst-case bound of 2 − 1/m. Finally, empirical studies show that our heuristic performs very well when compared with a lower bound of the optimal cost.     

Group Scheduling with Controllable Setup and Processing Times: Minimizing Total Weighted Completion Time

The following single machine scheduling problem is studied. A partition of a set of n jobs into g groups on the basis of group technology is given. The machine processes jobs of the same group contiguously, with a sequence independent setup time preceding the processing of each group. The setup times and the job processing times are controllable through the allocation of a continuously divisible or discrete resource to them. Each job uses the same amount of the resource. Each setup also uses the same amount of resource, which may be different from that for the jobs. Polynomial-time algorithms are constructed for variants of the problem of finding an optimal job sequence and resource values so as to minimize the total weighted job completion time, subject to given restrictions on resource consumption. The algorithms are based on a polynomial enumeration of the candidates for an optimal job sequence and solving the problem with a fixed job sequence by linear programming. This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grants Council of Hong Kong under grant number PolyU 5191/01E. In addition, the research of M.Y. Kovalyov was supported by INTAS under grant number 00-217.     

Multiresolution Network Models

Many existing statistical and machine learning tools for social network analysis focus on a single level of analysis. Methods designed for clustering optimize a global partition of the graph, whereas projection-based approaches (e.g., the latent space model in the statistics literature) represent in rich detail the roles of individuals. Many pertinent questions in sociology and economics, however, span multiple scales of analysis. Further, many questions involve comparisons across disconnected graphs that will, inevitably be of different sizes, either due to missing data or the inherent heterogeneity in real-world networks. We propose a class of network models that represent network structure on multiple scales and facilitate comparison across graphs with different numbers of individuals. These models differentially invest modeling effort within subgraphs of high density, often termed communities, while maintaining a parsimonious structure between said subgraphs. We show that our model class is projective, highlighting an ongoing discussion in the social network modeling literature on the dependence of inference paradigms on the size of the observed graph. We illustrate the utility of our method using data on household relations from Karnataka, India. Supplementary material for this article is available online.     

Robust quaternion matrix completion with applications to image inpainting

In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ?₁‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.     

Merit functions in vector optimization

We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence of error bounds for merit functions are also given. This research was supported by a direct grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong.