Asia-Pacific Journal of Operational Research

In this paper, we propose a novel nonconvex and nonsmooth optimization model for low-light or inhomogeneous image segmentation which is a hybrid of Mumford–Shah energy functional and Retinex theory. The given image is decomposed into the reflectance component and the illumination component by solving Retinex-based Mumford–Shah model with $L_1-𝜃L_2$ regularizer. Indeed, the existence of the $L_1-𝜃L_2$ regularizer means the nonsmooth term in the model is nonconvex. Thus, it is difficult to solve the proposed model directly. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed nonconvex and nonsmooth model. We apply a novel splitting technique in our algorithm to ensure that all subproblems admit closed-form solutions. Theoretically, we prove that the sequence generated by our proposed algorithm converges to a stationary point under mild conditions. Next, once the reflectance is obtained, the $K$-means clustering method is utilized to complete the segmentation. We compare the proposed Retinex-based method with other state-of-the-art segmentation methods under special lighting conditions. Experimental results show that the proposed method has better performance for both gray-scale images and color images efficiently in terms of the quantitative and qualitative results.

We consider a type of correlated queue in which the inter-arrival time between two consecutive customers linearly depends on the service time of the first customer. We first derive infinite systems of linear equations for the moments of the waiting time in two special cases, based on which we then develop several methods to calculate the moments of the waiting time by using MacLaurin series approximation, Padé approximation and truncation method. In addition, we show how the moments and covariances of the inter-departure times of the correlated queue can be calculated based on the moments of the waiting time. Finally, extensive numerical examples are provided to validate our methods.

In this paper, we investigate a scheduling problem with optional job rejection in a proportionate flow shop setting, where the job processing times are machine independent. A solution to our problem is characterized by (i) partitioning the set of jobs into a set of accepted jobs and a set of rejected jobs, and (ii) scheduling the accepted jobs in a proportionate flow shop setting. The aim is to find a solution to minimize the sum of total weighted late work of the accepted jobs and total rejection cost of the rejected jobs. When all jobs share a common due date, we show that the single-machine case is ${𝒩}{𝒫}$-hard by reduction from the Subset Sum problem. When the operations of all jobs have equal processing times, we solve the case in $O(n^3)$ time by reducing it into a linear assignment problem. For the general problem, we first provide a pseudo-polynomial-time algorithm via the dynamic programming method, then we convert it into a fully polynomial time approximation scheme. As a byproduct, we also resolve an open question in the literature.

The performance of a unit-load warehouse depends on numerous parameters such as storage area, layout, aisle configuration, width-to-depth ratio, the number and locations of dock doors, storage policy, etc. These parameters typically relate to the layout configuration, which can be either traditional (rectangle-shaped) or non-traditional (contour-line-shaped). In this paper, we analyze the performance of rectangle-shaped and contour-line-shaped storage areas within a unit-load warehouse having multiple dock doors. Expected distances traveled in rectangle-shaped storage areas are compared with expected distances in their counterpart contour-line-based storage areas when an ABC class-based storage policy is used to assign unit-loads. For a single product class, the expected-distance for a rectangle-shaped storage area is at most 6.07% greater than it is for the corresponding contour-line-shaped storage area. Depending on the skewness of the ABC curve or storage areas for multiple classes, the expected distance for rectangle-shaped storage areas can be no more than 0.59% greater than it is for the corresponding contour-line shaped storage areas when multiple dock doors are distributed with a specified distance between them.

This paper studies the convex isotonic regression with generalized order restrictions induced by a directed tree. The proposed model covers various intriguing optimization problems with shape or order restrictions, including the generalized nearly isotonic optimization and the total variation on a tree. Inspired by the success of the pool-adjacent-violator algorithm and its active-set interpretation, we propose an active-set-based recursive approach for solving the underlying model. Unlike the brute-force approach that traverses an exponential number of possible active-set combinations, our algorithm has a polynomial time computational complexity under mild assumptions.

In this paper, we consider two single-machine scheduling problems with deteriorating jobs and controllable processing time. The job processing time is a function of its starting time and resource allocation. For the convex resource function, a bicriteria analysis is provided, where the first (respectively, second) criteria is to minimize total weighted completion time (respectively, total resource consumption cost). The first problem is to minimize the weighted sum of the total weighted completion time and the total resource consumption cost. The second problem is to minimize the total weighted completion time subject to the total resource consumption cost is bound. These both problems are NP-hard, we propose some heuristic algorithms and a branch-and-bound algorithm to solve the problems.

This paper is concerned with the ${\ell }_{1,\infty }$-norm ball constrained multi-task learning problem, which has received extensive attention in many research areas such as machine learning, cognitive neuroscience, and signal processing. To address the challenges of solving large-scale multi-task Lasso problems, this paper develops an inexact semismooth Newton-based augmented Lagrangian (Ssnal) algorithm. When solving the inner problems in the Ssnal algorithm, the semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied. Theoretically, this paper presents the global and asymptotically superlinear local convergence of the Ssnal algorithm under standard conditions. Computationally, we derive an efficient procedure to construct the generalized Jacobian of the projector onto ${\ell }_{1,\infty }$-norm ball, which is an important component of the Ssnal algorithm, making the computational cost in the Ssn algorithm very cheap. Comprehensive numerical experiments on the multi-task Lasso problems demonstrate that the Ssnal algorithm is more efficient and robust than several existing state-of-the-art first-order algorithms.

The double-proximal gradient algorithm (DPGA) is a new variant of the classical difference-of-convex algorithm (DCA) for solving difference-of-convex (DC) optimization problems. In this paper, we propose an accelerated version of the double-proximal gradient algorithm for DC programming, in which the objective function consists of three convex modules (only one module is smooth). We establish convergence of the sequence generated by our algorithm if the objective function satisfies the Kurdyka–$Ł$ojasiewicz (K$Ł$) property and show that its convergence rate is not weaker than DPGA. Compared with DPGA, the numerical experiments on an image processing model show that the number of iterations of ADPGA is reduced by 43.57% and the running time is reduced by 43.47% on average.

Proximal alternating direction method of multipliers (PADMM) is a classical primal-dual splitting method for solving separable convex optimization problems with linear equality constraints, which have numerous applications in, e.g., signal and image processing, machine learning, and statistics. In this paper, we propose a new variant of PADMM, called PADMC, whose proximal centers are constructed by convex combinations of the iterates. PADMC is able to take advantage of problem structures and preserves the desirable properties of the classical PADMM. We establish iterate convergence as well as ${𝒪}(1/K)$ ergodic and ${𝒪}(1/\sqrt{K})$ nonergodic sublinear convergence rate results measured by function residual and feasibility violation, where $K$ denotes the iteration number. Moreover, we propose two fast variants of PADMC, one achieves faster ${𝒪}(1/K^2)$ ergodic convergence rate when one of the component functions is strongly convex, and the other ensures faster ${𝒪}(1/K)$ nonergodic convergence rate measured by constraint violation. Finally, preliminary numerical results on the LASSO and the elastic-net regularization problems are presented to demonstrate the performance of the proposed methods.

For smooth convex optimization problems, the optimal convergence rate of first-order algorithm is $O(1/k^2)$ in theory. This paper proposes three improved accelerated gradient algorithms with the gradient information at the latest point. For the step size, to avoid using the global Lipschitz constant and make the algorithm converge faster, new adaptive line search strategies are adopted. By constructing a descent Lyapunov function, we prove that the proposed algorithms can preserve the convergence rate of $O(1/k^2)$. Numerical experiments demonstrate that our algorithms perform better than some existing algorithms which have optimal convergence rate.