In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems. The convergence results of the PD methods have been shown in the longer version of the paper.

Design problem solving is a complex activity involving a number of subtasks and a number of alternative methods potentially available for each subtask. The structure of tasks has been a key concern of recent research in task-oriented methodologies for knowledge-based systems (Chandrasekaran 1986; Clancey 1985; Steels 1990; McDermott 1988). One way to conduct a task analysis is to develop a task structure (Chandrasekaran 1989) that lays out the relation between a task, applicable methods for it, the knowledge requirements for the methods, and the subtasks set up by them. I propose a task structure for design by analyzing a general class of methods that I call proposecritique-modify methods. The task structure is constructed by identifying a range of methods for each task.

Ajagekar, Akshay, Humble, Travis, You, Fengqi

Quantum computing (QC) has gained popularity due to its unique capabilities that are quite different from that of classical computers in terms of speed and methods of operations. This paper proposes hybrid models and methods that effectively leverage the complementary strengths of deterministic algorithms and QC techniques to overcome combinatorial complexity for solving large-scale mixed-integer programming problems. Four applications, namely the molecular conformation problem, job-shop scheduling problem, manufacturing cell formation problem, and the vehicle routing problem, are specifically addressed. Large-scale instances of these application problems across multiple scales ranging from molecular design to logistics optimization are computationally challenging for deterministic optimization algorithms on classical computers. To address the computational challenges, hybrid QC-based algorithms are proposed and extensive computational experimental results are presented to demonstrate their applicability and efficiency. The proposed QC-based solution strategies enjoy high computational efficiency in terms of solution quality and computation time, by utilizing the unique features of both classical and quantum computers.

Shang, Fanhua, Liu, Yuanyuan, Tong, Hanghang, Cheng, James, Cheng, Hong

Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint/mixed relaxations (i.e., l_{1}-norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable structured low-rank matrix factorization method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC) problems, or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP) problems. Specifically, we first present two small-scale matrix trace norm regularized bilinear structured factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods.

Liu, Ziyu, Ke, Guolin, Bian, Jiang, Liu, Tieyan

Multiclass decomposition splits a multiclass classification problem into a series of independent binary learners and recomposes them by combining their outputs to reconstruct the multiclass classification results. Three widely-used realizations of such decomposition methods are One-Versus-All (OVA), One-Versus-One (OVO), and Error-Correcting-Output-Code (ECOC). While OVA and OVO are quite simple, both of them assume all classes are orthogonal which neglect the latent correlation between classes in real-world. Error-Correcting-Output-Code (ECOC) based decomposition methods, on the other hand, are more preferable due to its integration of the correlation among classes. However, the performance of existing ECOC-based methods highly depends on the design of coding matrix and decoding strategy. Unfortunately, it is quite uncertain and time-consuming to discover an effective coding matrix with appropriate decoding strategy. To address this problem, we propose LightMC, an efficient dynamic multiclass decomposition algorithm. Instead of using fixed coding matrix and decoding strategy, LightMC uses a differentiable decoding strategy, which enables it to dynamically optimize the coding matrix and decoding strategy, toward increasing the overall accuracy of multiclass classification, via back propagation jointly with the training of base learners in an iterative way. Empirical experimental results on several public large-scale multiclass classification datasets have demonstrated the effectiveness of LightMC in terms of both good accuracy and high efficiency.